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65° angle with the ground. The base of the cable is 17 ft from the tower. What is the height of the tower to the nearest foot? 8 feet 15 feet 36 feet 40 feet 70. Which of the following has the same value as sin M? sin N tan M cos N cos M CHALLENGE AND EXTEND Algebra Find the value of x. Then find AB, BC, and AC. Round each to the nearest unit. 71. 72. 73. Multi-Step Prove the identity (tan A) 2 + 1 = 1 _ . (cos A) 2 74. A regular pentagon with 1 in. sides is inscribed in a circle. Find the radius of the circle rounded to the nearest hundredth. Each of the three trigonometric ratios has a reciprocal ratio, as defined below. These ratios are cosecant (csc), secant (sec), and cotangent (cot). csc A = 1 _ sin A sec A = 1 _ cot A = 1 _ cos A tan A Find each trigonometric ratio to the nearest hundredth. 75. csc Y 76. sec Z 77. cot Y SPIRAL REVIEW Find three ordered pairs that satisfy each function. (Previous course) 78. f (x) = 3x - 6 79. f (x) = -0.5x + 10 80. f (x) = x 2 - 4x + 2 Identify the property that justifies each statement. (Lesson 2-5) 81. 82. ̶̶ AB ≅ ̶̶ AB ≅ ̶̶ CD , and ̶̶ AB ̶̶ CD ≅ ̶̶ DE . So ̶̶ AB ≅ ̶̶ DE . 83. If ∠JKM ≅ ∠MLK, then ∠MLK ≅ ∠JKM. Find the geometric mean of each pair of numbers. (Lesson 8-1) 84. 3 and 27 85. 6 and 24 86. 8 and 32 532 532 Chapter 8 Right Triangles and Trigonometry �� Inverse Functions Algebra In Algebra, you learned that a function is a relation in which each element of the domain is paired with exactly one element of the range. If you switch the domain and range of a one-to-one function, you create an inverse function. See Skills Bank page S62 The function y = sin -1 x is the inverse of the function y = sin x. �� If you know the value of a trigonometric ratio, you can use the inverse trigonometric function to find the angle measure. You can do this either with a calculator or by looking at the graph of the function. �� Example Use the graphs above to find the value of x for 1 = sin x. Then write this expression using an inverse trigonometric function. 1 = sin x x = 90° Look at the graph of y = sin x. Find where the graph intersects the line y = 1 and read the corresponding x-coordinate. 90° = sin -1 (1) Switch the x- and y-values. Try This TAKS Grades 9–11 Obj. 2 Use the graphs above to find the value of x for each of the following. Then write each expression using an inverse trigonometric function. 1. 0 = sin x 4. 0 = cos x 2. 1 _ 2 = cos x 5. 0 = tan x 3. 1 = tan x 6. 1 _ 2 = sin x On Track for TAKS 533 533 �� 8-3 Solving Right Triangles TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as ... trigonometric ratios .... Objective Use trigonometric ratios to find angle measures in right triangles and to solve real-world problems. Also G.5.D, G.7.A, G.7.C, G.8.C, G.11.B Why learn this? You can convert the percent grade of a road to an angle measure by solving a right triangle. San Francisco, California, is famous for its steep streets. The steepness of a road is often expressed as a percent grade. Filbert Street, the steepest street in San Francisco, has a 31.5% grade. This means the road rises 31.5 ft over a horizontal distance of 100 ft, which is equivalent to a 17.5° angle. You can use trigonometric ratios to change a percent grade to an angle measure. E X A M P L E 1 Identifying Angles from Trigonometric Ratios Use the trigonometric ratio cos A = 0.6 to determine which angle of the triangle is ∠A. cos A = adj. leg_ hyp. cos ∠1 = 3.6_ 6 cos ∠2 = 4.8_ 6 = 0.6 = 0.8 Cosine is the ratio of the adjacent leg to the hypotenuse. The leg adjacent to ∠1 is 3.6. The hypotenuse is 6. The leg adjacent to ∠2 is 4.8. The hypotenuse is 6. Since cos A = cos ∠1, ∠1 is ∠A. Use the given trigonometric ratio to determine which angle of the triangle is ∠A. 1a. sin A = 8 _ 17 1b. tan A = 1.875 In Lesson 8-2, you learned that sin 30° = 0.5. Conversely, if you know that the sine of an acute angle is 0.5, you can conclude that the angle measures 30°. This is written as sin -1 (0.5) = 30°. If you know the sine, cosine, or tangent of an acute angle measure, you can use the inverse trigonometric functions to find the measure of the angle. Inverse Trigonometric Functions If sin A = x, then sin -1 x = m∠A. If cos A = x, then cos -1 x = m∠A. If tan A = x, then tan -1 x = m∠A. The expression sin -1 x is read “the inverse sine of x.” It does not mean 1 ____ . You can think of sin -1 x as “the angle whose sine is x.” sin x 534 534 Chapter 8 Right Triangles and Trigonometry �� E X A M P L E 2 Calculating Angle Measures from Trigonometric Ratios Use your calculator to find each angle measure to the nearest degree. A cos -1 (0.5) B sin -1 (0.45) C tan -1 (3.2) When using your calculator to find the value of an inverse trigonometric expression, you may need to press the [arc], [inv], or [2nd] key. cos -1 (0.5) = 60° sin -1 (0.45) ≈ 27° tan -1 (3.2) ≈ 73° Use your calculator to find each angle measure to the nearest degree. 2a. tan -1 (0.75) 2b. cos -1 (0.05) 2c. sin -1 (0.67) Using given measures to find the unknown angle measures or side lengths of a triangle is known as solving a triangle. To solve a right triangle, you need to know two side lengths or one side length and an acute angle measure. E X A M P L E 3 Solving Right Triangles Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Method 1: Method 2: By the Pythagorean Theorem, AC 2 = AB 2 + BC 2 . = (7.5) 2 + 5 2 = 81.25 So AC = √  81.25 ≈ 9.01. m∠A = tan -1 ( 5 _ ) ≈ 34° 7.5 Since the acute angles of a right triangle are complementary, m∠C ≈ 90° - 34° ≈ 56°. m∠A = tan -1 ( 5 _ ) ≈ 34° 7.5 Since the acute angles of a right triangle are complementary, m∠C ≈ 90° - 34° ≈ 56°. sin A = 5 _ AC , so AC = 5 _ . sin A 5 __ ≈ 9.01 ⎤ ⎡ tan -1 ( 5 _ ) ⎥ ⎢ sin ⎦ ⎣ AC ≈ 7.5 3. Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Solving Right Triangles Rounding can really make a difference! To find AC, I used the Pythagorean Theorem and got 15.62. Then I did it a different way. I used m∠A = tan -1 ( 10 __ find m∠A = 39.8056°, which I rounded to 40°. sin 40° = 10 ___ AC , so AC = 10 _____ ≈ 15.56. sin 40° 12 ) to Kendell Waters Marshall High School The difference in the two answers reminded me not to round values until the last step. 8- 3 Solving Right Triangles 535 535 �� E X A M P L E 4 Solving a Right Triangle in the Coordinate Plane The coordinates of the vertices of △JKL are J (-1, 2) , K (-1, -3) , and L (3, -3) . Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. Step 1 Find the side lengths. Plot points J, K, and L. JK = 5 KL = 4 By the Distance Formula, JL = √  ⎤ ⎦ ⎡ ⎣ 2 + (-3 - 2) 2 . 3 - (-1) √  4 2 + (-5) 2 = 16 + 25 = √  41 ≈ 6.40 = √  Step 2 Find the angle measures. m∠K = 90° m∠J = tan -1 ( 4 _ ) ≈ 39° 5 ̶̶ KL are ⊥. ̶̶ JK and ̶̶ KL is opp. ∠J, and ̶̶ JK is adj. to ∠J. m∠L ≈ 90° - 39° ≈ 51° The acute  of a rt. △ are comp. 4. The coordinates of the vertices of △RST are R (-3, 5) , S (4, 5) , and T (4, -2) . Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. E X A M P L E 5 Travel Application San Francisco’s Lombard Street is known as one of “the crookedest streets in the world.” The road’s eight switchbacks were built in the 1920s to make the steep hill passable by cars. If the hill has a percent grade of 84%, what angle does the hill make with a horizontal line? Round to the nearest degree. 84% = 84 _ 100 Change the percent grade to a fraction. An 84% grade means the hill rises 84 ft for every 100 ft of horizontal distance. Draw a right triangle to represent the hill. ∠A is the angle the hill makes with a horizontal line. m∠A = tan -1 ( 84 _ ) ≈ 40° 100 5. Baldwin St. in Dunedin, New Zealand, is the steepest street in the world. It has a grade of 38%. To the nearest degree, what angle does Baldwin St. make with a horizontal line? 536 536 Chapter 8 Right Triangles and Trigonometry �� THINK AND DISCUSS 1. Describe the steps you would use to solve △RST. 2. Given that cos Z = 0.35, write an equivalent statement using an inverse trigonometric function. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write a trigonometric ratio for ∠A. Then write an equivalent statement using an inverse trigonometric function. 8-3 Exercises Exercises GUIDED PRACTICE KEYWORD: MG7 8-3 KEYWORD: MG7 Parent . 534 Use the given trigonometric ratio to determine which angle of the triangle is ∠A. 1. sin A = 4 _ 5 2. tan A = 1 1 _ 3 3. cos A = 0.6 4. cos A = 0.8 5. tan A = 0.75 6. sin A = 0. Use your calculator to find each angle measure to the nearest degree. p. 535 7. tan -1 (2.1) 10. sin -1 (0.5) 8. cos -1 ( 1 _ ) 11. sin -1 (0.61) 3 9. cos -1 ( 5 _ ) 12. tan -1 (0.09. 535 Multi-Step Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 13. 14. 15. 536 Multi-Step For each triangle, find the side lengths to the nearest hundredth and the angle measures to the nearest degree. 16. D (4, 1) , E (4, -2) , F (-2, -2) 17. R (3, 3) , S (-2, 3) , T (-2, -3) 18. X (4, -6) , Y (-3, 1) , Z (-3, -6) 19. A (-1, 1) , B (1, 1) , C (1, 5) 8- 3 Solving Right Triangles 537 537 �� 20. Cycling A hill in the Tour de France p. 536 bike race has a grade of 8%. To the nearest degree, what is the angle that this hill makes with a horizontal line? Independent Practice Use the given trigonometric rat
io to determine which angle PRACTICE AND PROBLEM SOLVING For See Exercises Example 21–26 27–32 33–35 36–37 38 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S18 Application Practice p. S35 of the triangle is ∠A. 21. tan A = 5 _ 12 24. sin A = 5 _ 13 22. tan A = 2.4 25. cos A = 12 _ 13 23. sin A = 12 _ 13 26. cos A = 5 _ 13 Use your calculator to find each angle measure to the nearest degree. 27. sin -1 (0.31) 27. 28. tan -1 (1) 30. cos -1 (0.72) 31. tan -1 (1.55) 29. cos -1 (0.8) 32. sin -1 ( 9 _ ) 17 Multi-Step Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 33. 34. 35. Multi-Step For each triangle, find the side lengths to the nearest hundredth and the angle measures to the nearest degree. 36. A (2, 0) , B (2, -5) , C (1, -5) 37. M (3, 2) , N (3, -2) , P (-1, -2) 38. Building For maximum accessibility, a wheelchair ramp should have a slope between 1 __ 16 and 1 __ 20 . What is the range of angle measures that a ramp should make with a horizontal line? Round to the nearest degree. Complete each statement. If necessary, round angle measures to the nearest degree. Round other values to the nearest hundredth. ? ≈ 3.5 ̶̶̶̶ 39. tan 42. cos -1 ( 45. Critical Thinking Use trigonometric ratios to explain why the diagonal of ? 42° ≈ 0.74 ̶̶̶̶ ? 60° = 1 _ ̶̶̶̶ 2 40. sin 43. sin -1 ( ? ≈ 2 _ ̶̶̶̶ 3 ? ) ≈ 69° ̶̶̶̶ ? ) ≈ 12° ̶̶̶̶ 41. 44. a square forms a 45° angle with each of the sides. 46. Estimation You can use trigonometry to find angle measures when a protractor is not available. a. Estimate the measure of ∠P. b. Use a centimeter ruler to find RQ and PQ. c. Use your measurements from part b and an inverse trigonometric function to find m∠P to the nearest degree. d. How does your result in part c compare to your estimate in part a? 538 538 Chapter 8 Right Triangles and Trigonometry �� 46. This problem will prepare you for the Multi-Step TAKS Prep on page 542. An electric company wants to install a vertical utility pole at the base of a hill that has an 8% grade. a. To the nearest degree, what angle does the hill make with a horizontal line? b. What is the measure of the angle between the pole and the hill? Round to the nearest degree. c. A utility worker installs a 31-foot guy wire from the top of the pole to the hill. Given that the guy wire is perpendicular to the hill, find the height of the pole to the nearest inch. The side lengths of a right triangle are given below. Find the measures of the acute angles in the triangle. Round to the nearest degree. 48. 3, 4, 5 49. 5, 12, 13 50. 8, 15, 17 51. What if…? A right triangle has leg lengths of 28 and 45 inches. Suppose the length of the longer leg doubles. What happens to the measure of the acute angle opposite that leg? 52. Fitness As part of off-season training, the Houston Texans football team must sprint up a ramp with a 28% grade. To the nearest degree, what angle does this ramp make with a horizontal line? 53. The coordinates of the vertices of a triangle are A (-1, 0) , B (6, 1) , and C (0, 3) . a. Use the Distance Formula to find AB, BC, and AC. b. Use the Converse of the Pythagorean Theorem to show that △ABC is a right triangle. Identify the right angle. c. Find the measures of the acute angles of △ABC. Round to the nearest degree. Find the indicated measure in each rectangle. Round to the nearest degree. 54. m∠BDC 55. m∠STV Find the indicated measure in each rhombus. Round to the nearest degree. 56. m∠DGF 57. m∠LKN Fitness Running on a treadmill is slightly easier than running outdoors, since you don’t have to overcome wind resistance. Set the treadmill to a 1% grade to match the intensity of an outdoor run. 58. Critical Thinking Without using a calculator, compare the values of tan 60° and tan 70°. Explain your reasoning. The measure of an acute angle formed by a line with slope m and the x-axis can be found by using the expression tan -1 (m) . Find the measure of the acute angle that each line makes with the x-axis. Round to the nearest degree. 59. y = 3x + 5 60. y = 2 _ 3 x + 1 61. 5y = 4x + 3 8- 3 Solving Right Triangles 539 539 �� 62. /////ERROR ANALYSIS///// A student was asked to find m∠C. Explain the error in the student’s solution. 63. Write About It A student claims that you must know the three side lengths of a right triangle before you can use trigonometric ratios to find the measures of the acute angles. Do you agree? Why or why not? ̶̶ DC is an altitude of right △ABC. Use trigonometric ratios to find the missing lengths in the figure. Then use these lengths to verify the three relationships in the Geometric Mean Corollaries from Lesson 8-1. 64. 65. Which expression can be used to find m∠A? tan -1 (0.75) sin -1 ( 3 _ ) 5 cos -1 (0.8) tan -1 ( 4 _ ) 3 66. Which expression is NOT equivalent to cos 60°? 1 _ 2 sin 30° sin 60° _ tan 60° cos -1 ( 1 _ ) 2 67. To the nearest degree, what is the measure of the acute angle formed by Jefferson St. and Madison St.? 27° 31° 59° 63° 68. Gridded Response A highway exit ramp . To the nearest degree, has a slope of 3 __ 20 find the angle that the ramp makes with a horizontal line. CHALLENGE AND EXTEND Find each angle measure. Round to the nearest degree. 69. m∠J 70. m∠A Simply each expression. 71. cos -1 (cos 34°) ⎤ ⎦ ⎡ ⎣ tan -1 (1.5) 72. tan 73. sin ( sin -1 x) 74. A ramp has a 6% grade. The ramp is 40 ft long. Find the vertical distance that the ramp rises. Round your answer to the nearest hundredth. 540 540 Chapter 8 Right Triangles and Trigonometry ��Main St.Jefferson St.Madison St.2.7 mi1.4 mige07se_c08l03007aAB�� 75. Critical Thinking Explain why the expression sin -1 (1.5) does not make sense. 76. If you are given the lengths of two sides of △ABC and the measure of the included angle, you can use the formula 1 __ 2 bc sin A to find the area of the triangle. Derive this formula. (Hint: Draw an altitude from B to ratios to find the length of this altitude.) ̶̶ AC . Use trigonometric SPIRAL REVIEW The graph shows the amount of rainfall in a city for the first five months of the year. Determine whether each statement is true or false. (Previous course) 77. It rained more in April than it did in January, February, and March combined. 78. The average monthly rainfall for this five- month period was approximately 3.5 inches. 79. The rainfall amount increased at a constant rate each month over the five-month period. Use the diagram to find each value, given that △ABC ≅ △DEF. (Lesson 4-3) 80. x 81. y 82. DF Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. (Lesson 8-2) 83. sin 63° 84. cos 27° 85. tan 64° Using Technology Use a spreadsheet to complete the following. 1. In cells A2 and B2, enter values for the leg lengths of a right triangle. 2. In cell C2, write a formula to calculate c, the length of the hypotenuse. 3. Write a formula to calculate the measure of ∠A in cell D2. Be sure to use the Degrees function so that the answer is given in degrees. Format the value to include no decimal places. 4. Write a formula to calculate the measure of ∠B in cell E2. Again, be sure to use the Degrees function and format the value to include no decimal places. 5. Use your spreadsheet to check your answers for Exercises 48–50. 8- 3 Solving Right Triangles 541 541 �� SECTION 8A Trigonometric Ratios It’s Electrifying! Utility workers install and repair the utility poles and wires that carry electricity from generating stations to consumers. As shown in the figure, a crew of workers ̶̶ AC plans to install a vertical utility pole ̶̶ AB that is and a supporting guy wire perpendicular to the ground. 1. The utility pole is 30 ft tall. The crew finds that DC = 6 ft. What is the distance DB from the pole to the anchor point of the guy wire? 2. How long is the guy wire? Round to the nearest inch. 3. In the figure, ∠ABD is called the line angle. In order to choose the correct weight of the cable for the guy wire, the crew needs to know the measure of the line angle. Find m∠ABD to the nearest degree. 4. To the nearest degree, what is the measure of the angle formed by the pole and the guy wire? 5. What is the percent grade of the hill on which the crew is working? 542 542 Chapter 8 Right Triangles and Trigonometry �� SECTION 8A Quiz for Lessons 8-1 Through 8-3 8-1 Similarity in Right Triangles Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 1. 5 and 12 2. 2.75 and 44 and 15 _ 3. 5 _ 8 2 Find x, y, and z. 4. 5. 6. 7. A land developer needs to know the distance across a pond on a piece of property. What is AB to the nearest tenth of a meter? �� 8-2 Trigonometric Ratios Use a special right triangle to write each trigonometric ratio as a fraction. 8. tan 45° 9. sin 30° 10. cos 30° Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 11. sin 16° 12. cos 79° 13. tan 27° Find each length. Round to the nearest hundredth. 14. QR 15. AB 16. LM 8-3 Solving Right Triangles Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 17. 18. 19. 20. The wheelchair ramp at the entrance of the Mission Bay Library has a slope of 1 __ 18 . What angle does the ramp make with the sidewalk? Round to the nearest degree. Ready to Go On? 543 543 �� 8-4 Angles of Elevation and Depression TEKS G.11.C Similarity and the geometry of shape: ... apply ... triangle similarity relationships, su
ch as ... trigonometric ratios .... Also G.5.D Objective Solve problems involving angles of elevation and angles of depression. Who uses this? Pilots and air traffic controllers use angles of depression to calculate distances. Vocabulary angle of elevation angle of depression An angle of elevation is the angle formed by a horizontal line and a line of sight to a point above the line. In the diagram, ∠1 is the angle of elevation from the tower T to the plane P. An angle of depression is the angle formed by a horizontal line and a line of sight to a point below the line. ∠2 is the angle of depression from the plane to the tower. Since horizontal lines are parallel, ∠1 ≅ ∠2 by the Alternate Interior Angles Theorem. Therefore the angle of elevation from one point is congruent to the angle of depression from the other point. E X A M P L E 1 Classifying Angles of Elevation and Depression Classify each angle as an angle of elevation or angle of depression. A ∠3 ∠3 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression. B ∠4 ∠4 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation. Use the diagram above to classify each angle as an angle of elevation or angle of depression. 1a. ∠5 1b. ∠6 544 544 Chapter 8 Right Triangles and Trigonometry PT12ge07se_c08l04002aABAngle of depressionAngle of elevation3465ge07se_c08l04006aAB E X A M P L E 2 Finding Distance by Using Angle of Elevation An air traffic controller at an airport sights a plane at an angle of elevation of 41°. The pilot reports that the plane’s altitude is 4000 ft. What is the horizontal distance between the plane and the airport? Round to the nearest foot. Draw a sketch to represent the given information. Let A represent the airport and let P represent the plane. Let x be the horizontal distance between the plane and the airport. tan 41° = 4000 _ x x = 4000 _ tan 41° You are given the side opposite ∠A, and x is the side adjacent to ∠A. So write a tangent ratio. Multiply both sides by x and divide both sides by tan 41°. x ≈ 4601 ft Simplify the expression. 2. What if…? Suppose the plane is at an altitude of 3500 ft and the angle of elevation from the airport to the plane is 29°. What is the horizontal distance between the plane and the airport? Round to the nearest foot. E X A M P L E 3 Finding Distance by Using Angle of Depression A forest ranger in a 90-foot observation tower sees a fire. The angle of depression to the fire is 7°. What is the horizontal distance between the tower and the fire? Round to the nearest foot. Draw a sketch to represent the given information. Let T represent the top of the tower and let F represent the fire. Let x be the horizontal distance between the tower and the fire. The angle of depression may not be one of the angles in the triangle you are solving. It may be the complement of one of the angles in the triangle. By the Alternate Interior Angles Theorem, m∠F = 7°. tan 7° = 90 _ x Write a tangent ratio. x = 90 _ tan 7° Multiply both sides by x and divide both sides by tan 7°. x ≈ 733 ft Simplify the expression. 3. What if…? Suppose the ranger sees another fire and the angle of depression to the fire is 3°. What is the horizontal distance to this fire? Round to the nearest foot. 8- 4 Angles of Elevation and Depression 545 545 �� E X A M P L E 4 Aviation Application A pilot flying at an altitude of 2.7 km sights two control towers directly in front of her. The angle of depression to the base of one tower is 37°. The angle of depression to the base of the other tower is 58°. What is the distance between the two towers? Round to the nearest tenth of a kilometer. Step 1 Draw a sketch. Let P represent the plane and let A and B represent the two towers. Let x be the distance between the towers. Always make a sketch to help you correctly place the given angle measure. Step 2 Find y. By the Alternate Interior Angles Theorem, m∠CAP = 58°. In △APC, tan 58° = 2.7 _ y . So y = 2.7 _ ≈ 1.6871 km. tan 58° Step 3 Find z. By the Alternate Interior Angles Theorem, m∠CBP = 37°. In △BPC, tan 37° = 2.7 _ z . So z = 2.7 _ ≈ 3.5830 km. tan 37° Step 4 Find x. x = z - y x ≈ 3.5830 - 1.6871 ≈ 1.9 km So the two towers are about 1.9 km apart. 4. A pilot flying at an altitude of 12,000 ft sights two airports directly in front of him. The angle of depression to one airport is 78°, and the angle of depression to the second airport is 19°. What is the distance between the two airports? Round to the nearest foot. THINK AND DISCUSS 1. Explain what happens to the angle of elevation from your eye to the top of a skyscraper as you walk toward the skyscraper. 2. GET ORGANIZED Copy and complete the graphic organizer below. In each box, write a definition or make a sketch. 546 546 Chapter 8 Right Triangles and Trigonometry �� 8-4 Exercises Exercises KEYWORD: MG7 8-4 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. An angle of ? is measured from a horizontal line to a point above that line. ̶̶̶̶ (elevation or depression) 2. An angle of ? is measured from a horizontal line to a point below that line. ̶̶̶̶ (elevation or depression. 544 Classify each angle as an angle of elevation or angle of depression. 3. ∠1 4. ∠2 5. ∠3 6. ∠. Measurement When the angle of elevation to p. 545 the sun is 37°, a flagpole casts a shadow that is 24.2 ft long. What is the height of the flagpole to the nearest foot. 545 8. Aviation The pilot of a traffic helicopter sights an accident at an angle of depression of 18°. The helicopter’s altitude is 1560 ft. What is the horizontal distance from the helicopter to the accident? Round to the nearest foot. 546 9. Surveying From the top of a canyon, the angle of depression to the far side of the river is 58°, and the angle of depression to the near side of the river is 74°. The depth of the canyon is 191 m. What is the width of the river at the bottom of the canyon? Round to the nearest tenth of a meter. �� Independent Practice For See Exercises Example 10–13 14 15 16 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S19 Application Practice p. S35 PRACTICE AND PROBLEM SOLVING Classify each angle as an angle of elevation or angle of depression. 10. ∠1 11. ∠2 12. ∠3 13. ∠4 14. Geology To measure the height of a rock formation, a surveyor places her transit 100 m from its base and focuses the transit on the top of the formation. The angle of elevation is 67°. The transit is 1.5 m above the ground. What is the height of the rock formation? Round to the nearest meter. 8- 4 Angles of Elevation and Depression 547 547 1234ge07se_c08l04008a AB1423ge07sec08l04007a�� Space Shuttle Johnson Space Center, in Houston, is home to the Mission Control Center, the base of operations for all space shuttle missions. 15. Forestry A forest ranger in a 120 ft observation tower sees a fire. The angle of 15. depression to the fire is 3.5°. What is the horizontal distance between the tower and the fire? Round to the nearest foot. 16. Space Shuttle Marion is observing the launch of a space shuttle from the command center. When she first sees the shuttle, the angle of elevation to it is 16°. Later, the angle of elevation is 74°. If the command center is 1 mi from the launch pad, how far did the shuttle travel while Marion was watching? Round to the nearest tenth of a mile. Tell whether each statement is true or false. If false, explain why. 17. The angle of elevation from your eye to the top of a tree increases as you walk toward the tree. 18. If you stand at street level, the angle of elevation to a building’s tenth-story window is greater than the angle of elevation to one of its ninth-story windows. 19. As you watch a plane fly above you, the angle of elevation to the plane gets closer to 0° as the plane approaches the point directly overhead. 20. An angle of depression can never be more than 90°. � � � � Use the diagram for Exercises 21 and 22. 21. Which angles are not angles of elevation or angles of depression? 22. The angle of depression from the helicopter to the car is 30°. Find m∠1, m∠2, m∠3, and m∠4. 23. Critical Thinking Describe a situation in which the angle of depression to an object is decreasing. 24. An observer in a hot-air balloon sights a building that is 50 m from the balloon’s launch point. The balloon has risen 165 m. What is the angle of depression from the balloon to the building? Round to the nearest degree. 25. Multi-Step A surveyor finds that the angle of elevation to the top of a 1000 ft tower is 67°. a. To the nearest foot, how far is the surveyor from the base of the tower? b. How far back would the surveyor have to move so that the angle of elevation to the top of the tower is 55°? Round to the nearest foot. 26. Write About It Two students are using shadows to calculate the height of a pole. One says that it will be easier if they wait until the angle of elevation to the sun is exactly 45°. Explain why the student made this suggestion. 27. This problem will prepare you for the Multi-Step TAKS Prep on page 568. The pilot of a rescue helicopter is flying over the ocean at an altitude of 1250 ft. The pilot sees a life raft at an angle of depression of 31°. a. What is the horizontal distance from the helicopter to the life raft, rounded to the nearest foot? b. The helicopter travels at 150 ft/s. To the nearest second, how long will it take until the helicopter is directly over the raft? 548 548 Chapter 8 Right Triangles and Trigonometry 16º74ºge07sec08l04003_A1 mi �� 28. Mai is flying a plane at an altitude of 1600 ft. She sights a stadium at an angle of depression of 35°. What is Mai’s approximate horizontal distance from the stadium? 676 feet 1120 feet 1450 feet 2285 feet 29. Jeff finds that an office building casts a shadow that is 93 ft long when the angle
of elevation to the sun is 60°. What is the height of the building? 54 feet 81 feet 107 feet 161 feet 30. Short Response Jim is rafting down a river that runs through a canyon. He sees a trail marker ahead at the top of the canyon and estimates the angle of elevation from the raft to the marker as 45°. Draw a sketch to represent the situation. Explain what happens to the angle of elevation as Jim moves closer to the marker. CHALLENGE AND EXTEND 31. Susan and Jorge stand 38 m apart. From Susan’s position, the angle of elevation to the top of Big Ben is 65°. From Jorge’s position, the angle of elevation to the top of Big Ben is 49.5°. To the nearest meter, how tall is Big Ben? �� 32. A plane is flying at a constant altitude of 14,000 ft and a constant speed of 500 mi/h. The angle of depression from the plane to a lake is 6°. To the nearest minute, how much time will pass before the plane is directly over the lake? 33. A skyscraper stands between two school buildings. The two schools are 10 mi apart. From school A, the angle of elevation to the top of the skyscraper is 5°. From school B, the angle of elevation is 2°. What is the height of the skyscraper to the nearest foot? 34. Katie and Kim are attending a theater performance. Katie’s seat is at floor level. She looks down at an angle of 18° to see the orchestra pit. Kim’s seat is in the balcony directly above Katie. Kim looks down at an angle of 42° to see the pit. The horizontal distance from Katie’s seat to the pit is 46 ft. What is the vertical distance between Katie’s seat and Kim’s seat? Round to the nearest inch. SPIRAL REVIEW 35. Emma and her mother jog along a mile-long circular path in opposite directions. They begin at the same place and time. Emma jogs at a pace of 4 mi/h, and her mother runs at 6 mi/h. In how many minutes will they meet? (Previous course) 36. Greg bought a shirt that was discounted 30%. He used a coupon for an additional 15% discount. What was the original price of the shirt if Greg paid $17.85? (Previous course) Tell which special parallelograms have each given property. (Lesson 6-5) 37. The diagonals are perpendicular. 38. The diagonals are congruent. 39. The diagonals bisect each other. 40. Opposite angles are congruent. Find each length. (Lesson 8-1) 41. x 42. y 43. z 8- 4 Angles of Elevation and Depression 549 549 �� 8-4 Indirect Measurement Using Trigonometry A clinometer is a surveying tool that is used to measure angles of elevation and angles of depression. In this lab, you will make a simple clinometer and use it to find indirect measurements. Choose a tall object, such as a flagpole or tree, whose height you will measure. TEKS G.11.C Similarity and the geometry of shape: ... apply ... triangle similarity relationships, such as ... trigonometric ratios .... Also G.5.D, G.11.B Use with Lesson 8-4 Activity 1 Follow these instructions to make a clinometer. a. Tie a washer or paper clip to the end of a 6-inch string. b. Tape the string’s other end to the midpoint of the straight edge of a protractor. c. Tape a straw along the straight edge of the protractor. 2 Stand back from the object you want to measure. Use a tape measure to measure and record the distance from your feet to the base of the object. Also measure the height of your eyes above the ground. 3 Hold the clinometer steady and look through the straw to sight the top of the object you are measuring. When the string stops moving, pinch it against the protractor and record the acute angle measure. Try This 1. How is the angle reading from the clinometer related to the angle of elevation from your eye to the top of the object you are measuring? 2. Draw and label a diagram showing the object and the measurements you made. Then use trigonometric ratios to find the height of the object. 3. Repeat the activity, measuring the angle of elevation to the object from a different distance. How does your result compare to the previous one? 4. Describe possible measurement errors that can be made in the activity. 5. Explain why this method of indirect measurement is useful in real-world situations. 550 550 Chapter 8 Right Triangles and Trigonometry 8-5 Law of Sines and Law of Cosines TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as ... trigonometric ratios .... Also G.5.B, G.5.D, G.7.A, G.11.A Objective Use the Law of Sines and the Law of Cosines to solve triangles. Who uses this? Engineers can use the Law of Sines and the Law of Cosines to solve construction problems. Since its completion in 1370, engineers have proposed many solutions for lessening the tilt of the Leaning Tower of Pisa. The tower does not form a right angle with the ground, so the engineers have to work with triangles that are not right triangles. In this lesson, you will learn to solve any triangle. To do so, you will need to calculate trigonometric ratios for angle measures up to 180°. You can use a calculator to find these values. E X A M P L E 1 Finding Trigonometric Ratios for Obtuse Angles You will learn more about trigonometric ratios of angle measures greater than or equal to 90° in the Chapter Extension. Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. A sin 135° B tan 98° C cos 108° sin 135° ≈ 0.71 tan 98° ≈ -7.12 cos 108° ≈ -0.31 Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. 1a. tan 175° 1c. sin 160° 1b. cos 92° You can use the altitude of a triangle to find a relationship between the triangle’s side lengths. In △ABC, let h represent the length of the ̶̶ AB . altitude from C to , and sin B = h _ a . From the diagram, sin A = h _ b By solving for h, you find that h = b sin A and h = a sin B. So b sin A = a sin B, and sin A _ a You can use another altitude to show that these ratios equal sin C _ . c = sin B _ . b 8-5 Law of Sines and Law of Cosines 551 551 �� Theorem 8-5-1 The Law of Sines For any △ABC with side lengths a, b, and c, sin A _ a = sin C _ c = sin B _ . b You can use the Law of Sines to solve a triangle if you are given • two angle measures and any side length (ASA or AAS) or • two side lengths and a non-included angle measure (SSA). E X A M P L E 2 Using the Law of Sines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. A DF sin D _ EF sin 105° _ 18 = sin E _ DF = sin 32° _ DF Law of Sines Substitute the given values. DF sin 105° = 18 sin 32° Cross Products Property In a proportion with three parts, you can use any of the two parts together. DF = 18 sin 32° _ ≈ 9.9 sin 105° B m∠S sin T _ RS sin 75° _ 7 = sin S _ RT = sin S _ 5 sin S = 5 sin 75° _ m∠S ≈ sin -1 ( 5 sin 75° _ 7 7 ) ≈ 44° Divide both sides by sin 105°. Law of Sines Substitute the given values. Multiply both sides by 5. Use the inverse sine function to find m∠S. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 2a. NP 2b. m∠L 2c. m∠X 2d. AC The Law of Sines cannot be used to solve every triangle. If you know two side lengths and the included angle measure or if you know all three side lengths, you cannot use the Law of Sines. Instead, you can apply the Law of Cosines. 552 552 Chapter 8 Right Triangles and Trigonometry �� Theorem 8-5-2 The Law of Cosines For any △ABC with side lengths a, b, and c: a 2 = b 2 + c 2 - 2bc cos A b 2 = a 2 + c 2 - 2ac cos B c 2 = a 2 + b 2 - 2ab cos C The angle referenced in the Law of Cosines is across the equal sign from its corresponding side. You will prove one case of the Law of Cosines in Exercise 57. You can use the Law of Cosines to solve a triangle if you are given • two side lengths and the included angle measure (SAS) or • three side lengths (SSS). E X A M P L E 3 Using the Law of Cosines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. A BC BC 2 = AB 2 + AC 2 - 2 (AB) (AC) cos A = 14 2 + 9 2 - 2 (14 ) (9 ) cos 62° Law of Cosines Substitute the given BC 2 ≈ 158.6932 BC ≈ 12.6 values. Simplify. Find the square root of both sides. B m∠R ST 2 = RS 2 + RT 2 - 2 (RS) (RT) cos R Law of 9 2 = 4 2 + 7 2 -2 (4 ) (7 ) cos R 81 = 65 - 56 cos R 16 = -56 cos R cos R = - 16 _ 56 m∠R = cos -1 (- 16 _ 56 ) ≈ 107° Cosines Substitute the given values. Simplify. Subtract 65 from both sides. Solve for cos R. Use the inverse cosine function to find m∠R. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 3a. DE 3b. m∠K 3c. YZ 3d. m∠R 8-5 Law of Sines and Law of Cosines 553 553 �� E X A M P L E 4 Engineering Application The Leaning Tower of Pisa is 56 m tall. In 1999, the tower made a 100° angle with the ground. To stabilize the tower, an engineer considered attaching a cable from the top of the tower to a point that is 40 m from the base. How long would the cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree. Step 1 Find the length of the cable. AC 2 = AB 2 + BC 2 - 2 (AB) (BC) cos B = 40 2 + 56 2 - 2 (40 ) (56 ) cos 100° AC 2 ≈ 5513.9438 AC ≈ 74.3 m Law of Cosines Substitute the given values. Simplify. Find the square root of both sides. Do not round your answer until the final step of the computation. If a problem has multiple steps, store the calculated answers to each part in your calculator. Step 2 Find the measure of the angle the cable would make with the ground. sin A _ = sin B _ BC AC sin A _ ≈ sin 100° _ 56 74.2559 sin A ≈ 56 sin 100° _ m∠A ≈ sin -1 ( 56 sin 100° _ 74.2559 74.2559 ) ≈ 48° Law of Sines Substitute the calculated value for AC. Multiply both sides by 56. Use the inverse sine function to find m∠A. 4. What if…? Another engineer suggested using a cable attached from the top of the tower to a point 31 m from the base. How long would
this cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree. THINK AND DISCUSS 1. Tell what additional information, if any, is needed to find BC using the Law of Sines. 2. GET ORGANIZED Copy and complete the graphic organizer. Tell which law you would use to solve each given triangle and then draw an example. 554 554 Chapter 8 Right Triangles and Trigonometry 100º56 m40 mABCge07sec08l05002a�� 8-5 Exercises Exercises KEYWORD: MG7 8-5 KEYWORD: MG7 Parent GUIDED PRACTICE Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. p. 551 1. sin 100° 4. tan 141° 7. sin 147° 2. cos 167° 5. cos 133° 8. tan 164° 3. tan 92° 6. sin 150° 9. cos 156. 552 Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 10. RT 11. m∠B 12. m∠ 13. m∠Q 14. MN 15. AB p. 553 . 554 16. Carpentry A carpenter makes a triangular frame by joining three pieces of wood that are 20 cm, 24 cm, and 30 cm long. What are the measures of the angles of the triangle? Round to the nearest degree. Independent Practice Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. PRACTICE AND PROBLEM SOLVING For See Exercises Example 17–25 26–31 32–37 38 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S19 Application Practice p. S35 17. cos 95° 20. sin 132° 23. tan 139° 18. tan 178° 21. sin 98° 24. cos 145° 19. tan 118° 22. cos 124° 25. sin 128° Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 26. m∠C 27. PR 28. JL 29. EF 30. m∠J 31. m∠X 8-5 Law of Sines and Law of Cosines 555 555 �� Surveying Many modern surveys are done with GPS (Global Positioning System) technology. GPS uses orbiting satellites as reference points from which other locations are established. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 32. AB 33. m∠Z 34. m∠R 35. EF 36. LM 37. m∠G 38. Surveying To find the distance across a lake, a surveyor locates points A, B, and C as shown. What is AB to the nearest tenth of a meter, and what is m∠B to the nearest degree? Use the figure for Exercises 39–42. Round lengths to the nearest tenth and angle measures to the nearest degree. 39. m∠A = 74°, m∠B = 22°, and b = 3.2 cm. Find a. 40. m∠C = 100°, a = 9.5 in., and b = 7.1 in. Find c. 41. a = 2.2 m, b = 3.1 m, and c = 4 m. Find m∠B. 42. a = 10.3 cm, c = 8.4 cm, and m∠A = 45°. Find m∠C. 43. Critical Thinking Suppose you are given the three angle measures of a triangle. Can you use the Law of Sines or the Law of Cosines to find the lengths of the sides? Why or why not? 44. What if…? What does the Law of Cosines simplify to when the given angle is a right angle? 45. Orienteering The map of a beginning orienteering course is shown at right. To the nearest degree, at what angle should a team turn in order to go from the first checkpoint to the second checkpoint? Multi-Step Find the perimeter of each triangle. Round to the nearest tenth. 46. 47. 48. 49. The ambiguous case of the Law of Sines occurs when you are given an acute angle measure and when the side opposite this angle is shorter than the other given side. In this case, there are two possible triangles. Find two possible values for m∠C to the nearest degree. (Hint: The inverse sine function on your calculator gives you only acute angle measures. Consider this angle and its supplement.) 556 556 Chapter 8 Right Triangles and Trigonometry 6 km4 km3 kmSecondcheckpointFirstcheckpointStart?ge07se_c08L05003atopo mapGeometry 2007 SEHolt Rinehart WinstonKaren Minot(415)883-6560�� 50. This problem will prepare you for the Multi-Step TAKS Prep on page 568. Rescue teams at two heliports, A and B, receive word of a fire at F. a. What is m∠AFB? b. To the nearest mile, what are the distances from each heliport to the fire? c. If a helicopter travels 150 mi/h, how much time is saved by sending a helicopter from A rather than B? Identify whether you would use the Law of Sines or Law of Cosines as the first step when solving the given triangle. 51. 52. 53. 54. The coordinates of the vertices of △RST are R (0, 3) , S (3, 1) , and T (-3, -1) . a. Find RS, ST, and RT. b. Which angle of △RST is the largest? Why? c. Find the measure of the largest angle in △RST to the nearest degree. 55. Art Jessika is creating a pattern for a piece of stained glass. Find BC, AB, and m∠ABC. Round lengths to the nearest hundredth and angle measures to the nearest degree. 56. /////ERROR ANALYSIS///// Two students were asked to find x in △DEF. Which solution is incorrect? Explain the error. 57. Complete the proof of the Law of Cosines for the case when △ABC is an acute triangle. Given: △ABC is acute with side lengths a, b, and c. Prove: a 2 = b 2 + c 2 - 2bc cos A Proof: Draw the altitude from C to ̶̶ AB . Let h be the length of this altitude. ̶̶ AB into segments of lengths x and y. By the Pythagorean Theorem, It divides a 2 = a. to get c. expression for b 2 to get d. Therefore a 2 = b 2 + c 2 - 2bc cos A by f. ? , and b. ̶̶̶̶ ? . Rearrange the terms to get - 2cx. Substitute the ̶̶̶̶ ? = h 2 + x 2 . Substitute y = c - x into the first equation ̶̶̶̶ ? . From the diagram, cos A = x __ . So x = e. ̶̶̶̶ b ? . ̶̶̶̶ ? . ̶̶̶̶ 58. Write About It Can you use the Law of Sines to solve △EFG? Explain why or why not. 8-5 Law of Sines and Law of Cosines 557 557 �� 59. Which of these is closest to the length of ̶̶ AB ? 5.5 centimeters 7.5 centimeters 14.4 centimeters 22.2 centimeters 60. Which set of given information makes it possible to find x using the Law of Sines? m∠T = 38°, RS = 8.1, ST = 15.3 RS = 4, m∠S = 40°, ST = 9 m∠R = 92°, m∠S = 34°, ST = 7 m∠R = 105°, m∠S = 44°, m∠T = 31° 61. A surveyor finds that the face of a pyramid makes a 135° angle with the ground. From a point 100 m from the base of the pyramid, the angle of elevation to the top is 25°. ̶̶ XY ? How long is the face of the pyramid, 48 meters 81 meters 124 meters 207 meters CHALLENGE AND EXTEND 62. Multi-Step Three circular disks are placed next to each other as shown. The disks have radii of 2 cm, 3 cm, and 4 cm. The centers of the disks form △ABC. Find m∠ACB to the nearest degree. 63. Line ℓ passes through points (-1, 1) and (1, 3) . Line m passes through points (-1, 1) and (3, 2) . Find the measure of the acute angle formed by ℓ and m to the nearest degree. 64. Navigation The port of Bonner is 5 mi due south of the port of Alston. A boat leaves the port of Alston at a bearing of N 32° E and travels at a constant speed of 6 mi/h. After 45 minutes, how far is the boat from the port of Bonner? Round to the nearest tenth of a mile. SPIRAL REVIEW Write a rule for the nth term in each sequence. (Previous course) 65. 3, 6, 9, 12, 15, … 66. 3, 5, 7, 9, 11, … 67. 4, 6, 8, 10, 12, … State the theorem or postulate that justifies each statement. (Lesson 3-2) 68. ∠1 ≅ ∠8 69. ∠4 ≅ ∠5 70. m∠4 + m∠6 = 180° 71. ∠2 ≅ ∠7 Use the given trigonometric ratio to determine which angle of the triangle is ∠A. (Lesson 8-3) 73. sin A = 15 _ 72. cos A = 15 _ 17 17 74. tan A = 1.875 558 558 Chapter 8 Right Triangles and Trigonometry ��YX135°25°100 mge07se_c08l05005aA2 cm4 cm3 cmBCge07se_c08l05006aaAB�� 8-6 Vectors TEKS G.7.A Dimensionality and the geometry of location: use ... two-dimensional coordinate systems to represent ... line segments, and figures. Objectives Find the magnitude and direction of a vector. Use vectors and vector addition to solve realworld problems. Who uses this? By using vectors, a kayaker can take water currents into account when planning a course. (See Example 5.) The speed and direction an object moves can be represented by a vector. A vector is a quantity that has both length and direction. You can think of a vector as a directed line segment. The vector below may be named  AB or  v . Vocabulary vector component form magnitude direction equal vectors parallel vectors resultant vector Also G.1.B, G.7.C, G.11.C A vector can also be named using component form. The component form 〈x, y〉 of a vector lists the horizontal and vertical change from the initial point to   CD is 〈2, 3〉. the terminal point. The component form of E X A M P L E 1 Writing Vectors in Component Form Write each vector in component form. A B   EF The horizontal change from E to F is 4 units. The vertical change from E to F is -3 units. So the component form of  EF is 〈4, -3〉.   PQ with P (7, -5) and Q (4, 3)  PQ = 〈〉 Subtract the coordinates of the initial point  PQ = 〈4 - 7, 3 - (-5) 〉  PQ = 〈-3, 8〉 from the coordinates of the terminal point. Substitute the coordinates of the given points. Simplify. Write each vector in component form. 1a.  u 1b. the vector with initial point L (-1, 1) and terminal point M (6, 2) 8-6 Vectors 559 559 �� The magnitude of a vector is its length. The magnitude of a vector is written  AB ⎟ or ⎜ ⎜  v ⎟ . When a vector is used to represent speed in a given direction, the magnitude of the vector equals the speed. For example, if a vector rep
resents the course a kayaker paddles, the magnitude of the vector is the kayaker’s speed. E X A M P L E 2 Finding the Magnitude of a Vector Draw the vector 〈4, -2〉 on a coordinate plane. Find its magnitude to the nearest tenth. Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Then (4, -2) is the terminal point. Step 2 Find the magnitude. Use the Distance Formula. ⎜〈4, -2〉⎟ = √  (4 - 0) 2 + (-2 - 0) 2 = √  20 ≈ 4.5 2. Draw the vector 〈-3, 1〉 on a coordinate plane. Find its magnitude to the nearest tenth. The direction of a vector is the angle that it makes with a horizontal line. This angle is measured counterclockwise from the positive x-axis.  AB is 60°. The direction of See Lesson 4-5, page 252, to review bearings. The direction of a vector can also be given as a bearing relative to the compass directions north, south, east, and west. of N 30° E.  AB has a bearing E X A M P L E 3 Finding the Direction of a Vector A wind velocity is given by the vector 〈2, 5〉. Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree. Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Step 2 Find the direction. Draw right triangle ABC as shown. ∠A is the angle formed by the vector and the x-axis, and tan A = 5 _ 2 . So m∠A = tan -1 ( 5 _ ) ≈ 68°. 2 3. The force exerted by a tugboat is given by the vector 〈7, 3〉. Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree. 560 560 Chapter 8 Right Triangles and Trigonometry �� Two vectors are equal vectors if they have the same  v .  u = magnitude and the same direction. For example, Equal vectors do not have to have the same initial point and terminal point. Two vectors are parallel vectors if they have the same direction or if they have opposite directions. They may have different magnitudes. For example, vectors are always parallel vectors.  x . Equal  w ǁ  BA  AB ≠ Note that since these vectors do not have the same direction. E X A M P L E 4 Identifying Equal and Parallel Vectors Identify each of the following. A equal vectors  AB =  GH Identify vectors with the same magnitude and direction. B parallel vectors  AB ǁ  GH and  CD ǁ  EF Identify vectors with the same or opposite directions. Identify each of the following. 4a. equal vectors 4b. parallel vectors The resultant vector is the vector that represents the sum of two given vectors. To add two vectors geometrically, you can use the head-to-tail method or the parallelogram method. Vector Addition METHOD EXAMPLE Head-to-Tail Method Place the initial point (tail) of the second vector on the terminal point (head) of the first vector. The resultant is the vector that joins the initial point of the first vector to the terminal point of the second vector. Parallelogram Method Use the same initial point for both of the given vectors. Create a parallelogram by adding a copy of each vector at the terminal point (head) of the other vector. The resultant vector is a diagonal of the parallelogram formed. 8-6 Vectors 561 561 �� To add vectors numerically, add their components. If  v = 〈 x 2 , y 2 〉, then  〉.  〉 and E X A M P L E 5 Sports Application A kayaker leaves shore at a bearing of N 55° E and paddles at a constant speed of 3 mi/h. There is a 1 mi/h current moving due east. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Step 1 Sketch vectors for the kayaker and the current. Component form gives the horizontal and vertical change from the initial point to the terminal point of the vector. Step 2 Write the vector for the kayaker in component form. The kayaker’s vector has a magnitude of 3 mi/h and makes an angle of 35° with the x-axis. , so x = 3 cos 35° ≈ 2.5. cos 35° = x _ 3 y _ , so y = 3 sin 35° ≈ 1.7. sin 35° = 3 The kayaker’s vector is 〈2.5, 1.7〉. Step 3 Write the vector for the current in component form. Since the current moves 1 mi/h in the direction of the x-axis, it has a horizontal component of 1 and a vertical component of 0. So its vector is 〈1, 0〉. Step 4 Find and sketch the resultant vector AB . Add the components of the kayaker’s vector and the current’s vector. 〈2.5, 1.7〉 + 〈1, 0〉 = 〈3.5, 1.7〉 The resultant vector in component form is 〈3.5, 1.7〉. Step 5 Find the magnitude and direction of the resultant vector. The magnitude of the resultant vector is the kayak’s actual speed. ⎜〈3.5, 1.7〉⎟ = √ (3.5 - 0)2 + (1.7 - 0)2 ≈ 3.9 mi/h The angle measure formed by the resultant vector gives the kayak’s actual direction. tan A = 1.7_ 3.5 3.5) ≈ 26°, or N 64° E. , so A = tan -1(1.7_ 5. What if…? Suppose the kayaker in Example 5 instead paddles at 4 mi/h at a bearing of N 20° E. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. 562 562 Chapter 8 Right Triangles and Trigonometry ��〈��〉�� THINK AND DISCUSS 1. Explain why the segment with endpoints (0, 0) and (1, 4) is not a vector. 2. Assume you are given a vector in component form. Other than the Distance Formula, what theorem can you use to find the vector’s magnitude? 3. Describe how to add two vectors numerically. 4. GET ORGANIZED Copy and complete the graphic organizer. 8-6 Exercises Exercises KEYWORD: MG7 8-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. 2. ? vectors have the same magnitude and direction. (equal, parallel, or resultant) ̶̶̶̶ ? vectors have the same or opposite directions. (equal, parallel, or resultant) ̶̶̶̶ 3. The ? of a vector indicates the vector’s size. (magnitude or direction) ̶̶̶̶ Write each vector in component form. p. 559 4.  AC with A (1, 2) and C (6, 5) 5. the vector with initial point M (-4, 5) and terminal point N (4, -3) 6.  PQ Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. p. 560 7. 〈1, 4〉 8. 〈-3, -2〉 9. 〈5, -3. 560 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 10. A river’s current is given by the vector 〈4, 6〉. 11. The velocity of a plane is given by the vector 〈5, 1〉. 12. The path of a hiker is given by the vector 〈6, 3〉. Identify each of the following. p. 561 13. equal vectors in diagram 1 14. parallel vectors in diagram 1 15. equal vectors in diagram 2 16. parallel vectors in diagram 2 8-6 Vectors 563 563 �� . 562 17. Recreation To reach a campsite, a hiker first walks for 2 mi at a bearing of N 40° E. Then he walks 3 mi due east. What are the magnitude and direction of his hike from his starting point to the campsite? Round the distance to the nearest tenth of a mile and the direction to the nearest degree. PRACTICE AND PROBLEM SOLVING Independent Practice Write each vector in component form. For See Exercises Example 18. JK with J (-6, -7) and K (3, -5)  18–20 21–23 24–26 27–30 31 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S19 Application Practice p. S35 19.  EF with E (1.5, -3) and F (-2, 2.5) 20.  w Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 21. 〈-2, 0〉 22. 〈1.5, 1.5〉 23. 〈2.5, -3.5〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 24. A boat’s velocity is given by the vector 〈4, 1.5〉. 25. The path of a submarine is given by the vector 〈3.5, 2.5〉. 26. The path of a projectile is given by the vector 〈2, 5〉. Identify each of the following. 27. equal vectors in diagram 1 28. parallel vectors in diagram 1 29. equal vectors in diagram 2 30. parallel vectors in diagram 2 31. Aviation The pilot of a single-engine airplane flies at a constant speed of 200 km/h at a bearing of N 25° E. There is a 40 km/h crosswind blowing southeast (S 45° E) . What are the plane’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Find each vector sum. 32. 〈1, 2〉 + 〈0, 6〉 34. 〈0, 1〉 + 〈7, 0〉 33. 〈-3, 4〉 + 〈5, -2〉 35. 〈8, 3〉 + 〈-2, -1〉 36. Critical Thinking Is vector addition commutative? That is, is  u +  v equal to  v +  u ? Use the head-to-tail method of vector addition to explain why or why not. 564 564 Chapter 8 Right Triangles and Trigonometry 40°2 mi3 miCampsiteNSWEge07se_c08L06002ahiking mapGeometry 2007 SEHolt Rinehart WinstonKaren Minot(415)883-6560�� 37. This problem will prepare you for the Multi-Step TAKS Prep on page 568. A helicopter at H must fly at 50 mi/h in the direction N 45° E to reach the site of a flood victim F. There is a 41 mi/h wind in the direction N 53° W.  HX he should use so that The pilot needs to the know the velocity vector  HF . his resultant vector will be a. What is m∠F ? (Hint: Consider a vertical line through F.) b. Use the Law of Cosines to find the magnitude of  HX to the nearest tenth. c. Use the Law of Sines to find m∠FHX to the nearest degree. d. What is the direction of  HX ? Write each vector in component form. Round values to the nearest tenth. 38. magnitude 15, direction 42° 39. magnitude 7.2, direction 9° 40. magnitude 12.1, direction N 57° E 41. magnitude 5.8, direction N 22° E 42. Physics A classroom has a window near the ceiling, and a long pole must be used to close it. a. Carla holds the pole at a 45° angle to the floor and applies 10 lb of force to the upper edge of the window. Find the vertical component of the vector representing the force on the window. Round to the nearest tenth. b. Taneka also applies 10 lb of force
to close the window, but she holds the pole at a 75° angle to the floor. Find the vertical component of the force vector in this case. Round to the nearest tenth. c. Who will have an easier time closing the window, Carla or Taneka? (Hint: Who applies more vertical force?) 43. Probability The numbers 1, 2, 3, and 4 are written on slips of paper and placed in a hat. Two different slips of paper are chosen at random to be the x- and y-components of a vector. a. What is the probability that the vector will be equal to 〈1, 2〉? b. What is the probability that the vector will be parallel to 〈1, 2〉? 44. Estimation Use the vector 〈4, 6〉 to complete the following. a. Draw the vector on a sheet of graph paper. b. Estimate the vector’s direction to the nearest degree. c. Use a protractor to measure the angle the vector makes with a horizontal line. d. Use the vector’s components to calculate its direction. e. How did your estimate in part b compare to your measurement in part c and your calculation in part d? Multi-Step Find the magnitude of each vector to the nearest tenth and the direction of each vector to the nearest degree. 45.  u 47.  w 46.  v 48.  z 8-6 Vectors 565 565 �� 49. Football Write two vectors in component form to represent the pass pattern that Jason is told to run. Find the resultant vector and show that Jason’s move is equivalent to the vector. For each given vector, find another vector that has the same magnitude but a different direction. Then find a vector that has the same direction but a different magnitude. 50. 〈-3, 6〉 51. 〈12, 5〉 52. 〈8, -11〉 Multi-Step Find the sum of each pair of vectors. Then find the magnitude and direction of the resultant vector. Round the magnitude to the nearest tenth and the direction to the nearest degree. 53.  u = 〈1, 2〉,  v = 〈4.8, -3.1〉  v = 〈2.5, -1〉  u = 〈-2, 7〉, 54. 55.  u = 〈6, 0〉,  v = 〈-2, 4〉 56.  u = 〈-1.2, 8〉,  v = 〈5.2, -2.1〉 57. Math History In 1827, the mathematician August Ferdinand Möbius published a book in which he introduced directed line segments (what we now call vectors). He showed how to perform scalar multiplication of vectors. For example, consider a hiker who walks along a path given by the vector walks twice as far in the same direction is given by the vector 2 a. Write the component form of the vectors b. Find the magnitude of c. Find the direction of d. Given the component form of a vector, explain how to find  v . How do they compare?  v . How do they compare?  v . The path of another hiker who  v and 2  v and 2  v and 2  v .  v . the components of the vector k  v , where k is a constant. e. Use scalar multiplication with k = -1 to write the negation of a vector  v in component form. 58. Critical Thinking A vector Another vector possible directions and magnitudes for the resultant vector.  u points due west with a magnitude of u units.  v points due east with a magnitude of v units. Describe three Math History August Ferdinand Möbius is best known for experimenting with the Möbius strip, a three-dimensional figure that has only one side and one edge. 59. Write About It Compare a line segment, a ray, and a vector. 566 566 Chapter 8 Right Triangles and Trigonometry �� 60. Which vector is parallel to 〈2, 1〉?  u  v  w  z 61. The vector 〈7, 9〉 represents the velocity of a helicopter. What is the direction of this vector to the nearest degree? 38° 52° 128° 142° 62. A canoe sets out on a course given by the vector 〈5, 11〉. What is the length of the canoe’s course to the nearest unit? 6 8 12 16 63. Gridded Response  AB has an initial point of (-3, 6) and a terminal point of (-5, -2) . Find the magnitude of  AB to the nearest tenth. CHALLENGE AND EXTEND Recall that the angle of a vector’s direction is measured counterclockwise from the positive x-axis. Find the direction of each vector to the nearest degree. 64. 〈-2, 3〉 65. 〈-4, 0〉 66. 〈-5, -3〉 67. Navigation The captain of a ship is planning to sail in an area where there is a 4 mi/h current moving due east. What speed and bearing should the captain maintain so that the ship’s actual course (taking the current into account) is 10 mi/h at a bearing of N 70° E? Round the speed to the nearest tenth and the direction to the nearest degree. 68. Aaron hikes from his home to a park by walking 3 km at a bearing of N 30° E, then 6 km due east, and then 4 km at a bearing of N 50° E. What are the magnitude and direction of the vector that represents the straight path from Aaron’s home to the park? Round the magnitude to the nearest tenth and the direction to the nearest degree. SPIRAL REVIEW Solve each system of equations by graphing. (Previous course) x - y = -5 ⎧ 69. ⎨ ⎩ y = 3x + 1 x - 2y = 0 ⎧ 70. ⎨ ⎩ 2y + x = 8 x + y = 5 ⎧ 71. ⎨ ⎩ 3y + 15 = 2x Given that △JLM ∼ △NPS, the perimeter of △JLM is 12 cm, and the area of △JLM is 6 cm 2 , find each measure. (Lesson 7-5) 72. the perimeter of △NPS 73. the area of △NPS Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. (Lesson 8-5) 74. BC 75. m∠B 76. m∠C 8-6 Vectors 567 567 �� SECTION 8B Applying Trigonometric Ratios Help Is on the Way! Rescue helicopters were first used in the 1950s during the Korean War. The helicopters made it possible to airlift wounded soldiers to medical stations. Today, helicopters are used to rescue injured hikers, flood victims, and people who are stranded at sea. 1. The pilot of a helicopter is searching for an injured hiker. While flying at an altitude of 1500 ft, the pilot sees smoke at an angle of depression of 14°. Assuming that the smoke is a distress signal from the hiker, what is the helicopter’s horizontal distance to the hiker? Round to the nearest foot. 2. The pilot plans to fly due north at 100 mi/h from the helicopter’s current position H to the location of the smoke S. However there is a 30 mi/h wind in the direction N 57° W. The pilot needs to know  HA that he should use so that the velocity vector his resultant vector will be then use the Law of Cosines to find the magnitude of  HA to the nearest mile per hour.  HS . Find m∠S and 3. Use the Law of Sines to find the direction of  HA to the nearest degree. 568 568 Chapter 8 Right Triangles and Trigonometry �� SECTION 8B Quiz for Lessons 8-4 Through 8-6 8-4 Angles of Elevation and Depression 1. An observer in a blimp sights a football stadium at an angle of depression of 34°. The blimp’s altitude is 1600 ft. What is the horizontal distance from the blimp to the stadium? Round to the nearest foot. 2. When the angle of elevation of the sun is 78°, a building casts a shadow that is 6 m long. What is the height of the building to the nearest tenth of a meter? 8-5 Law of Sines and Law of Cosines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 3. m∠A 4. GH 5. XZ 6. UV 7. m∠F 8. QS 8-6 Vectors Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 9. 〈3, 1〉 10. 〈-2, -4〉 11. 〈0, 5〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 12. A wind velocity is given by the vector 〈2, 1〉. 13. The current of a river is given by the vector 〈5, 3〉. 14. The force of a spring is given by the vector 〈4, 4〉. 15. To reach an island, a ship leaves port and sails for 6 km at a bearing of N 32° E. It then sails due east for 8 km. What are the magnitude and direction of the voyage directly from the port to the island? Round the distance to the nearest tenth of a kilometer and the direction to the nearest degree. Ready to Go On? 569 569 �� EXTENSION Trigonometry and EXTENSION the Unit Circle TEKS G.11.A Similarity and the geometry of shape: use and extend ... transformations to expore ... geometric figures. Also G.5.B, G.5.C, G.7.A, G.10.A, G.11.C Objective Define trigonometric ratios for angle measures greater than or equal to 90°. Rotations are used to extend the concept of trigonometric ratios to angle measures greater than or equal to 90°. Consider a ray with its endpoint at the origin, pointing in the direction of the positive x-axis. Rotate the ray counterclockwise around the origin. The acute angle formed by the ray and the nearest part of the positive or negative x-axis is called the reference angle . The rotated ray is called the terminal side of that angle. Vocabulary reference angle unit circle Angle measure: 135° Reference angle: 45° Angle measure: 345° Reference angle: 15° Angle measure: 435° Reference angle: 75° E X A M P L E 1 Finding Reference Angles Sketch each angle on the coordinate plane. Find the measure of its reference angle. A 102° B 236° Reference angle: 180° - 102° = 78° Reference angle: 236° - 180° = 56° Sketch each angle on the coordinate plane. Find the measure of its reference angle. 1a. 309° 1b. 410° The unit circle is a circle with a radius of 1 unit, centered at the origin. It can be used to find the trigonometric ratios of an angle. Consider the acute angle θ. Let P (x, y) be the point where the terminal side of θ intersects the unit circle. Draw a vertical line from P to the x-axis. Since cos θ = x __ 1 y __ 1 , the coordinates of P can be written and sin θ = as (cos θ, sin θ) . Thus if you know the coordinates of a point on the unit circle, you can find the trigonometric ratios for the associated angle. In trigonometry, the Greek letter theta, θ, is often used to represent angle measures. 570 570 Chapter 8 Right Triangles and Trigonometry �� E X A M P L E 2 Finding Trigonometric Ratios Find each trigonometric ratio. A cos 150° Sketch the angle on the coordinate plane. The reference angle is 30°. cos 30° = √  3 _ 2 sin 30° = 1 _ 2 Be sure to use the correct sign when assigning coordin
ates to a point on the unit circle. Let P (x, y) be the point where the terminal side of the angle intersects the unit circle. Since P is in Quadrant II, its x-coordinate is negative, and its y-coordinate is positive. So the coordinates of P are (- √  3 ___ 2 , 1 __ 2 ) . The cosine of 150° is the x-coordinate of P, so cos 150° = - √  3 ___ 2 . B tan 315° Sketch the angle on the coordinate plane. The reference angle is 45°. cos 45° = √  2 _ 2 sin 45° = √  2 _ 2 Since P (x, y) is in Quadrant IV, its y-coordinate is negative. So the coordinates of P are ( √  2 ___ 2 , - √  2 ___ 2 ) . Remember that tan θ = sin θ _ . So tan 315° = sin 315° _ = cos θ cos 315° - √  2 ___ 2 _ √  2 ___ 2 = -1. Find each trigonometric ratio. 2a. cos 240° 2b. sin 135° EXTENSION Exercises Exercises Sketch each angle on the coordinate plane. Find the measure of its reference angle. 1. 125° 2. 216° 3. 359° Find each trigonometric ratio. 4. cos 225° 7. tan 135° 10. sin 90° 5. sin 120° 8. cos 420° 11. cos 180° 6. cos 300° 9. tan 315° 12. sin 270° 13. Critical Thinking Given that cos θ = 0.5, what are the possible values for θ between 0° and 360°? 14. Write About It Explain how you can use the unit circle to find tan 180°. 15. Challenge If sin θ ≈ -0.891, what are two values of θ between 0° and 360°? Chapter 8 Extension 571 571 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary angle of depression . . . . . . . . . 544 equal vectors . . . . . . . . . . . . . . . 561 sine . . . . . . . . . . . . . . . . . . . . . . . . 525 angle of elevation . . . . . . . . . . . 544 geometric mean . . . . . . . . . . . . 519 tangent . . . . . . . . . . . . . . . . . . . . 525 component form . . . . . . . . . . . 559 magnitude . . . . . . . . . . . . . . . . . 560 trigonometric ratio . . . . . . . . . 525 cosine . . . . . . . . . . . . . . . . . . . . . 525 parallel vectors . . . . . . . . . . . . . 561 vector . . . . . . . . . . . . . . . . . . . . . . 559 direction . . . . . . . . . . . . . . . . . . . 560 resultant vector . . . . . . . . . . . . . 561 Complete the sentences below with vocabulary words from the list above. 1. The ? of a vector gives the horizontal and vertical change from the initial point ̶̶̶̶ to the terminal point. 2. Two vectors with the same magnitude and direction are called 3. If a and b are positive numbers, then √  ab is the ? . ̶̶̶̶ ? of a and b. ̶̶̶̶ 4. A(n) ? is the angle formed by a horizontal line and a line of sight to a point ̶̶̶̶ above the horizontal line. 5. The sine, cosine, and tangent are all examples of a(n) ? . ̶̶̶̶ 8-1 Similarity in Right Triangles (pp. 518–523) E X A M P L E S EXERCISES ■ Find the geometric mean of 5 and 30. 6. Write a similarity Let x be the geometric mean. x 2 = (5)(30) = 150 Def. of geometric mean x = √150 = 5 √6 Find the positive square root. statement comparing the three triangles. TEKS G.5.B, G.5.D, G.8.A, G.8.C, G.11.A, G.11.B, G.11.C Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 7. 1 _ 4 8. 3 and 17 and 100 √  33 is the geometric mean of 3 and 3 + x. Find x, y, and z. 9. 10. ■ Find x, y, and z. ( √  33 ) 2 = 3 (3 + x) 33 = 9 + 3x 24 = 3x x = 8 y 2 = (3) (8) y 2 = 24 y = √  24 = 2 √  6 y is the geometric mean of 3 and 8. 11. z 2 = (8) (11) z 2 = 88 z = √  88 = 2 √  22 z is the geometric mean of 8 and 11. 572 572 Chapter 8 Right Triangles and Trigonometry �� 8-2 Trigonometric Ratios (pp. 525–532) TEKS G.5.B, G.5.D, G.8.C, G.11.B, G.11.C E X A M P L E S EXERCISES Find each length. Round to the nearest hundredth. Find each length. Round to the nearest hundredth. 12. UV ■ EF sin 75° = EF _ 8.1 EF = 8.1 (sin 75°) EF ≈ 7.82 cm ■ AB tan 34° = 4.2 _ AB AB tan 34° = 4.2 AB = 4.2 _ tan 34° AB ≈ 6.23 in. Since the opp. leg and hyp. are involved, use a sine ratio. 13. PR 14. XY 15. JL Since the opp. and adj. legs are involved, use a tangent ratio. 8-3 Solving Right Triangles (pp. 534–541) TEKS G.5.D, G.7.A, G.7.C, G.8.C, G.11.B, G.11.C E X A M P L E EXERCISES ■ Find the unknown measures in △LMN. Round lengths to the nearest hundredth and angle measures to the nearest degree. Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 16. The acute angles of a right triangle are 17. complementary. So m∠N = 90° - 61° = 29°. sin L = MN _ Write a trig. ratio. LN sin 61° = 8.5 _ LN LN = 8.5 _ sin 61° tan L = MN _ LM tan 61° = 8.5 _ LM LM = 8.5 _ tan 61° ≈ 9.72 ≈ 4.71 Substitute the given 18. 19. values. Solve for LN. Write a trig. ratio. Substitute the given values. Solve for LM. Study Guide: Review 573 573 �� 8-4 Angles of Elevation and Depression (pp. 544–549) TEKS G.5.D, G.11.C E X A M P L E S EXERCISES Classify each angle as an angle of elevation or angle of depression. ■ A pilot in a plane spots a forest fire on the ground at an angle of depression of 71°. The plane’s altitude is 3000 ft. What is the horizontal distance from the plane to the fire? Round to the nearest foot. tan 71° = 3000_ XF XF = 3000_ tan 71° XF ≈ 1033 ft ■ A diver is swimming at a depth of 63 ft below sea level. He sees a buoy floating at sea level at an angle of elevation of 47°. How far must the diver swim so that he is directly beneath the buoy? Round to the nearest foot. tan 47° = 63 _ XD XD = 63 _ tan 47° XD ≈ 59 ft 20. ∠1 21. ∠2 22. When the angle of elevation to the sun is 82°, a monument casts a shadow that is 5.1 ft long. What is the height of the monument to the nearest foot? 23. A ranger in a lookout tower spots a fire in the distance. The angle of depression to the fire is 4°, and the lookout tower is 32 m tall. What is the horizontal distance to the fire? Round to the nearest meter. 8-5 Law of Sines and Law of Cosines (pp. 551–558) TEKS G.5.B, G.5.D, G.7.A, G.11.A, G.11.C E X A M P L E S EXERCISES Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. ■ m∠B Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 24. m∠Z sin B _ AC sin B _ 6 = sin C _ AB = sin 88° _ 8 Law of Sines Substitute the given values. 25. MN sin B = 6 sin 88° _ m∠B = sin -1 ( 6 sin 88° _ 8 8 ) ≈ 49° Multiply both sides by 6. 574 574 Chapter 8 Right Triangles and Trigonometry �� Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. ■ HJ Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 26. EF Use the Law of Cosines. 27. m∠Q HJ 2 = GH 2 + G J 2 - 2 (GH) (GJ) cos G =10 ) (11 ) cos 32° H J 2 ≈ 34.4294 HJ ≈ 5.9 Simplify. Find the square root. 8-6 Vectors (pp. 559–567) TEKS G.1.B, G.7.A, G.7.C, G.11.C E X A M P L E S EXERCISES ■ Draw the vector 〈-1, 4〉 on a coordinate plane. Find its magnitude to the nearest tenth. ⎜〈-1, 4〉⎟ = √ (-1)2 + (4)2 Write each vector in component form. 28.  AB with A (5, 1) and B (-2, 3) 29.  MN with M (-2, 4) and N (-1, -2) = √17 ≈ 4.1 30.  RS ■ The velocity of a jet is given by the vector 〈4, 3〉. Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree. In △PQR, tan P = 3 _ 4 , so m∠P = tan -1 ( 3 _ ) ≈ 37°. 4 ■ Susan swims across a river at a bearing of N 75° E at a speed of 0.5 mi/h. The river’s current moves due east at 1 mi/h. Find Susan’s actual speed to the nearest tenth and her direction to the nearest degree. cos 15° = x _ 0.5 y _ 0.5 sin 15° = , so x ≈ 0.48. , so y ≈ 0.13. Susan’s vector is 〈0.48, 0.13〉. The current is 〈1, 0〉. Susan’s actual speed is the magnitude of the resultant vector, 〈1.48, 0.13〉. ⎜〈1.48, 0.13〉⎟ = Her direction is tan -1 ( 0.13 _ √  (1.48) 2 + (0.13) 2 ≈ 1.5 mi/h ) ≈ 5°, or N 85° E. 1.48 Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 31. 〈-5, -3〉 32. 〈-2, 0〉 33. 〈4, -4〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 34. The velocity of a helicopter is given by the vector 〈4, 5〉. 35. The force applied by a tugboat is given by the vector 〈7, 2〉. 36. A plane flies at a constant speed of 600 mi/h at a bearing of N 55° E. There is a 50 mi/h crosswind blowing due east. What are the plane’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Study Guide: Review 575 575 �� Find x, y, and z. 1. 2. 3. Use a special right triangle to write each trigonometric ratio as a fraction. 4. cos 60° 5. sin 45° 6. tan 60° Find each length. Round to the nearest hundredth. 7. PR 8. AB 9. FG 10. Nate built a skateboard ramp that covers a horizontal distance of 10 ft. The ramp rises a total of 3.5 ft. What angle does the ramp make with the ground? Round to the nearest degree. 11. An observer at the top of a skyscraper sights a tour bus at an angle of depression of 61°. The skyscraper is 910 ft tall. What is the horizontal distance from the base of the skyscraper to the tour bus? Round to the nearest foot. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 12. m∠B 13. RS 14. m∠M Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 15. 〈1, 3〉 16. 〈-4, 1〉 17. 〈2, -3〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 18. The velocity of a plane is given by the vector 〈3, 5〉. 19. A wind velocity is given by the vector 〈4, 1〉. 20. Kate is rowing across a river. She sets out at a bearing of N 40° E and paddles at a constant rate of 3.5 mi/h. There is a 2 mi/h current moving due east. What are Kate’s actual speed and direction? Round the speed to the nearest tenth an
d the direction to the nearest degree. 576 576 Chapter 8 Right Triangles and Trigonometry �� FOCUS ON SAT MATHEMATICS SUBJECT TESTS Though you can use a calculator on the SAT Mathematics Subject Tests, it may be faster to answer some questions without one. Remember to use test-taking strategies before you press buttons! 4. A swimmer jumps into a river and starts swimming directly across it at a constant velocity of 2 meters per second. The speed of the current is 7 meters per second. Given the current, what is the actual speed of the swimmer to the nearest tenth? (A) 0.3 meters per second (B) 1.7 meters per second (C) 5.0 meters per second (D) 7.3 meters per second (E) 9.0 meters per second 5. What is the approximate measure of the vertex angle of the isosceles triangle below? (A) 28.1° (B) 56.1° (C) 62.0° (D) 112.2° (E) 123.9° The SAT Mathematics Subject Tests each consist of 50 multiple-choice questions. You are not expected to have studied every topic on the SAT Mathematics Subject Tests, so some questions may be unfamiliar. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. 1. Let P be the acute angle formed by the line -x + 4y = 12 and the x-axis. What is the approximate measure of ∠P? (A) 14° (B) 18° (C) 72° (D) 76° (E) 85° 2. In right triangle DEF, DE = 15, EF = 36, and DF = 39. What is the cosine of ∠F? (A) 5 _ 12 (B) 12 _ 5 (C) 5 _ 13 (D) 12 _ 13 (E) 13 _ 12 3. A triangle has angle measures of 19°, 61°, and 100°. What is the approximate length of the side opposite the 100° angle if the side opposite the 61° angle is 8 centimeters long? (A) 2.5 centimeters (B) 3 centimeters (C) 9 centimeters (D) 12 centimeters (E) 13 centimeters College Entrance Exam Practice 577 577 �� Any Question Type: Estimate Once you find the answer to a test problem, take a few moments to check your answer by using estimation strategies. By doing so, you can verify that your final answer is reasonable. Gridded Response Find the geometric mean of 38 and 12 to the nearest hundredth. Let x be the geometric mean. x 2 = (38) (12) = 456 Def. of geometric mean x ≈ 21.35 Find the positive square root. Now use estimation to check that this answer is reasonable. x 2 ≈ (40) (10) = 400 Round 38 to 40 and round 12 to 10. x ≈ 20 Find the positive square root. The estimate is close to the calculated answer, so 21.35 is a reasonable answer. Multiple Choice Which of the following is equal to sin X ? 0.02 0.41 0.91 2.44 Use a trigonometric ratio to find the answer. sin X = YZ _ XZ sin X = 9 _ 22 ≈ 0.41 The sine of an ∠ is opp. leg _ hyp. . Substitute the given values and simplify. Now use estimation to check that this answer is reasonable. sin X ≈ 10 _ 20 ≈ 0.5 Round 9 to 10 and round 22 to 20. The estimate is close to the calculated answer, so B is a reasonable answer. 578 578 Chapter 8 Right Triangles and Trigonometry �� An extra minute spent checking your answers can result in a better test score. Item C Multiple Choice In △QRS, what is the ̶̶ SQ to the nearest tenth of a measure of centimeter? Read each test item and answer the questions that follow. Item A Gridded Response A cell phone tower casts a shadow that is 121 ft long when the angle of elevation to the sun is 48°. How tall is the cell phone tower? Round to the nearest foot. 1. A student estimated that the answer should be slightly greater than 121 by comparing tan 48° and tan 45°. Explain why this estimation strategy works. 2. Describe how to use the inverse tangent function to estimate whether an answer of 134 ft makes sense. Item B   BC has an initial point of Multiple Choice (-1, 0) and a terminal point of (4, 2) . What are the magnitude and direction of   BC ? 5.39; 22° 5.39; 68° 6.39; 22° 6.39; 68° 3. A student correctly found the magnitude of  BC as √  29 . The student then calculated the value of this radical as 6.39. Explain how to use perfect squares to estimate the value of √  29 . Is 6.39 a reasonable answer? 4. A student calculated the measure of the angle the vector forms with a horizontal line as 68°. Use estimation to explain why this answer is not reasonable. 9.3 centimeters 10.5 centimeters 30.1 centimeters 61.7 centimeters 5. A student calculated the answer as 30.1 cm. The student then used the diagram to estimate that SQ is more than half of RQ. So the student decided that his answer was reasonable. Is this estimation method a good way to check your answer? Why or why not? 6. Describe how to use estimation and the Pythagorean Theorem to check your answer to this problem. Item D Multiple Choice The McCleods have a variable interest rate on their mortgage. The rate is 2.625% the first year and 4% the following year. The average interest rate is the geometric mean of these two rates. To the nearest hundredth of a percent, what is the average interest rate for their mortgage? 1.38% 3.24% 3.89% 10.50% 7. Describe how to use estimation to show that choices F and J are unreasonable. 8. To find the answer, a student uses the equation x 2 = (2.625) (4) . Which compatible numbers should the student use to quickly check the answer? TAKS Tackler 579 579 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–8 Multiple Choice 1. What is the length of ̶̶ UX to the nearest centimeter? 6. △ABC has vertices A (-2, -2) , B (-3, 2) , and C (1, 3) . Which translation produces an image with vertices at the coordinates (-2, -2) , (2, -1) , and (-1, -6) ? (x, y) → (x + 1, y - 4) (x, y) → (x + 2, y - 8) (x, y) → (x - 3, y - 5) (x, y) → (x - 4, y + 1) 7. △ABC is a right triangle in which m∠A = 30° and m∠B = 60°. Which of the following are possible lengths for the sides of this triangle? AB = √  3 , AC = √  2 , and BC = 1 AB = 4, AC = 2, and BC = 2 √  3 AB = 6 √  3 , AC = 27, and BC = 3 √  3 AB = 8, AC = 4 √  3 , and BC = 4 8. Based on the figure below, which of the following similarity statements must be true? △PQR ∼ △TSR △PQR ∼ △RTQ △PQR ∼ △TSQ △PQR ∼ △TQP 9. ABCD is a rhombus with vertices A (1, 1) and C (3, 4) . Which of the following lines is parallel to diagonal ̶̶ BD ? 2x - 3y = 12 2x + 3y = 12 3x + 2y = 12 3x - 4y = 12 3 centimeters 7 centimeters 9 centimeters 13 centimeters 2. △ABC is a right triangle. m∠A = 20°, m∠B = 90°, AC = 8, and AB = 3. Which expression can be used to find BC? 3 _ tan 70° 8 _ sin 20° 8 tan 20° 3 cos 70° 3. A slide at a park is 25 ft long, and the top of the slide is 10 ft above the ground. What is the approximate measure of the angle the slide makes with the ground? 21.8° 23.6° 66.4° 68.2° 4. Which of the following vectors is equal to the vector with an initial point at (2, -1) and a terminal point at (-2, 4) ? 〈-4, -5〉〈-4, 5〉〈5, -4〉〈5, 4〉 5. Which statement is true by the Addition Property of Equality? If 3x + 6 = 9y, then x + 2 = 3y. If t = 1 and s = t + 5, then s = 6. If k + 1 = ℓ + 2, then 2k + 2 = 2ℓ + 4. If a + 2 = 3b, then a + 5 = 3b + 3. 580 580 Chapter 8 Right Triangles and Trigonometry �� 10. Which of the following is NOT equivalent to sin 60°? cos 30° √  3 _ 2 (cos 60°) (tan 60°) tan 30° _ sin 30° 11. ABCDE is a convex pentagon. ∠A ≅ ∠B ≅ ∠C, ∠D ≅ ∠E, and m∠A = 2m∠D. What is the measure of ∠C? 67.5° 135° 154.2° 225° 12. Which of the following sets of lengths can represent the side lengths of an obtuse triangle? 4, 7.5, and 8.5 7, 12, and 13 9.5, 16.5, and 35 36, 75, and 88 �� Be sure to correctly identify any pairs of parallel lines before using the Alternate Interior Angles Theorem or the Same-Side Interior Angles Theorem. 13. What is the value of x? 22.5 45 90 135 Gridded Response 14. Find the next item in the pattern below. 1, 3, 7, 13, 21, … 15. In △XYZ, ∠X and ∠Z are remote interior angles of exterior ∠XYT. If m∠X = (x + 15) °, m∠Z = (50 - 3x) °, and m∠XYT = (4x - 25) °, what is the value of x? 16. In △ABC and △DEF, ∠A ≅ ∠F. If EF = 4.5, DF = 3, ̶̶ AB would let you and AC = 1.5, what length for conclude that △ABC ∼ △FED? STANDARDIZED TEST PREP Short Response 17. A building casts a shadow that is 85 ft long when the angle of elevation to the sun is 34°. a. What is the height of the building? Round to the nearest inch and show your work. b. What is the angle of elevation to the sun when the shadow is 42 ft 6 in. long? Round to the nearest tenth of a degree and show your work. 18. Use the figure to find each of the following. Round to the nearest tenth of a centimeter and show your work. a. the length of b. the length of ̶̶ DC ̶̶ AB Extended Response 19. Tony and Paul are taking a vacation with their cousin, Greg. Tony and Paul live in the same house. Paul will go directly to the vacation spot, but Tony has to pick up Greg. Tony travels 90 miles at a bearing of N 25° E to get to his cousin’s house. He then travels due east for 50 miles to get to the vacation spot. Paul travels on one highway to get from his house to the vacation spot. For each of the following, explain in words how you found your answer and round to the nearest tenth. a. Write the vectors in component form for the route from Tony and Paul’s house to their cousin’s house and the route from their cousin’s house to the vacation spot. b. What are the direction and magnitude of Paul’s direct route from his house to the vacation spot? c. Tony and Paul leave the house at the same time and arrive at the vacation spot at the same time. If Tony traveled at an average speed of 50 mi/h, what was Paul’s average speed? Cumulative Assessment, Chapters 1–8 581 581 �� T E X A S TAKS Grades 9–11 Obj. 10 �� Reunion Tower The 55-story Reunion Tower is one of the most recognized buildings in the Dallas skyline. Built as a part of the Hyatt Regency Hotel, the tower is topped by a geodesic dome that houses a revolving restaurant. �� The tower itself consists o
f four concrete cylinders arranged in a �� triangular pattern. The center cylinder contains an elevator, which �� takes visitors on a 68-second ride to the top of the tower. �� Choose one or more strategies to solve each problem. 1. The building’s observation deck, the Lookout, is on the fifty-third floor, approximately 540 feet above street level. The deck is equipped with telescopes that offer close-up views of the Dallas area. Using one of the telescopes, a visitor spots a sculpture in a nearby park. The angle of depression to the sculpture is 10°. To the nearest foot, how far is the sculpture from the base of Reunion Tower? For 2–4, use the table. 2. At noon on May 15, the shadow of Reunion Tower is 150 ft long. Find the height of the tower to the nearest foot. 3. How long is the shadow of the tower at noon on October 15? Round to the nearest foot. 4. On which of the dates shown is the tower’s shadow the longest? What is the length of the shadow to the nearest foot? 582 582 Chapter 8 Right Triangles and Trigonometry Elevation of the Sun in Dallas, Texas Date January 15 February 15 March 15 April 15 May 15 June 15 July 15 August 15 September 15 October 15 November 15 December 15 Angle of Elevation at Noon (°) 36 44 54 66 75 79 77 70 60 48 39 34 Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Lyndon B. Johnson’s Birthplace Lyndon Baines Johnson (1908–1973), the thirtysixth president of the United States, was born in Stonewall, Texas, and moved to what is now Johnson City at the age of 5. Visitors to the Lyndon B. Johnson National Historical Park can tour the home where President Johnson spent much of his childhood. Choose one or more strategies to solve each problem. 1. In 1973, the National Park Service finished restoring President Johnson’s house to its appearance during Johnson’s childhood. The blueprint below shows the layout of the house after the restoration. Suppose there is a border of wallpaper along the edge of the ceiling around the dining room. About how long is the border? 2. During the restoration of the house, the wooden floor of the parlor, dining room, and kitchen was replaced with new wood. About how many square feet of wood was used? 3. Estimate the percentage of the square footage of the home that is occupied by porches. 4. In 1934, the Johnson family spent $666.09 for improvements to their home. The total cost consisted of materials and labor. The materials cost $416.09 more than the labor. How much did they spend on labor for the improvements? �� Problem Solving on Location 583 583 Extending Perimeter, Circumference, and Area 9A Developing Geometric Formulas 9-1 Developing Formulas for Triangles and Quadrilaterals Lab Develop π 9-2 Developing Formulas for Circles and Regular Polygons 9-3 Composite Figures Lab Develop Pick’s Theorem for Area of Lattice Polygons 9B Applying Geometric Formulas 9-4 Perimeter and Area in the Coordinate Plane 9-5 Effects of Changing Dimensions Proportionally 9-6 Geometric Probability Lab Use Geometric Probability to Estimate π KEYWORD: MG7 ChProj This map of Texas counties shows the elevations of different regions. 584 584 Chapter 9 Vocabulary Match each term on the left with a definition on the right. 1. area A. a polygon that is both equilateral and equiangular 2. kite 3. perimeter 4. regular polygon B. a quadrilateral with exactly one pair of parallel sides C. the number of nonoverlapping unit squares of a given size that exactly cover the interior of a figure D. a quadrilateral with exactly two pairs of adjacent congruent sides E. the distance around a closed plane figure Convert Units Use multiplication or division to change from one unit of measure to another. 5. 12 mi = yd Length 6. 7.3 km = m 7. 6 in. = ft 8. 15 m = mm Metric 1 kilometer = 1000 meters 1 meter = 100 centimeters 1 centimeter = 10 millimeters Customary 1 mile = 1760 yards 1 mile = 5280 feet 1 yard = 3 feet 1 foot = 12 inches Pythagorean Theorem Find x in each right triangle. Round to the nearest tenth, if necessary. 9. 11. 10. Measure with Customary and Metric Units Measure each segment to the nearest eighth of an inch and to the nearest half of a centimeter. 12. 14. 13. Solve for a Variable Solve each equation for the indicated variable. 15. A = 1_ 2 17. A = 1_ 2(b 1 + b 2)h for b 1 bh for b 16. P = 2b + 2h for h 18. A = 1_ 2 d 1d 2 for d 1 Extending Perimeter, Circumference, and Area 585 585 �� Key Vocabulary/Vocabulario apothem apotema center of a circle centro de un circulo center of a regular polygon centro de un poligono regular central angle of a regular polygon ángulo central de un poligono circle circulo composite figure figuras compuestas geometric probability probabilidad geométrica Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. How can you use the everyday meaning of the word center to understand the term center of a circle ? 2. The word composite means “of separate parts.” What do you think the term composite figure means? 3. What does the word probability mean? How do you think geometric probability differs from theoretical probability? 4. The word apothem begins with the root apo-, which means “away from.” The apothem of a regular polygon is measured “away from” the center to the midpoint of a side. What do you think is true about the apothem and the side of the polygon? Geometry TEKS G.3.C Geometric structure* use logical reasoning to prove statements are true ... G.3.E Geometric structure* use deductive reasoning to prove a statement Les. 9-1 ★ ★ 9-2 Geo. Lab Les. 9-2 Les. 9-3 9-3 Geo. Lab Les. 9-4 Les. 9-5 Les. 9-6 9-6 Geo. Lab G.5.A Geometric patterns* use ... geometric patterns to develop ★ ★ ★ algebraic expressions representing geometric properties G.5.B Geometric patterns* use ... geometric patterns to make generalizations about geometric properties, including ... ratios in similar figures ... G.7.A Dimensionality and the geometry of location* use ... two- dimensional coordinate systems to represent ... figures G.7.B Dimensionality and the geometry of location* use slopes ... to investigate geometric relationships ... ★ ★ ★ ★ G.8.A Congruence and the geometry of size* find areas of ... circles, ★ ★ ★ ★ ★ ★ and composite figures G.8.C Congruence and the geometry of size* ... use the ★ ★ Pythagorean Theorem G.11.D Similarity and the geometry of shape* describe the effect on perimeter, area, ... when one or more dimensions of a figure are changed ... ★ * Knowledge and skills are written out completely on pages TX28–TX35. 586 586 Chapter 9 Study Strategy: Memorize Formulas Throughout a geometry course, you will learn many formulas, theorems, postulates, and corollaries. You may be required to memorize some of these. In order not to become overwhelmed by the amount of information, it helps to use flash cards. In a right triangle, the two sides that form the right angle are the legs . The side across from the right angle that stretches from one leg to the other is the hypotenuse . In the diagram, a and b are the lengths of the shorter sides, or legs, of the right triangle. The longest side is called the hypotenuse and has length c. Theorem 1-6-1 Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a 2 + b 2 = c 2 To create a flash card, write the name of the formula or theorem on the front of the card. Then clearly write the appropriate information on the back of the card. Be sure to include a labeled diagram. Front Back �� Try This 1. Choose a lesson from this book that you have already studied, and make flash cards of the formulas or theorems from the lesson. 2. Review your flash cards by looking at the front of each card and trying to recall the information on the back of the card. Extending Perimeter, Circumference, and Area 587 587 �� Literal Equations Algebra A literal equation contains two or more variables. Formulas you have used to find perimeter, circumference, area, and side relationships of right triangles are examples of literal equations. See Skills Bank page S59 If you want to evaluate a formula for several different values of a given variable, it is helpful to solve for the variable first. Example Danielle plans to use 50 feet of fencing to build a dog run. Use the formula P = 2ℓ + 2w to find the length ℓ when the width w is 4, 5, 6, and 10 feet. Solve the equation for ℓ. First solve the formula for the variable. P = 2ℓ + 2w Write the original equation. P - 2w = 2ℓ P - 2w _ 2 = ℓ Subtract 2w from both sides. Divide both sides by 2. Use your result to find ℓ for each value of w. ℓ = P - 2w _ = 2 ℓ = P - 2w _ = 2 50 - 2 (4) _ 2 50 - 2 (5) _ 2 = 21 ft = 20 ft ℓ = P - 2w _ = 2 50 - 2 (6) _ 2 = 19 ft ℓ = P - 2w _ = 2 50 - 2 (10) _ 2 = 15 ft Substitute 50 for P and 4 for w. Substitute 50 for P and 5 for w. Substitute 50 for P and 6 for w. Substitute 50 for P and 10 for w. Try This TAKS Grades 9–11 Obj. 1 1. A rectangle has a perimeter of 24 cm. Use the formula P = 2ℓ + 2w to find the width when the length is 2, 3, 4, 6, and 8 cm. 2. A right triangle has a hypotenuse of length c = 65 ft. Use the Pythagorean Theorem to find the length of leg a when the length of leg b is 16, 25, 33, and 39 feet. 3. The perimeter of △ABC is 112 in. Write an expression for a in terms of b and c, and use it to complete the following table. a b 48 36 14 c 35 36 50 588 588 Chapter 9 Extending Perimeter, Circum
ference, and Area �� 9-1 Developing Formulas for Triangles and Quadrilaterals TEKS G.5.A Geometric patterns: use ... geometric patterns to develop algebraic expressions representing geometric properties. Also G.1.B, G.3.C, G.3.E, G.8.C Objectives Develop and apply the formulas for the areas of triangles and special quadrilaterals. Solve problems involving perimeters and areas of triangles and special quadrilaterals. Why learn this? You can use formulas for area to help solve puzzles such as the tangram. A tangram is an ancient Chinese puzzle made from a square. The pieces can be rearranged to form many different shapes. The area of a figure made with all the pieces is the sum of the areas of the pieces. Postulate 9-1-1 Area Addition Postulate The area of a region is equal to the sum of the areas of its nonoverlapping parts. Recall that a rectangle with base b and height h has an area of A = bh. You can use the Area Addition Postulate to see that a parallelogram has the same area as a rectangle with the same base and height. � � � � A triangle is cut off one side and translated to the other side. Area Parallelogram The area of a parallelogram with base b and height h is A = bh. Remember that rectangles and squares are also parallelograms. The area of a square with side s is A = s 2 , and the perimeter is P = 4s. E X A M P L E 1 Finding Measurements of Parallelograms The height of a parallelogram is measured along a segment perpendicular to a line containing the base. Find each measurement. A the area of the parallelogram Step 1 Use the Pythagorean Theorem to find the height h Step 2 Use h to find the area of the parallelogram. A = bh A = 6 (4) A = 24 in 2 Substitute 6 for b and 4 for h. Area of a parallelogram Simplify. 9- 1 Developing Formulas for Triangles and Quadrilaterals 589 589 �� Find each measurement. B the height of a rectangle in which b = 5 cm and A = (5 x 2 - 5x) cm 2 A = bh 5 x 2 - 5x = 5h 5 ( x 2 - x) = 5h x 2 - x = h Area of a rectangle Substitute 5 x 2 - 5x for A and 5 for b. Factor 5 out of the expression for A. Divide both sides by 5. h = ( x 2 - x) cm Sym. Prop. of = C the perimeter of the rectangle, in which A = 12 x ft 2 Step 1 Use the area and the height to find the base. A = bh 12x = b (6) 2x = b Area of a rectangle Substitute 12x for A and 6 for h. Divide both sides by 6. The perimeter of a rectangle with base b and height h is P = 2b + 2h, or P = 2 (b + h) . Step 2 Use the base and the height to find the perimeter. P = 2b + 2h P = 2 (2x) + 2 (6) P = (4x + 12) ft. Substitute 2x for b and 6 for h. Perimeter of a rectangle Simplify. 1. Find the base of a parallelogram in which h = 56 yd and A = 28 yd 2 . To understand the formula for the area of a triangle or trapezoid, notice that two congruent triangles or two congruent trapezoids fit together to form a parallelogram. Thus the area of a triangle or trapezoid is half the area of the related parallelogram. � � �� Area Triangles and Trapezoids The area of a triangle with base b and height h is A = 1 __ 2 bh. The area of a trapezoid with bases b 1 and b 2 and height h is A = 1 __ 2 ( b 1 + b 2 ) h, or __ Finding Measurements of Triangles and Trapezoids Find each measurement. A the area of a trapezoid in which b 1 = 9 cm, b 2 = 12 cm, and h = 3 cm 9 + 12) 3 2 A = 31.5 cm 2 Area of a trapezoid Substitute 9 for b 1 , 12 for b 2 , and 3 for h. Simplify. 590 590 Chapter 9 Extending Perimeter, Circumference, and Area �� Find each measurement. B the base of the triangle, in which A = x 2 in 2 bx bh 2x = b b = 2x in. Area of a triangle Substitute x 2 for A and x for h. Divide both sides by x. Multiply both sides by 2. Sym. Prop. of = C Area of a trapezoid b 2 of the trapezoid, in which A = 8 ft 3 + b 2 ) (2 ft Sym. Prop. of = Substitute 8 for A, 3 for b 1 , and 2 for h. Multiply 1 __ 2 Subtract 3 from both sides. by 2. 2. Find the area of the triangle. A kite or a rhombus with diagonals d 1 and d 2 can be divided into two congruent triangles with a base of d 1 and a height of 1 __ 2 d 2 . area of each triangle total area �� Area Rhombuses and Kites The area of a rhombus or kite with diagonals d 1 and d 2 is A = 1 __ Finding Measurements of Rhombuses and Kites Find each measurement. A Area of a kite d 2 of a kite in which d 1 = 16 cm and A = 48 cm 48 = 1 _ (16 cm Solve for d 2 . Sym. Prop. of = Substitute 48 for A and 16 for d 1 . 9- 1 Developing Formulas for Triangles and Quadrilaterals 591 591 �� Find each measurement. B the area of the rhombus 6x + 4) (10x + 10) 2 A = 1 _ (60 x 2 + 100x + 40) 2 Substitute (6x + 4) for d 1 and (10x + 10) for d 2 . Multiply the binomials (FOIL). The diagonals of a rhombus or kite are perpendicular, and the diagonals of a rhombus bisect each other. A = (30 x 2 + 50x + 20) in 2 Distrib. Prop. C the area of the kite Step 1 The diagonals d 1 and d 2 form four right triangles. Use the Pythagorean Theorem to find x and y. 9 2 + x 2 = 41 2 x 2 = 1600 x = 40 9 2 + y 2 = 15 2 y 2 = 144 y = 12 Step 2 Use d 1 and d 2 to find the area. d 1 is equal to x + y, which is 52. Half of d 2 is equal to 9, so d 2 is equal to 1852) (18) 2 A = 468 ft 2 Substitute 52 for d 1 and 18 for d 2 . Area of a kite Simplify. 3. Find d 2 of a rhombus in which d 1 = 3x m and A = 12xy Games Application The pieces of a tangram are arranged in a square in which s = 4 cm. Use the grid to find the perimeter and area of the red square. Perimeter: Each side of the red square is the diagonal of a square of the grid. Each grid square has a side length of 1 cm, so the diagonal is √  2 cm. The perimeter of the red square is P = 4s = 4 √  2 cm. Area: Method 1 The red square is also a rhombus. The diagonals d 1 and d 2 each measure 2 cm. So its area is 2) (2) = 2 cm 2 . 2 2 Method 2 The side length of the red square is √  2 cm, so the area is cm. 2 4. In the tangram above, find the perimeter and area of the large green triangle. 592 592 Chapter 9 Extending Perimeter, Circumference, and Area �� THINK AND DISCUSS 1. Explain why the area of a triangle is half the area of a parallelogram with the same base and height. 2. Compare the formula for the area of a trapezoid with the formula for the area of a rectangle. 3. GET ORGANIZED Copy and complete the graphic organizer. Name all the shapes whose area is given by each area formula and sketch an example of each shape. 9-1 Exercises Exercises GUIDED PRACTICE Find each measurement. KEYWORD: MG7 9-1 KEYWORD: MG7 Parent . 589 1. the area of the parallelogram 2. the height of the rectangle, in which A = 10 x 2 ft 2 3. the perimeter of a square in which A = 169 cm . the area of the trapezoid 5. the base of the triangle, in which p. 590 A = 58.5 in 2 6. b 1 of a trapezoid in which A = (48x + 68) in 2 , h = 8 in., and b 2 = (9x + 12) in. the area of the rhombus p. 591 8. d 2 of the kite, in which A = 187.5 m 2 9. d 2 of a kite in which A = 12 x 2 y 3 cm 2 , d 1 = 3xy cm 10. Art The stained-glass window shown is a rectangle p. 592 with a base of 4 ft and a height of 3 ft. Use the grid to find the area of each piece. 9- 1 Developing Formulas for Triangles and Quadrilaterals 593 593 �� Independent Practice For See Exercises Example 11–13 14–16 17–19 20–22 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S20 Application Practice p. S36 PRACTICE AND PROBLEM SOLVING Find each measurement. 11. the height of the parallelogram, 12. the perimeter of the rectangle in which A = 7.5 m 2 13. the area of a parallelogram in which b = (3x + 5) ft and h = (7x - 1) ft 13. 14. the area of the triangle 15. the height of the trapezoid, in which A = 280 cm 2 16. the area of a triangle in which b = (x + 1) ft and h = 8x ft 17. the area of the kite 18. d 2 of the rhombus, in which A = (3 x 2 + 6x) m 2 19. the area of a kite in which d 1 = (6x + 5) ft and d 2 = (4x + 8) ft Crafts In origami, a square base is the starting point for the creation of many figures, such as a crane. In the pattern for the square base, ABCD is a square, and E, F, G, and H are the midpoints of the sides. If AB = 6 in., find the area of each shape. 20. rectangle ABFH 21. △AEJ 22. trapezoid ABFJ � � � � � � � � � Multi-Step Find the area of each figure. Round to the nearest tenth, if necessary. 23. 24. 25. Write each area in terms of x. 26. equilateral triangle 27. 30°-60°-90° triangle 28. 45°-45°-90° triangle 594 594 Chapter 9 Extending Perimeter, Circumference, and Area �� 29. This problem will prepare you for the Multi-Step TAKS Prep on page 614. A sign manufacturer makes yield signs by cutting an equilateral triangle from a square piece of aluminum with the dimensions shown. a. Find the height of the yield sign to the nearest tenth. b. Find the area of the sign to the nearest tenth. c. How much material is left after a sign is made? Find the missing measurements for each rectangle. Base b Height h Area A Perimeter P 12 17 30. 31. 32. 33. 16 11 136 216 50 66 34. The perimeter of a rectangle is 72 in. The base is 3 times the height. Find the area of the rectangle. 35. The area of a triangle is 50 cm 2 . The base of the triangle is 4 times the height. Find the height of the triangle. 36. The perimeter of an isosceles trapezoid is 40 ft. The bases of the trapezoid are 11 ft and 19 ft. Find the area of the trapezoid. Use the conversion table for Exercises 37–42. 37. 1 yd 2 = 38. 1 m 2 = 39. 1 cm 2 = 40. 1 mi 2 = ? ft 2 ̶̶̶̶ ? mm 2 ̶̶̶̶ ? cm 2 ̶̶̶̶ ? in 2 ̶̶̶̶ History 41. A triangle has a base of 3 yd and a height of 8 yd. Find the area in square feet. 42. A r
hombus has diagonals 500 yd and 800 yd in length. Find the area in square miles. Conversion Factors Metric 1 km = 1000 m 1 m = 100 cm 1 cm = 10 mm Customary 1 mi = 1760 yd 1 mi = 5280 ft 1 yd = 3 ft 1 ft = 12 in. President James Garfield was a classics professor and a major general in the Union Army. He was assassinated in 1881. Source: www.whitehouse.gov 43. The following proof of the Pythagorean Theorem was discovered by President James Garfield in 1876 while he was a member of the House of Representatives. a. Write the area of the trapezoid in terms of a and b. b. Write the areas of the three triangles in terms of a, b, and c. c. Use the Area Addition Postulate to write an equation relating your results from parts a and b. Simplify the equation to prove the Pythagorean Theorem. 44. Use the diagram to prove the formula for the area of a rectangle, given the formula for the area of a square. Given: Rectangle with base b and height h Prove: The area of the rectangle is A = bh. Plan: Use the formula for the area of a square to find the areas of the outer square and the two squares inside the figure. Write and solve an equation for the area of the rectangle. 9- 1 Developing Formulas for Triangles and Quadrilaterals 595 595 �� Prove each area formula. 45. Given: Parallelogram with area A = bh Prove: The area of the triangle is A = 1 __ 2 bh. 46. Given: Triangle with area A = 1__ 2bh Prove: The area of the trapezoid is . 2 47. Measurement Choose an appropriate unit of measurement and measure the base and height of each parallelogram. a. Find the area of each parallelogram. Give your answer with the correct precision. b. Which has the greatest area? 48. Hobbies Tina is making a kite according to the plans at right. The fabric weighs about 40 grams per square meter. The diagonal braces, or spars, weigh about 20 grams per meter. Estimate the weight of the kite. 49. Home Improvement Tom is buying tile for a 12 ft by 18 ft rectangular kitchen floor. He needs to buy 15% extra in case some of the tiles break. The tiles are squares with 4 in. sides that come in cases of 100. How many cases should he buy? �� 50. Critical Thinking If the maximum error in the given measurements of the rectangle is 0.1 cm, what is the greatest possible error in the area? Explain. �� 51. Write About It A square is also a parallelogram, a rectangle, and a rhombus. Prove that the area formula for each shape gives the same result as the formula for the area of a square. 52. Which expression best represents the area of the rectangle? 2x + 2 (x - c) x (x - c) x 2 + (x - c) 2 2x (x - c) 53. The length of a rectangle is 3 times the width. The perimeter is 48 inches. Which system of equations can be used to find the dimensions of the rectangle? ℓ = w + 3 2 (ℓ + w) = 48 ℓ = 3w 2ℓ + 6w = 48 ℓ = 3w 2 (ℓ + w) = 48 ℓ = w + 3 2ℓ + 6w = 48 596 596 Chapter 9 Extending Perimeter, Circumference, and Area �� 54. A 16- by 18-foot rectangular section of a wall will be covered by square tiles that measure 2 feet on each side. If the tiles are not cut, how many of them will be needed to cover the section of the wall? 288 144 72 17 55. The area of trapezoid HJKM is 90 square centimeters. Which is closest to the length of 10 centimeters ̶̶ JK ? 11.7 centimeters 10.5 centimeters 16 centimeters 56. Gridded Response A driveway is shaped like a parallelogram with a base of 28 feet and a height of 17 feet. Covering the driveway with crushed stone will cost $2.75 per square foot. How much will it cost to cover the driveway with crushed stone? �� CHALLENGE AND EXTEND Multi-Step Find h in each parallelogram. 57. �� 58. �� 59. Algebra A rectangle has a perimeter of (26x + 16) cm and an area of (42 x 2 + 51x + 15) cm 2 . Find the dimensions of the rectangle in terms of x. 60. Prove that the area of any quadrilateral with perpendicular diagonals is 1 __ 2 d 1 d 2 . 61. Gardening A gardener has 24 feet of fencing to enclose a rectangular garden. a. Let x and y represent the side lengths of the rectangle. Solve the perimeter formula 2x + 2y = 24 for y, and substitute the expression into the area formula A = xy. b. Graph the resulting function on a coordinate plane. What are the domain and range of the function? c. What are the dimensions of the rectangle that will enclose the greatest area? d. Write About It How would you find the dimensions of the rectangle with the least perimeter that would enclose a rectangular area of 100 square feet? SPIRAL REVIEW Determine the range of each function for the given domain. (Previous course) 62. f (x) = x - 3, domain: -4 ≤ x ≤ 6 63. f (x) = - x 2 + 2, domain: -2 ≤ x ≤ 2 Find the perimeter and area of each figure. Express your answers in terms of x. (Lesson 1-5) 64. �� 65. � �� Write each vector in component form. (Lesson 8-6) 66.  LM with L (4, 3) and M (5, 10) 67.  ST with S (-2, -2) and T (4, 6) 9- 1 Developing Formulas for Triangles and Quadrilaterals 597 597 9-2 Develop π The ratio of the circumference of a circle to its diameter is defined as π. All circles are similar, so this ratio is the same for all circles: π = circumference __ . diameter Use with Lesson 9-2 TEKS G.5.A Geometric patterns: use numeric and geometric patterns to develop algebraic expressions representing geometric properties. Activity 1 1 Use your compass to draw a large circle on a piece of cardboard and then cut it out. 2 Use a measuring tape to measure the circle’s diameter and circumference as accurately as possible. 3 Use the results from your circle to estimate π. Compare your answers with the results of the rest of the class. Try This 1. Do you think it is possible to draw a circle whose ratio of circumference to diameter is not π ? Why or why not? 2. How does knowing the relationship between circumference, diameter, and π help you determine the formula for circumference? 3. Use a ribbon to make a π measuring tape. Mark off increments of π inches or π cm on your ribbon as accurately as possible. How could you use this π measuring tape to find the diameter of a circular object? Use your π measuring tape to measure 5 circular objects. Give the circumference and diameter of each object. 598 598 Chapter 9 Extending Perimeter, Circumference, and Area Archimedes used inscribed and circumscribed polygons to estimate the value of π. His “method of exhaustion” is considered to be an early version of calculus. In the figures below, the circumference of the circle is less than the perimeter of the larger polygon and greater than the perimeter of the smaller polygon. This fact is used to estimate π. Activity 2 1 Construct a large square. Construct the perpendicular bisectors of two adjacent sides. 2 Use your compass to draw an inscribed circle as shown. 3 Connect the midpoints of the sides to form a square that is inscribed in the circle. 4 Let P 1 represent the perimeter of the smaller square, P 2 represent the perimeter of the larger square, and C represent the circumference of the circle. Measure the squares to find P 1 and P 2 and substitute the values into the inequality below. P 1 < C < P 2 5 Divide each expression in the inequality by the diameter of the circle. Why does this give you an inequality in terms of π ? Complete the inequality below. ? < π < ̶̶̶̶̶ ? ̶̶̶̶̶ Try This 4. Use the perimeters of the inscribed and circumscribed regular hexagons to write an inequality for π. Assume the diameter of each circle is 2 units. 5. Compare the inequalities you found for π. What do you think would be true about your inequality if you used regular polygons with more sides? How could you use inscribed and circumscribed regular polygons to estimate π ? 6. An alternate definition of π is the area of a circle with radius 1. How could you use this definition and the figures above to estimate the value of π? 9- 2 Geometry Lab 599 599 9-2 Developing Formulas for Circles and Regular Polygons TEKS G.8.A Congruence and the geometry of size: find areas of regular polygons, circles, ... Also G.5.A, G.8.C Objectives Develop and apply the formulas for the area and circumference of a circle. Develop and apply the formula for the area of a regular polygon. Vocabulary circle center of a circle center of a regular polygon apothem central angle of a regular polygon Who uses this? Drummers use drums of different sizes to produce different notes. The pitch is related to the area of the top of the drum. (See Example 2.) A circle is the locus of points in a plane that are a fixed distance from a point called the center of the circle . A circle is named by the symbol ⊙ and its center. ⊙A has radius r = AB and diameter d = CD. The irrational number π is defined as the ratio of the circumference C to the diameter d, or π = C__ d Solving for C gives the formula C = πd. Also d = 2r, so C = 2πr. . You can use the circumference of a circle to find its area. Divide the circle and rearrange the pieces to make a shape that resembles a parallelogram. The base of the parallelogram is about half the circumference, or πr, and the height is close to the radius r. So A ≅ πr · r = πr 2 . The more pieces you divide the circle into, the more accurate the estimate will be. Circumference and Area Circle A circle with diameter d and radius r has circumference C = πd or C = 2πr and area Finding Measurements of Circles Find each measurement. A the area of ⊙P in terms of π A = πr 2 A = π (8) 2 A = 64π cm 2 Area of a circle Divide the diameter by 2 to find the radius, 8. Simplify. 600 600 Chapter 9 Extending Perimeter, Circumference, and Area �� Find each measurement. B the radius of ⊙X in which C = 24π in. C = 2πr 24π = 2πr r = 12 in. Circumference of a circle Substitute 24π for C. Divide both sides by 2π. C the circumference of ⊙S in which A = 9x 2 π cm 2 Step 1 Use the given area to solve for r. A = π r 2 9x 2 π = π r 2 9x 2 = r 2 3x = r Area of a circle
Substitute 9x 2 π for A. Divide both sides by π. Take the square root of both sides. Step 2 Use the value of r to find the circumference. C = 2πr C = 2π (3x) Substitute 3x for r. C = 6xπ cm Simplify. 1. Find the area of ⊙A in terms of π in which C = (4x - 6) π m. E X A M P L E 2 Music Application A drum kit contains three drums with diameters of 10 in., 12 in., and 14 in. Find the area of the top of each drum. Round to the nearest tenth. The π key gives the best possible approximation for π on your calculator. Always wait until the last step to round. 10 in. diameter A = π ( 5 2 ) r = 10 _ 2 ≅ 78.5 in 2 12 in. diameter 14 in. diameter = 5 A = π ( 6 2 ) r = 12 _ 2 ≅ 113.1 in 2 = 6 A = π (7) 2 r = 14 _ 2 = 7 ≅ 153.9 in 2 2. Use the information above to find the circumference of each drum. The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a side. A central angle of a regular polygon has its vertex at the center, and its sides pass through consecutive vertices. ____ n . Each central angle measure of a regular n-gon is 360° To find the area of a regular n-gon with side length s and apothem a, divide it into n congruent isosceles triangles. area of each triangle: 1 _ as 2 total area of the polygon: A = n ( 1 _ as) , or A = 1 _ 2 2 aP The perimeter is P = ns. Area Regular Polygon The area of a regular polygon with apothem a and perimeter P is A = 1 _ 2 aP. 9- 2 Developing Formulas for Circles and Regular Polygons 601 601 �� E X A M P L E 3 Finding the Area of a Regular Polygon Find the area of each regular polygon. Round to the nearest tenth. A a regular hexagon with side length 6 m The perimeter is 6 (6) = 36 m. The hexagon can be divided into 6 equilateral triangles with side length 6 m. By the 30°-60°-90° Triangle Theorem, the apothem is 3 √  3 m3 √  3 ) (36) 2 aP Area of a regular polygon Substitute 3 √  3 for a and 36 for P. The tangent of an angle in a right triangle is the ratio of the opposite leg length to the adjacent leg length. See page 525. A = 54 √  3 ≅ 93.5 m 2 Simplify. B a regular pentagon with side length 8 in. Step 1 Draw the pentagon. Draw an isosceles triangle with its vertex at the center of the ____ 5 = 72°. pentagon. The central angle is 360° Draw a segment that bisects the central angle and the side of the polygon to form a right triangle. Step 2 Use the tangent ratio to find the apothem. tan 36° = 4 _ a 4 _ a = tan 36° The tangent of an angle is opp. leg _______ . adj. leg Solve for a. Step 3 Use the apothem and the given side length to find the area. aP tan 36° 2 A ≅ 110.1 in 2 ) (40) Area of a regular polygon The perimeter is 8 (5) = 40 in. Simplify. Round to the nearest tenth. 3. Find the area of a regular octagon with a side length of 4 cm. THINK AND DISCUSS 1. Describe the relationship between the circumference of a circle and π. 2. Explain how you would find the central angle of a regular polygon with n sides. 3. GET ORGANIZED Copy and complete the graphic organizer. 602 602 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-2 Exercises Exercises KEYWORD: MG7 9-2 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Describe how to find the apothem of a square with side length s Find each measurement. p. 600 2. the circumference of ⊙C 3. the area of ⊙A in terms of π 4. the circumference of ⊙P in which A = 36π in . Food A pizza parlor offers pizzas with diameters of 8 in., 10 in., and 12 in. p. 601 Find the area of each size pizza. Round to the nearest tenth Find the area of each regular polygon. Round to the nearest tenth. p. 602 6. 7. Independent Practice For See Exercises Example 10–12 13 14–17 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S20 Application Practice p. S36 8. an equilateral triangle with an apothem of 2 ft 9. a regular dodecagon with a side length of 5 m PRACTICE AND PROBLEM SOLVING Find each measurement. Give your answers in terms of π. 10. the area of ⊙M 11. the circumference of ⊙Z 12. the diameter of ⊙G in which C = 10 ft. 13. Sports A horse trainer uses circular pens that are 35 ft, 50 ft, and 66 ft in diameter. Find the area of each pen. Round to the nearest tenth. Find the area of each regular polygon. Round to the nearest tenth, if necessary. 14. 15. 16. a regular nonagon with a perimeter of 144 in. 17. a regular pentagon with an apothem of 2 ft. 9- 2 Developing Formulas for Circles and Regular Polygons 603 603 �� Biology Dendroclimatologists study tree rings for evidence of changes in weather patterns over time. Find the central angle measure of each regular polygon. (Hint: To review polygon names, see page 382.) 18. equilateral triangle 19. square 20. pentagon 21. hexagon 22. heptagon 23. octagon 24. nonagon 25. decagon Find the area of each regular polygon. Round to the nearest tenth. 26. 29. 27. 30. 28. 31. 32. Biology You can estimate a tree’s age in years by using the formula a = r __ w , where r is the tree’s radius without bark and w is the average thickness of the tree’s rings. The circumference of a white oak tree is 100 in. The bark is 0.5 in. thick, and the average width of a ring is 0.2 in. Estimate the tree’s age. 33. /////ERROR ANALYSIS///// A circle has a circumference of 2π in. Which calculation of the area is incorrect? Explain. Find the missing measurements for each circle. Give your answers in terms of π. Diameter d Radius r Area A Circumference C 6 34. 35. 36. 37. 100 17 36 π 38. Multi-Step Janet is designing a garden around a gazebo that is a regular hexagon with side length 6 ft. The garden will be a circle that extends 10 feet from the vertices of the hexagon. What is the area of the garden? Round to the nearest square foot. 39. This problem will prepare you for the Multi-Step TAKS Prep on page 614. A stop sign is a regular octagon. The signs are available in two sizes: 30 in. or 36 in. a. Find the area of a 30 in. sign. Round to the nearest tenth. b. Find the area of a 36 in. sign. Round to the nearest tenth. c. Find the percent increase in metal needed to make a 36 in. sign instead of a 30 in. sign. 604 604 Chapter 9 Extending Perimeter, Circumference, and Area �� 40. Measurement A trundle wheel is used to measure distances by rolling it on the ground and counting its number of turns. If the circumference of a trundle wheel is 1 meter, what is its diameter? 41. Critical Thinking Which do you think would seat more people, a 4 ft by 6 ft rectangular table or a circular table with a diameter of 6 ft? How many people would you sit at each table? Explain your reasoning. 42. Write About It The center of each circle in the figure lies on the number line. Describe the relationship between the circumference of the largest circle and the circumferences of the four smaller circles. 43. Find the perimeter of the regular octagon to the nearest centimeter. 5 40 20 68 44. Which of the following ratios comparing a circle’s circumference C to its diameter d gives the value of π? C _ d 4C _ d 2 d _ C d _ 2C 45. Alisa has a circular tabletop with a 2-foot diameter. She wants to paint a pattern on the table top that includes a 2-foot-by-1-foot rectangle and 4 squares with sides 0.5 foot long. Which information makes this scenario impossible? There will be no room left on the tabletop after the rectangle has been painted. A 2-foot-long rectangle will not fit on the circular tabletop. Squares cannot be painted on the circle. There will not be enough room on the table to fit all the 0.5-foot squares. CHALLENGE AND EXTEND 46. Two circles have the same center. The radius of the larger circle is 5 units longer than the radius of the smaller circle. Find the difference in the circumferences of the two circles. 47. Algebra Write the formula for the area of a circle in terms of its circumference. 48. Critical Thinking Show that the formula for the area of a regular n-gon approaches the formula for the area of a circle as n gets very large. SPIRAL REVIEW Write an equation for the linear function represented by the table. (Previous course) 49. x y -2 0 -19 -13 5 2 10 17 50. x y -3 2 0 4 9 -1 -5 -10 Find each value. (Lesson 4-8) 51. m∠B 52. AB Find each measurement. (Lesson 9-1) 53. d 2 of a kite if A = 14 cm 2 and d 1 = 20 cm 54. the area of a trapezoid in which b 1 = 3 yd, b 2 = 6 yd, and h = 4 yd 9- 2 Developing Formulas for Circles and Regular Polygons 605 605 �� 9-3 Composite Figures TEKS G.8.A Congruence and the geometry of size: find areas of ... composite figures. Objectives Use the Area Addition Postulate to find the areas of composite figures. Use composite figures to estimate the areas of irregular shapes. Vocabulary composite figure Who uses this? Landscape architects must compute areas of composite figures when designing gardens. (See Example 3.) A composite figure is made up of simple shapes, such as triangles, rectangles, trapezoids, and circles. To find the area of a composite figure, find the areas of the simple shapes and then use the Area Addition Postulate. E X A M P L E 1 Finding the Areas of Composite Figures by Adding Find the shaded area. Round to the nearest tenth, if necessary. A B Divide the figure into rectangles. Divide the figure into parts. The base of the triangle is √  10. 2 2 - 4. 8 2 = 9 ft. area of top rectangle: A = bh = 12 (15) = 180 c m 2 area of bottom rectangle: A = bh = 9 (27) = 243 c m 2 shaded area: 180 + 243 = 423 c m 2 area of triangle: A = 1 __ 2 bh = 1 __ 2 (9) (4.8) = 21.6 ft 2 area of rectangle: A = bh = 9 (3) = 27 ft 2 area of half circle: A = 1 __ 2 π
r 2 = 1 __ 2 π (4. 5 2 ) = 10.125π f t 2 shaded area: 21.6 + 27 + 10.125π ≈ 80.4 ft 2 1. Find the shaded area. Round to the nearest tenth, if necessary. 606 606 Chapter 9 Extending Perimeter, Circumference, and Area �� Sometimes you need to subtract to find the area of a composite figure. E X A M P L E 2 Finding the Areas of Composite Figures by Subtracting Find the shaded area. Round to the nearest tenth, if necessary. A B Subtract the area of the triangle from the area of the rectangle. area of rectangle: A = bh = 18 (36) = 648 m 2 area of triangle: A = 1 __ 2 bh = 1 __ 2 (36) (9) = 162 m 2 area of figure: A = 648 - 162 = 486 m 2 The two half circles have the same area as one circle. Subtract the area of the circle from the area of the rectangle. area of the rectangle: A = bh = 33 (16) = 528 f t 2 area of circle ) = 64π f t 2 area of figure: A = 528 - 64π ≈ 326.9 ft 2 2. Find the shaded area. Round to the nearest tenth, if necessary. E X A M P L E 3 Landscaping Application Katie is using the given plan to convert part of her lawn to a xeriscape garden. A newly planted xeriscape uses 17 gallons of water per square foot per year. How much water will the garden require in one year? To find the area of the garden in square feet, divide the garden into parts. The area of the top rectangle is 28.5 (7.5) = 213.75 f t 2 . The area of the center trapezoid is 1 __ 2 (12 + 18) (6) = 90 f t 2 . The area of the bottom rectangle is 12 (6) = 72 f t 2 . The total area of the garden is 213.75 + 90 + 72 = 375.75 f t 2 . The garden will use 375.75 (17) = 6387.75 gallons of water per year. Landscaping The rainwater harvesting system at the Lady Bird Johnson Wildflower Center in Austin, Texas, collects approximately 10,200 gallons of water per inch of rain. Approximately 300,000 gallons are collected annually. 3. The lawn that Katie is replacing requires 79 gallons of water per square foot per year. How much water will Katie save by planting the xeriscape garden? 9- 3 Composite Figures 607 607 ��ge07sec09l03002aaABeckmann28.5 ft12 ft10.5 ft7.5 ft6 ft19.5 ftge07se_c09l03003aABeckmann28.5 ft12 ft6 ft7.5 ft6 ft18 ft To estimate the area of an irregular shape, you can sometimes use a composite figure. First, draw a composite figure that resembles the irregular shape. Then divide the composite figure into simple shapes. E X A M P L E 4 Estimating Areas of Irregular Shapes Use a composite figure to estimate the shaded area. The grid has squares with side lengths of 1 cm. Draw a composite figure that approximates the irregular shape. Find the area of each part of the composite figure. area of triangle a: bh = 1 _ A = 1 _ (3) (1) = 1.5 c m 2 2 2 area of parallelogram b: A = bh = 3 (1) = 3 c m 2 area of trapezoid c: A = 1 _ (3 + 2) (1) = 2.5 c m 2 2 area of triangle d: A = 1 _ (2) (1) = 1 c m 2 2 area of composite figure: 1.5 + 3 + 2.5 + 1 = 8 cm 2 The shaded area is about 8 c m 2 . 4. Use a composite figure to estimate the shaded area. The grid has squares with side lengths of 1 ft. THINK AND DISCUSS 1. Describe a composite figure whose area you could find by using subtraction. 2. Explain how to find the area of an irregular shape by using a composite figure. 3. GET ORGANIZED Copy and complete the graphic organizer. Use the given composite figure. 608 608 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-3 Exercises Exercises KEYWORD: MG7 9-3 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Draw a composite figure that is made up of two rectangles Multi-Step Find the shaded area. Round to the nearest tenth, if necessary. p. 606 2. p. 607 3. 5. Interior Decorating Barbara is getting carpet p. 607 installed in her living room and hallway. The cost of installation is $6 per square yard. What is the total cost of installing the carpet. 608 Use a composite figure to estimate each shaded area. The grid has squares with side lengths of 1 in. 7. 8. PRACTICE AND PROBLEM SOLVING Independent Practice Multi-Step Find the shaded area. Round to the nearest tenth, if necessary. For See Exercises Example 9. 9–10 11–12 13 14–15 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S20 Application Practice p. S36 11. 10. 12. 9- 3 Composite Figures 609 609 �� 13. Drama Pat is painting a stage backdrop for a play. The paint he is using covers 90 square feet per quart. How many quarts of paint should Pat buy? Use a composite figure to estimate each shaded area. The grid has squares with side lengths of 1 m. 14. 15. Find the area of each figure first by adding and then by subtracting. Compare your answers. 16. 17. Find the area of each figure. Give your answers in terms of π. 18. 19. 20. 21. Geography Use the grid on the map of Lake Superior to estimate the area of the surface of the lake. Each square on the grid has a side length of 100 miles. 22. Critical Thinking A trapezoid can be divided into a rectangle and two triangles. Show that the area formula for a trapezoid gives the same result as the sum of the areas of the rectangle and triangles. 23. This problem will prepare you for the Multi-Step TAKS Prep on page 614. A school crossing sign has the dimensions shown. a. Find the area of the sign. b. A manufacturer has a rectangular sheet of metal measuring 45 in. by 105 in. Draw a figure that shows how 6 school crossing signs can be cut from this sheet of metal. c. How much metal will be left after the six signs are made? 610 610 Chapter 9 Extending Perimeter, Circumference, and Area 30 ft15 ft22 ftge07sec09l03005a��Lake SuperiorWISCONSINMINN.CANADAMICHIGAN ge07se_c09l03006a�� Math History � � � � � � Hippocrates attempted to use lunes to solve a �� problem that has since been proven impossible: constructing a square with the same area as a given circle. Multi-Step Use a ruler and compass to draw each figure and then find the area. 24. A rectangle with a base length of b = 3 cm and a height of h = 4 cm has a circle with a radius of r = 1 cm removed from the interior. 25. A square with a side length of s = 4 in. shares a side with a triangle with a height of h = 5 in. and a base length of b = 4 in. and shares another side with a half circle with d = 4 in. 26. A circle with a radius of r = 5 cm has a right triangle with a base of b = 8 cm and a height of h = 6 cm removed from its interior. � 27. Multi-Step A lune is a crescent-shaped figure bounded by two intersecting circles. Find the shaded area in each of the first three diagrams, and then use your results to � find the area of the lune. � � � � �� Estimation Trace each irregular shape and draw a composite figure that approximates it. Measure the composite figure and use it to estimate the area of the irregular shape. 28. 29. 30. Write About It Explain when you would use addition to find the area of a composite figure and when you would use subtraction. 31. Which equation can be used to find the area of the composite figure? A = bh + 1 _ (h) 2 2 A = bh + h 2 A = h + 2b + h 2 A = h + 2b + 1 _ h 2 2 32. Use a ruler to measure the dimensions of the composite figure to the nearest tenth of a centimeter. Which of the following best represents the area of the composite figure? 4 c m 2 19 cm 2 22 cm 2 42 c m 2 9- 3 Composite Figures 611 611 �� 33. Find the area of the unshaded part of the rectangle. 1800 m 2 2250 m 2 2925 m 2 4725 m 2 CHALLENGE AND EXTEND 34. An annulus is the region between two circles that have the same center. Write the formula for the area of the annulus in terms of the outer radius R and the inner radius r. 35. Draw two composite figures with the same area: one made up of two rectangles and the other made up of a rectangle and a triangle. 36. Draw a composite figure that has a total area of 10π c m 2 and is made up of a rectangle and a half circle. Label the dimensions of your figure. SPIRAL REVIEW Find each sale price. (Previous course) 37. 20% off a regular price of $19.95 38. 15% off a regular price of $34.60 Find the length of each segment. (Lesson 7-4) 39. ̶̶ BC 40. ̶̶ CD Find the area of each regular polygon. Round to the nearest tenth. (Lesson 9-2) 41. an equilateral triangle with a side length of 3 cm 42. a regular hexagon with an apothem of 4 √  3 m Q: What math classes did you take in high school? A: In high school I took Algebra 1, Geometry, Algebra 2, and Trigonometry. KEYWORD: MG7 Career Q: What math classes did you take in college? A: In college I took Precalculus, Calculus, and Statistics. Q: What technical materials do you write? A: I write training manuals for computer software packages. Q: How do you use math? A: Some manuals I write are for math programs, so I use a lot of formulas to describe patterns and measurements. Q: What are your future plans? A: After I get a few more years experience writing manuals, I would like to train others who use these programs. Anessa Liu Technical writer 612 612 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-3 Use with Lesson 9-3 Develop Pick’s Theorem for Area of Lattice Polygons A lattice polygon is a polygon drawn on graph paper so that all its vertices are on intersections of grid lines, called lattice points. The lattice points of the grid at right are shown in red. In this lab, you will discover a formula called Pick’s Theorem, which is used to find the area of lattice polygons. Activity TEKS G.8.A Congruence and the geometry of size: find areas of ... composite figures. 1 Find the area of each figure. Create a table like the one below with a row for each shape to record your answers. The first one is don
e for you. 2 Count the number of lattice points on the boundary of each figure. Record your answers in the table. 3 Count the number of lattice points in the interior of each figure. Record your answers in the table. Figure Area Number of Lattice Points On Boundary In Interior 2.5 5 1 A B C D E F Try This 1. Make a Conjecture What do you think is true about the relationship between the area of a figure and the number of lattice points on the boundary and in the interior of the figure? Write your conjecture as a formula in terms of the number of lattice points on the boundary B and the number of lattice points in the interior I. 2. Test your conjecture by drawing at least three different figures on graph paper and by finding their areas. 3. Estimate the area of the curved figure by using a lattice polygon. 4. Find the shaded area in the figure by subtracting. Test your formula on this figure. Does your formula work for figures with holes in them? 9- 3 Geometry Lab 613 613 �� SECTION 9A Developing Geometric Formulas Traffic Signs Traffic signs are usually made of reflective aluminum. A manufacturer of traffic signs begins with a rectangular sheet of aluminum that measures 60 in. by 90 in. 1. A railroad crossing sign is a circle with a diameter of 30 in. The manufacturer can make 6 of these signs from the sheet of aluminum by arranging the signs as shown. How much aluminum is left over once the signs have been made? 2. A stop sign is a regular octagon. The manufacturer can use the sheet of aluminum to make 6 stop signs as shown. How much aluminum is left over in this case? 3. A yield sign is an equilateral triangle with sides 30 in. long. By arranging the triangles as shown, the manufacturer can use the sheet of aluminum to make 10 yield signs. How much aluminum is left over when yield signs are made? 4. The making of which type of sign results in the least amount of waste? 614 614 Chapter 9 Extending Perimeter, Circumference, and Area �� SECTION 9A Quiz for Lessons 9-1 Through 9-3 9-1 Developing Formulas for Triangles and Quadrilaterals Find each measurement. 1. the area of the parallelogram 2. the base of the rectangle, in which A = (24 x 2 + 8x) m 2 3. d 1 of the kite, in which A = 126 ft 2 4. the area of the rhombus 5. The tile mosaic shown is made up of 1 cm squares. Use the grid to find the perimeter and area of the green triangle, the blue trapezoid, and the yellow parallelogram. 9-2 Developing Formulas for Circles and Regular Polygons Find each measurement. 6. the circumference of ⊙R in terms of π 7. the area of ⊙E in terms of π Find the area of each regular polygon. Round to the nearest tenth. 8. a regular hexagon with apothem 6 ft 9. a regular pentagon with side length 12 m 9-3 Composite Figures Find the shaded area. Round to the nearest tenth, if necessary. 10. 11. 12. Shelby is planting grass in an irregularly shaped garden as shown. The grid has squares with side lengths of 1 yd. Estimate the area of the garden. Given that grass cost $6.50 per square yard, find the cost of the grass. Ready to Go On? 615 615 �� 9-4 Perimeter and Area in the Coordinate Plane TEKS G.7.A Dimensionality and the geometry of location: use ... two-dimensional coordinate systems to represent ... figures. Also G.7.B, G.8.A Objective Find the perimeters and areas of figures in a coordinate plane. Why learn this? You can use figures in a coordinate plane to solve puzzles like the one at right. (See Example 4.) In Lesson 9-3, you estimated the area of irregular shapes by drawing composite figures that approximated the irregular shapes and by using area formulas. Another method of estimating area is to use a grid and count the squares on the grid. E X A M P L E 1 Estimating Areas of Irregular Shapes in the Coordinate Plane Estimate the area of the irregular shape. Method 1: Draw a composite Method 2: Count the number of figure that approximates the irregular shape and find the area of the composite figure. squares inside the figure, estimating half squares. Use a ■ for a whole for a half square. square and a The area is approximately 4 + 6.5 + 5 + 4 + 5 + 3.5 + 3 + 3 + 2 = 36 unit s 2 . There are approximately 31 whole squares and 13 half squares, so the area is about 31 + 1 __ 2 (13) = 37.5 unit s 2 . 1. Estimate the area of the irregular shape. 616 616 Chapter 9 Extending Perimeter, Circumference, and Area �� E X A M P L E 2 Finding Perimeter and Area in the Coordinate Plane Draw and classify the polygon with vertices A (-4, 1) , B (2, 4) , C (4, 0) , and D (-2, -3) . Find the perimeter and area of the polygon. ( x 2 - x 1 ) The distance from ( x 1 , y 1 ) to ( x 2 , y 2 ) in a coordinate plane is d = √  2 2 + ( y 2 - y 1 ) , and the slope of the line containing the y 2 - y 1 ______ x 2 - x 1 . points is m = See pages 44 and 182. Step 1 Draw the polygon. slope of Step 2 ABCD appears to be a rectangle. To verify this, use slopes to show that the sides are perpendicular. ̶̶ = 3 _ AB : - (-4) ̶̶ BC : 0 - 4 _ = -4 _ 4 - 2 2 ̶̶ = -3 _ CD : -3 - 0 _ = 1 _ -6 -2 - 4 2 1- (-3) = 4 _ _ -2 -4 - (-2) slope of slope of slope of ̶̶ DA : = -2 = -2 ̶̶ CD be the base and The consecutive sides are perpendicular, so ABCD is a rectangle. ̶̶ BC be the height of the rectangle. Step 3 Let Use the Distance Formula to find each side length. b = CD = √  h = BC = √  perimeter of ABCD: P = 2b + 2h = 2 (3 √  5 ) + 2 (2 √  5 ) = 10 √  5 units area of ABCD: A = bh = (3 √  5 ) (2 √  5 ) = 30 units 2 . (-2 - 4) 2 + (-3 - 0) 2 = √  45 = 3 √  5 (4 - 2) 2 + (0 - 4) 2 = √  20 = 2 √  5 2. Draw and classify the polygon with vertices H (-3, 4) , J (2, 6) , K (2, 1) , and L (-3, -1) . Find the perimeter and area of the polygon. For a figure in a coordinate plane that does not have an area formula, it may be easier to enclose the figure in a rectangle and subtract the areas of the parts of the rectangle that are not included in the figure. E X A M P L E 3 Finding Areas in the Coordinate Plane by Subtracting Find the area of the polygon with vertices W (1, 4) , X (4, 2) , Y (2, -3) , and Z (-4, 0) . Draw the polygon and enclose it in a rectangle. area of the rectangle: A = bh = 8 (7) = 56 units 2 area of the triangles: bh = 1 _ a: A = 1 _ (5) (4) = 10 units 2 2 2 bh = 1 _ b: A = 1 _ (3) (2) = 3 units 2 2 2 bh = 1 _ c: A = 1 _ (2) (5) = 5 units 2 2 2 bh = 1 _ d: A = 1 _ (6) (3) = 9 units 2 2 2 The area of the polygon is 56 - 10 - 3 - 5 - 9 = 29 units 2 . 3. Find the area of the polygon with vertices K (-2, 4) , L (6, -2) , M (4, -4) , and N (-6, -2) . 9- 4 Perimeter and Area in the Coordinate Plane 617 617 �� E X A M P L E 4 Problem-Solving Application In the puzzle, the two figures are made up of the same pieces, but one figure appears to have a larger area. Use coordinates to show that the area does not change when the pieces are rearranged. Understand the Problem The parts of the puzzle appear to form two triangles with the same base and height that contain the same shapes, but one appears to have an area that is larger by one square unit. Make a Plan Find the areas of the shapes that make up each figure. If the corresponding areas are the same, then both figures have the same area by the Area Addition Postulate. To explain why the area appears to increase, consider the assumptions being made about the figure. Each figure is assumed to be a triangle with a base of 8 units and a height of 3 units. Both figures are divided into several smaller shapes. Solve Find the area of each shape. Top figure Bottom figure red triangle: bh = 1 _ A = 1 _ (5) (2) = 5 unit s 2 2 2 blue triangle: bh = 1 _ A = 1 _ (3) (1) = 1.5 units 2 2 2 green rectangle: A = bh = (3) (1) = 3 units 2 red triangle: bh = 1 _ A = 1 _ (5) (2) = 5 u nits 2 2 2 blue triangle: bh = 1 _ A = 1 _ (3) (1) = 1.5 u nits 2 2 2 green rectangle: A = bh = (3) (1) = 3 units 2 yellow rectangle: A = bh = (2) (1) = 2 units 2 yellow rectangle: A = bh = (2) (1) = 2 units 2 The areas are the same. Both figures have an area of 5 + 1.5 + 3 + 2 = 11.5 units 2 . If the figures were triangles, their areas would be A = 1 __ 2 (8) (3) = 12 units 2 . By the Area Addition Postulate, the area is only 11.5 units 2 , so the figures must not be triangles. Each figure is a quadrilateral whose shape is very close to a triangle. Look Back The slope of the hypotenuse of the red triangle is 2 __ 5 . The slope of the hypotenuse of the blue triangle is 1 __ 3 . Since the slopes are unequal, the hypotenuses do not form a straight line. This means the overall shapes are not triangles. 4. Create a figure and divide it into pieces so that the area of the figure appears to increase when the pieces are rearranged. 618 618 Chapter 9 Extending Perimeter, Circumference, and Area ��1234 THINK AND DISCUSS 1. Describe two ways to estimate the area of an irregular shape in a coordinate plane. 2. Explain how you could use the Distance Formula to find the area of a special quadrilateral in a coordinate plane. 3. GET ORGANIZED Copy the graph and the graphic organizer. Complete the graphic organizer by writing the steps used to find the area of the parallelogram. 9-4 Exercises Exercises GUIDED PRACTICE Estimate the area of each irregular shape. p. 616 1. 2. KEYWORD: MG7 9-4 KEYWORD: MG7 Parent . 617 Multi-Step Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 3. V (-3, 0) , W (3, 0) , X (0, 3) 4. F (2, 8) , G (4, 4) , H (2, 0) 5. P (-2, 5) , Q (8, 5) , R (8, 1) , S (-2, 1) 6. A (-4, 2) , B (-2, 6) , C (6, 6) , D (8, 2 Find the area of each polygon with the given vertices. p. 617 7. S (3, 8) , T (8, 3) , U (2, 1) 8. L (3, 5) , M (6, 8) , N (9, 6) , P (5, 0. 618 9. Find the area and perimeter of each polygon shown. Use your results to d
raw a polygon with a perimeter of 12 units and an area of 4 un its 2 and a polygon with a perimeter of 12 units and an area of 3 un its 2 . 9- 4 Perimeter and Area in the Coordinate Plane 619 619 �� Independent Practice For See Exercises Example 10–11 12–15 16–17 18 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S21 Application Practice p. S36 PRACTICE AND PROBLEM SOLVING Estimate the area of each irregular shape. 10. 11. Multi-Step Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 12. H (-3, -3) , J (-3, 3) , K (5, 3) 13. L (7, 5) , M (5, 0) , N (3, 5) , P (5, 10) 14. X (2, 1) , Y (5, 3) , Z (7, 1) 15. A (-3, 5) , B (2, 7) , C (2, 1) , D (-3, 3) Find the area of each polygon with the given vertices. 16. A (9, 9) , B (4, -4) , C (-4, 1) 17. T (-4, 4) , U (5, 3) , V (4, -5) , W (-5, 1) 18. In which two figures do the rectangles cover the same area? Explain your reasoning. Algebra Graph each set of lines to form a triangle. Find the area and perimeter. 19. y = 2, x = 5, and y = x 20. y = -5, x = 2, and y = -2x + 7 21. Transportation The graph shows the speed of a boat versus time. a. If the base of each square on the graph represents 1 hour and the height represents 20 miles per hour, what is the area of one square on the graph? Include units in your answer. b. Estimate the shaded area in the graph. c. Critical Thinking Use your results from part a to interpret the meaning of the area you found in part b. (Hint: Look at the units.) 22. Write About It Explain how to find the perimeter of the polygon with vertices A (2, 3) , B (4, 0) , C (3, -2) , D (-1, -1) , and E (-2, 0) . 23. This problem will prepare you for the Multi-Step TAKS Prep on page 638. A carnival game uses a 10-by-10 board with three targets. Each player throws a dart at the board and wins a prize if it hits a target. a. One target is a parallelogram as shown. Find its area. b. What should the coordinates be for points C and H so that the triangular target △ABC and the kite-shaped target EFGH have the same area as the parallelogram? 620 620 Chapter 9 Extending Perimeter, Circumference, and Area �� 24. A circle with center (0, 0) passes through the point (3, 4) . What is the area of the circle to the nearest tenth of a square unit? 15.7 25.0 31.4 78.5 25. △ABC with vertices A (1, 1) and B (3, 5) has an area of 10 unit s 2 . Which is NOT a possible location of the third vertex? C (-4, 1) C (7, 3) C (6, 1) C (3, -3) 26. Extended Response Mike estimated the area of the irregular figure to be 64 units 2 . a. Explain why his answer is not very accurate. b. Explain how to use a composite figure to estimate the area. c. Explain how to estimate the area by averaging the areas of two squares. CHALLENGE AND EXTEND Algebra Estimate the shaded area under each curve. 28. y = x 2 for 0 ≤ x ≤ 3 27. y = 2 x for 0 ≤ x ≤ 3 29. y = √  x for 0 ≤ x ≤ 9 30. Estimation Use a composite figure and the Distance Formula to estimate the perimeter of the irregular shape. 31. Graph a regular octagon on the coordinate plane with vertices on the x-and y-axes and on the lines y = x and y = -x so that the distance between opposite vertices is 2 units. Find the area and perimeter of the octagon. SPIRAL REVIEW Solve and graph each compound inequality. (Previous course) 32. -4 < x + 3 < 7 ̶̶ BC , ∠DCA ≅ ∠ACB Prove: ∠DAC ≅ ∠BAC (Lesson 4-6) 35. Given: ̶̶ DC ≅ 33. 0 < 2a + 4 < 10 34. 12 ≤ -2m + 10 ≤ 20 Find each measurement. (Lesson 9-2) 36. the area of ⊙C if the circumference is 16π cm 37. the diameter of ⊙H if the area is 121π f t 2 9- 4 Perimeter and Area in the Coordinate Plane 621 621 �� 9-5 Effects of Changing Dimensions Proportionally TEKS G.11.D Similarity and the geometry of shape: describe the effect on perimeter, area,... when one or more dimensions of a figure are changed .... Also G.5.A, G.5.B Objectives Describe the effect on perimeter and area when one or more dimensions of a figure are changed. Apply the relationship between perimeter and area in problem solving. Why learn this? You can analyze a graph to determine whether it is misleading or to explain why it is misleading. (See Example 4.) In the graph, the height of each DVD is used to represent the number of DVDs shipped per year. However as the height of each DVD increases, the width also increases, which can create a misleading effect. E X A M P L E 1 Effects of Changing One Dimension Describe the effect of each change on the area of the given figure. A The height of the parallelogram is doubled. double the height: A = bh = 12 (18) = 216 cm 2 original dimensions: A = bh = 12 (9) = 108 cm 2 Notice that 216 = 2 (108) . If the height is doubled, the area is also doubled. B The base length of the triangle with vertices A(1, 1) , B(6, 1) , and C (3, 5) is multiplied by 1 __ . 2 Draw the triangle in a coordinate plane and find the base and height. original dimensions: bh = 1 _ A = 1 _ (5) (4) = 10 units 2 2 2 base multiplied by 1 __ 2 : bh = 1 _ A = 1 _ (2.5) (4) = 5 units 2 2 2 Notice that 5 = 1 __ 2 (10) . If the base length is multiplied by 1 __ 2 , the area is multiplied by 1 __ 2 . 1. The height of the rectangle is tripled. Describe the effect on the area. 622 622 Chapter 9 Extending Perimeter, Circumference, and Area ge07se_c09l05001a AB�� E X A M P L E 2 Effects of Changing Dimensions Proportionally Describe the effect of each change on the perimeter or circumference and the area of the given figure. A The base and height of a rectangle with base 8 m and height 3 m are If the radius of a circle or the side length of a square is changed, the size of the entire figure changes proportionally. both multiplied by 5. original dimensions: P = 2 (8) + 2 (3) = 22 m A = 83 = 24 m 2 dimensions multiplied by 5: P = 2 (40) + 2 (15) = 110 m A = 40 (15) = 600 m 2 The perimeter is multiplied by 5. The area is multiplied by 5 2 , or 25. B The radius of ⊙A is multiplied by 1 __ . 3 original dimensions: C = 2π (9) = 18π in. A = π (9) 2 = 81π in 2 dimensions multiplied by 1 __ 3 : C = 2π (3) = 6π in. A = π (3) 2 = 9π in 2 P = 2b + 2h A = bh 5 (8) = 40; 5 (3) = 15 5 (22) = 110 25 (24) = 600 C = 2πr A = π r 2 1 __ (9) = 3 3 The perimeter is multiplied by 1 __ 3 . The area is multiplied by ( 1 __ 3 ) , or 1 __ 9 . 2 1 __ (18π) = 6π 3 1 __ (81π) = 9π 9 2. The base and height of the triangle with vertices P (2, 5) , Q (2, 1) and R (7, 1) are tripled. Describe the effect on its area and perimeter. When all the dimensions of a figure are changed proportionally, the figure will be similar to the original figure. Effects of Changing Dimensions Proportionally Change in Dimensions Perimeter or Circumference Area All dimensions multiplied by a Changes by a factor of a Changes by a factor of Effects of Changing Area A A square has side length 5 cm. If the area is tripled, what happens to the side length? The area of the original square is A = s 2 = 5 2 = 25 cm 2 . If the area is tripled, the new area is 75 cm 2 . s 2 = 75 s = √  75 = 5 √  3 Set the new area equal to s 2 . Take the square root of both sides and simplify. Notice that 5 √  3 = √  3 (5) . The side length is multiplied by √  3 . 9- 5 Effects of Changing Dimensions Proportionally 623 623 �� B A circle has a radius of 6 in. If the area is doubled, what happens to the circumference? The original area is A = π r 2 = 36π in 2 , and the circumference is C = 2πr = 12π in. If the area is doubled, the new area is 72π in 2 . π r 2 = 72π r 2 = 72 r 2 = √  72 = 6 √  2 C = 2πr = 2π ( 6 √  2 ) = 12 √  2 π Set the new area equal to π r 2 . Divide both sides by π. Take the square root of both sides and simplify. Substitute 6 √  2 for r and simplify. Notice that 12 √  2 π = √  2 (12π) . The circumference is multiplied by √  2 . 3. A square has a perimeter of 36 mm. If the area is multiplied by 1 __ 2 , what happens to the side length? E X A M P L E 4 Entertainment Application The graph shows that DVD shipments totaled about 182 million in 2000, 364 million in 2001, and 685 million in 2002. The height of each DVD is used to represent the number of DVDs shipped. Explain why the graph is misleading. The height of the DVD representing shipments in 2002 is about 3.8 times the height of the DVD representing shipments in 2002. This means that the area of the DVD is multiplied by about 3.8 2 , or 14.4, so the area of the larger DVD is about 14.4 times the area of the smaller DVD. The graph gives the misleading impression that the number of shipments in 2002 was more than 14 times the number in 2000, but it was actually closer to 4 times the number shipped in 2000. 4. Use the information above to create a version of the graph that is not misleading. THINK AND DISCUSS 1. Discuss how changing both dimensions of a rectangle affects the area and perimeter. 2. GET ORGANIZED Copy and complete the graphic organizer. 624 624 Chapter 9 Extending Perimeter, Circumference, and Area ��20002001YearDVDs shipped (millions)2002100200300400500600ge06se_c09105002aDVD Shipments 9-5 Exercises Exercises KEYWORD: MG7 9-5 KEYWORD: MG7 Parent GUIDED PRACTICE Describe the effect of each change on the area of the given figure. p. 622 1. The height of the triangle is doubled. 2. The height of a trapezoid with base lengths 12 cm and 18 cm and height 5 cm is multiplied by 1 __ . 623 Describe the effect of each change on the perimeter or circumference and the area of the given figure. 3. The base and height of a triangle with base 12 in. and height 6 in. are both tripled. 4. The base and height of the rectangle are both multiplied by 1 __ . A square has an area of 36 m 2 . If the a
rea is doubled, what happens to the p. 623 side length? 6. A circle has a diameter of 14 ft. If the area is tripled, what happens to the circumference. Business A restaurant has a weekly ad in a local newspaper that is 2 inches wide p. 624 and 4 inches high and costs $36.75 per week. The cost of each ad is based on its area. If the owner of the restaurant decides to double the width and height of the ad, how much will the new ad cost? Independent Practice Describe the effect of each change on the area of the given figure. PRACTICE AND PROBLEM SOLVING For See Exercises Example 8–9 10–11 12–13 14 1 2 3 4 8. The height of the triangle with vertices (1, 5) , (2, 3) , and (-1, -6) is multiplied by 4. 9. The base of the parallelogram is multiplied by 2 __ 3 . Describe the effect of each change on the perimeter or circumference and the area of the given figure. TEKS TEKS TAKS TAKS 10. The base and height of the triangle are both doubled. Skills Practice p. S21 Application Practice p. S36 11. The radius of the circle with center (0, 0) that passes through (5, 0) is multiplied by 3 __ 5 . 12. A circle has a circumference of 16π mm. If you multiply the area by 1 __ 3 , what happens to the radius? 13. A square has vertices (3, 2) , (8, 2,) (8, 7) , and (3, 7) . If you triple the area, what happens to the side length? 14. Entertainment Two televisions have rectangular screens with the same ratio of base to height. One has a 32 in. diagonal, and the other has a 36 in. diagonal. a. What is the ratio of the height of the larger screen to that of the smaller screen? b. What is the ratio of the area of the larger screen to that of the smaller screen? 9- 5 Effects of Changing Dimensions Proportionally 625 625 �� Describe the effect of each change on the area of the given figure. 15. The diagonals of a rhombus are both multiplied by 8. 16. The circumference of a circle is multiplied by 2.4. 17. The base of a rectangle is multiplied by 4, and the height is multiplied by 7. 18. The apothem of a regular octagon is tripled. 19. The diagonal of a square is divided by 4. 20. One diagonal of a kite is multiplied by 1 _ . 7 21. 21. The perimeter of an equilateral triangle is doubled. 22. 22. Find the area of the trapezoid. Describe the effect of each change on the area. a. a. The length of the top base is doubled. b. The length of both bases is doubled. c. The height is doubled. d. Both bases and the height are doubled. 23. Geography A map has the scale 1 inch = 10 miles. On the map, the area of Big Bend National Park in Texas is about 12.5 square inches. Estimate the actual area of the park in acres. (Hint: 1 square mile = 640 acres) 24. Critical Thinking If you want to multiply the dimensions of a figure so that the area is 50% of the original area, what is your scale factor? Multi-Step For each figure in the coordinate plane, describe the effect on the area that results from each change. a. Only the x-coordinates of the vertices are multiplied by 3. b. Only the y-coordinates of the vertices are multiplied by 3. c. Both the x- and y-coordinates of the vertices are multiplied by 3. 25. 26. 27. Geography Geography The altitude in Big Bend National Park ranges from approximately 1800 feet along the Rio Grande to 7800 feet in the Chisos Mountains. 28. Write About It How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram? if the parallelogram does have to be similar to the original parallelogram? 29. This problem will prepare you for the Multi-Step TAKS Prep on page 638. To win a prize at a carnival, a player must toss a beanbag onto a circular disk with a diameter of 8 in. a. The organizer of the game wants players to win twice as often, so he changes the disk so that it has twice the area. What is the diameter of the new disk? b. Suppose the organizer wants players to win half as often. What should be the disk’s diameter in this case? 626 626 Chapter 9 Extending Perimeter, Circumference, and Area �� 30. Which of the following describes the effect on the area of a square when the side length is doubled? The area remains constant. The area is reduced by a factor of 1 __ . 2 The area is doubled. The area is increased by a factor of 4. 31. If the area of a circle is increased by a factor of 4, what is the change in the diameter of the circle? The diameter is 1 __ of the original diameter. 2 The diameter is 2 times the original diameter. The diameter is 4 times the original diameter. The diameter is 16 times the original diameter. 32. Tina and Kieu built rectangular play areas for their dogs. The play area for Tina’s dog is 1.5 times as long and 1.5 times as wide as the play area for Kieu’s dog. If the play area for Kieu’s dog is 60 square feet, how big is the play area for Tina’s dog? 40 ft 2 90 ft 2 135 ft 2 240 ft 2 33. Gridded Response Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled. Find the perimeter of the new triangle in inches. CHALLENGE AND EXTEND 34. Algebra A square has a side length of (2x + 5) cm . If the side length is multiplied by 5, what is the area of the new square? 35. Algebra A circle has a diameter of 6 in. If the circumference is multiplied by (x + 3) , what is the area of the new circle? 36. Write About It How could you change the dimensions of the composite figure to double the area if the resulting figure does not have to be similar to the original figure? if the resulting figure does have to be similar to the original figure? SPIRAL REVIEW Write an equation that can be used to determine the value of the variable in each situation. (Previous course) 37. Steve can make 2 tortillas per minute. He makes t tortillas in 36 minutes. 38. A car gets 25 mi/gal. At the beginning of a trip of m miles, the car’s gas tank contains 13 gal of gas. At the end of the trip, the car has 8 gal of gasoline left. Find the measure of each interior and each exterior angle of each regular polygon. Round to the nearest tenth, if necessary. (Lesson 6-1) 39. heptagon 40. decagon 41. 14-gon Find the area of each polygon with the given vertices. (Lesson 9-4) 42. L (-1, 1) , M (5, 2) , and N (1, -5) 43. A (-4, 2) , M (-2, 4) , C (4, 2) and D (2, -4) 9- 5 Effects of Changing Dimensions Proportionally 627 627 �� Probability Probability An experiment is an activity in which results are observed. Each result of an experiment is called an outcome. The sample space is the set of all outcomes of an experiment. An event is any set of outcomes. See Skills Bank page S77 The probability of an event is a number from 0 to 1 that tells you how likely the event is to happen. The closer the probability is to 0, the less likely the event is to happen. The closer it is to 1, the more likely the event is to happen. An experiment is fair if all outcomes are equally likely. The theoretical probability of an event is the ratio of the number of outcomes in the event to the number of outcomes in the sample space. P (E) = number of outcomes in event E ___ number of possible outcomes Example 1 A fair number cube has six faces, numbered 1 through 6. An experiment consists of rolling the number cube. A What is the sample space of the experiment? The sample space has 6 possible outcomes. The outcomes are 1, 2, 3, 4, 5, and 6. B What is the probability of the event “rolling a 4”? The event “rolling a 4” contains only 1 outcome. The probability is P (E) = number of outcomes in event E___ number of possible outcomes = 1_ . 6 C What are the outcomes in the event “rolling an odd number”? What is the probability of rolling an odd number? The event “rolling an odd number” contains 3 outcomes. The outcomes are 1, 3, and 5. The probability is P (E) = number of outcomes in event E = 3 _ = 1 _ ___ . 2 6 number of possible outcomes If two events A and B have no outcomes in common, then the probability that A or B will happen is P (A) + P (B) . The complement of an event is the set of outcomes that are not in the event. If the probability of an event is p, then the probability of the complement of the event is 1 - p. 628 628 Chapter 9 Extending Perimeter, Circumference, and Area Example 2 The tiles shown below are placed in a bag. An experiment consists of drawing a tile at random from the bag. A What is the sample space of the experiment? The sample space has 9 possible outcomes. The outcomes are 1, 2, 3, 4, A, B, C, D, E, and F. B What is the probability of choosing a 3 or a vowel? The event “choosing a 3” contains only 1 outcome. The probability is P (A) = number of outcomes in event A ___ number of possible outcomes = 1 _ . 9 The event “choosing a vowel” has 2 outcomes, A and E. The probability is P (B) = number of outcomes in event B ___ number of possible outcomes = The probability of choosing a 3 or a vowel is 1 _ . 3 9 9 9 C What is the probability of not choosing a letter? The event “choosing a letter” contains 5 outcomes, A, B, C, D, and E. The probability is P (E) = number of outcomes in event E ___ number of possible outcomes = 5 _ . 9 The event of not choosing a letter is the complement of the event of choosing a letter. The probability of not choosing a letter is 1 - 5 __ 9 = 4 __ 9 . Try This TAKS Grades 9–11 Obj. 8, 9 An experiment consists of randomly choosing one of the given shapes. 1. What is the probability of choosing a circle? 2. What is the probability of choosing a shape whose area is 36 cm 2 ? 3. What is the probability of choosing a quadrilateral or a triangle? 4. What is the probability of not choosing a triangle? On Track for TAKS 629 629 �� 9-6 Geometric Probability TEKS G.8.A Congruence and the geometry of size: find areas of ... regular polygons, circles, and composite figures. Objectives Calculate geometric probabilities. Use geometric probability to predict results in realworld
situations. Vocabulary geometric probability Why learn this? You can use geometric probability to estimate how long you may have to wait to cross a street. (See Example 2.) Remember that in probability, the set of all possible outcomes of an experiment is called the sample space. Any set of outcomes is called an event. If every outcome in the sample space is equally likely, the theoretical probability of an event is P = _____________________________ number of outcomes in the event number of outcomes in the sample space . Geometric probability is used when an experiment has an infinite number of outcomes. In geometric probability , the probability of an event is based on a ratio of geometric measures such as length or area. The outcomes of an experiment may be points on a segment or in a plane figure. Three models for geometric probability are shown below. Model Length Angle Measure Area Geometric Probability Example Sample space All points on ̶̶̶ AD All points in the circle All points in the rectangle ̶̶ BC All points in the All points in the triangle All points on shaded region P = BC _ AD P = measure of angle __ 360° P = area of triangle __ area of rectangle Event Probability E X A M P L E 1 Using Length to Find Geometric Probability A point is chosen randomly on ̶̶ AD . Find the probability of each event. If an event has a probability p of occurring, the probability of the event not occurring is 1 - p. A The point is on ̶̶ AC . P = AC _ AD = 7 _ 12 B The point is not on ̶̶ AB . First find the probability that the point is on P ( ̶̶ AB ) = AB _ AD = 4 _ 12 = 1 _ 3 ̶̶ AB . Subtract from 1 to find the probability that the point is not on P (not on ̶̶ = 2 _ AB ) = 1 - 1 _ 3 3 ̶̶ AB . 630 630 Chapter 9 Extending Perimeter, Circumference, and Area �� A point is chosen randomly on Find the probability of each event. ̶̶ AD . C The point is on P ( ̶̶ AB or ̶̶ CD ) = P ( ̶̶ ̶̶ CD . AB or ̶̶ AB ) + P ( ̶̶ CD ) = 4 _ 12 + 5 _ 12 = 9 _ 12 = 3 _ 4 1. Use the figure above to find the probability that the point is ̶̶ BD . on E X A M P L E 2 Transportation Application A stoplight has the following cycle: green for 25 seconds, yellow for 5 seconds, and red for 30 seconds. A What is the probability that the light will be yellow when you arrive? To find the probability, draw a segment to represent the number of seconds that each color light is on. P = 5 _ 60 = 1 _ 12 ≈ 0.08 The light is yellow for 5 out of every 60 seconds. B If you arrive at the light 50 times, predict about how many times you will have to stop and wait more than 10 seconds. In the model, the event of stopping and waiting more than 10 seconds is represented by a segment that starts at C and ends 10 units from D. The probability of stopping and waiting more than 10 seconds is P = 20 __ 60 = 1 __ 3 . If you arrive at the light 50 times, you will probably stop and wait more than 10 seconds about 1 __ 3 (50) ≈ 17 times. 2. Use the information above. What is the probability that the light will not be red when you arrive? E X A M P L E 3 Using Angle Measures to Find Geometric Probability Use the spinner to find the probability of each event. A the pointer landing on red = 2 _ 9 P = 80 _ 360 The angle measure in the red region is 80°. B the pointer landing on purple or blue = 135 _ 360 75 +60 _ 360 = 3 _ 8 P = The angle measure in the purple region is 75°. The angle measure in the blue region is 60°. In Example 3C, you can also find the probability of the pointer landing on yellow, and subtract from 1. C the pointer not landing on yellow P = 360 - 100 _ 360 = 13 _ 18 = 260 _ 360 The angle measure in the yellow region is 100°. Substract this angle measure from 360°. 3. Use the spinner above to find the probability of the pointer landing on red or yellow. 9- 6 Geometric Probability 631 631 �� Geometric Probability I like to write a probability as a percent to see if my answer is reasonable. The probability of the pointer landing on red is 80° ____ = 2 __ ≈ 22%. 9 360° The angle measure is close to 90°, which is 25% of the circle, so the answer is reasonable. Jeremy Denton Memorial High School E X A M P L E 4 Using Area to Find Geometric Probability Find the probability that a point chosen randomly inside the rectangle is in each given shape. Round to the nearest hundredth. A the equilateral triangle The area of the triangle is A = 1 _ aP 2 = 1 _ (6) (36 √  3 ) ≈ 187 m 2 . 2 The area of the rectangle is A = bh = 45 (20) = 900 m 2 . The probability is P = 187 _ 900 ≈ 0.21. B the trapezoid The area of the trapezoid is 3 + 12) (10) = 75 m 2 . 2 The area of the rectangle is A = bh = 45 (20) = 900 m 2. The probability is P = 75 _ 900 ≈ 0.08. C the circle The area of the circle is ) = 36π ≈ 113.1 m 2 . The area of the rectangle is A = bh = 45 (20) = 900 m 2 . The probability is P = 113.1 _ ≈ 0.13. 900 4. Use the diagram above. Find the probability that a point chosen randomly inside the rectangle is not inside the triangle, circle, or trapezoid. Round to the nearest hundredth. 632 632 Chapter 9 Extending Perimeter, Circumference, and Area �� THINK AND DISCUSS 1. Explain why the ratio used in theoretical probability cannot be used to find geometric probability. 2. A spinner is one-half red and one-third blue, and the rest is yellow. How would you find the probability of the pointer landing on yellow? 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, give an example of the geometric probability model. 9-6 Exercises Exercises KEYWORD: MG7 9-6 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Give an example of a model used to find geometric probability. 630 A point is chosen randomly on each event. ̶̶ WZ . Find the probability of 2. The point is on 4. The point is on ̶̶ XZ . ̶̶̶ WX or ̶̶ YZ . 3. The point is not on ̶̶̶ WY . 5. The point is on ̶̶ XY . . 631 Transportation A bus comes to a station once every 10 minutes and waits at the station for 1.5 minutes. 6. Find the probability that the bus will be at the station when you arrive. 7. If you go to the station 20 times, predict about how many times you will have to wait less than 3 minutes Use the spinner to find the probability of each event. p. 631 8. the pointer landing on green 9. the pointer landing on orange or blue 10. the pointer not landing on red 11. the pointer landing on yellow or blue . 632 Multi-Step Find the probability that a point chosen randomly inside the rectangle is in each shape. Round to the nearest hundredth. 12. the triangle 13. the trapezoid 14. the square 15. the part of the rectangle that does not include the square, triangle, or trapezoid 9- 6 Geometric Probability 633 633 �� Independent Practice For See Exercises Example 16–19 20–22 23–26 27–30 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S21 Application Practice p. S36 PRACTICE AND PROBLEM SOLVING A point is chosen randomly on Find the probability of each event. Round to the nearest hundredth. ̶̶̶ HM . 16. The point is on 18. The point is on ̶̶ JK . ̶̶ HJ or ̶̶ KL . 17. The point is not on 19. The point is not on ̶̶̶ LM . ̶̶ JK or ̶̶̶ LM . Communications A radio station gives a weather report every 15 minutes. Each report lasts 45 seconds. Suppose you turn on the radio at a random time. 20. Find the probability that the weather report will be on when you turn on the radio. 21. Find the probability that you will have to wait more than 5 minutes to hear the weather report. 22. If you turn on the radio at 50 random times, predict about how many times you will have to wait less than 1 minute before the start of the next weather report. Use the spinner to find the probability of each event. 23. the pointer landing on red 24. the pointer landing on yellow or blue 25. the pointer not landing on green 26. the pointer landing on red or green Multi-Step Find the probability that a point chosen randomly inside the rectangle is in each shape. Round to the nearest hundredth, if necessary. 27. the equilateral triangle 28. the square 29. the part of the circle that does not include the square 30. the part of the rectangle that does not include the square, circle, or triangle 31. /////ERROR ANALYSIS///// In the spinner at right, the angle measure of the red region is 90°. The angle measure of the yellow region is 135°, and the angle measure of the blue region is 135°. Which value of the probability of the spinner landing on yellow is incorrect? Explain. Algebra A point is chosen randomly inside rectangle ABCD with vertices A (2, 8) , B (15, 8) , C (15, 1) , and D (2, 1) . Find the probability of each event. Round to the nearest hundredth. 32. The point lies in △KLM with vertices K (4, 3) , L (5, 7) , and M (9, 5) . 33. The point does not lie in ⊙P with center P (2, 5) and radius 3. (Hint: draw the rectangle and circle.) 634 634 Chapter 9 Extending Perimeter, Circumference, and Area �� Algebra A point is chosen at random in the coordinate plane such that -5 ≤ x ≤ 5 and -5 ≤ y ≤ 5. Find the probability of each event. Round to the nearest hundredth. Sports 34. The point is inside the parallelogram. 35. The point is inside the circle. 36. The point is inside the triangle or the circle. 37. The point is not inside the triangle, the parallelogram, or the circle. 38. Sports The point value of each region of an Olympic archery target is shown in the diagram. The outer diameter of each ring is 12.2 cm greater than the inner diameter. a. What is the probability of hitting the center? b. What is the probability of hitting a blue or black r
ing? c. What is the probability of scoring higher than five points? d. Write About It In an actual event, why might the probabilities be different from those you calculated in parts a, b, and c? Olympic archers stand 70 m from their targets. From that distance, the target appears about the size of the head of a thumbtack held at arm’s length. Source: www.olympic.org A point is chosen randomly in each figure. Describe an event with a probability of 1 __ . 2 39. 41. 40. 42. If a fly lands randomly on the tangram, what is the probability that it will land on each of the following pieces? a. the blue parallelogram b. the medium purple triangle c. the large yellow triangle d. Write About It Do the probabilities change if you arrange the tangram pieces differently? Explain. 43. Critical Thinking If a rectangle is divided into 8 congruent regions and 4 of them are shaded, what is the probability that you will randomly pick a point in the shaded area? Does it matter which four regions are shaded? Explain. 44. This problem will prepare you for the Multi-Step TAKS Prep on page 638. A carnival game board consists of balloons that are 3 inches in diameter and are attached to a rectangular board. A player who throws a dart at the board wins a prize if the dart pops a balloon. a. Find the probability of winning if there are 40 balloons on the board. b. How many balloons must be on the board for the probability of winning to be at least 0.25? 9- 6 Geometric Probability 635 635 �� 45. What is the probability that a ball thrown randomly at the backboard of the basketball goal will hit the inside rectangle? 0.14 0.21 0.26 0.27 46. Point B is between A and C. If AB = 18 inches and BC = 24 inches, what is the probability that a point chosen at random is on ̶̶ AB ? 0.18 0.43 0.57 0.75 47. A skydiver jumps from an airplane and parachutes down to the 70-by-100-meter rectangular field shown. What is the probability that he will miss all three targets? 0.014 0.180 0.089 0.717 �� 48. Short Response A spinner is divided into 12 congruent regions, colored red, blue, and green. Landing on red is twice as likely as landing on blue. Landing on blue and landing on green are equally likely. a. What is the probability of landing on green? Show your work or explain in words how you got your answer. b. How many regions of the spinner are colored green? Explain your reasoning. CHALLENGE AND EXTEND 49. If you randomly choose a point on the grid, what is the probability that it will be in a red region? 50. You are designing a target that is a square inside an 18 ft by 24 ft rectangle. What size should the square be in order for the target to have a probability of 1 __ 3 ? to have a probability of 3 __ 4 ? 51. Recreation How would you design a spinner so that 1 point is earned for landing on yellow, 3 points for landing on blue and 6 points for landing on red? Explain. SPIRAL REVIEW Simplify each expression. (Previous course) 52. (3 x 2 y) (4 x 3 y 2 ) 55. Given: A (0, 4) , B (4, 6) , C (4, 2) , D (8, 8) , and E (8, 0) 2 53. (2 m 5 ) Prove: △ABC ∼ △ADE (Lesson 7-6) 54. -8 a 4 b 6 _ 2a b 3 Find the shaded area. Round to the nearest tenth, if necessary. (Lesson 9-3) 56. 57. 636 636 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-6 Use with Lesson 9-6 Activity Use Geometric Probability to Estimate π In this lab, you will use geometric probability to estimate π. The squares in the grid below are the same width as the diameter of a penny: 0.75 in., or 19.05 mm. TEKS G.8.A Congruence and the geometry of size: find areas of ... regular polygons, circles, and composite figures. 1 Toss a penny onto the grid 20 times. Let x represent the number of times the penny lands touching or covering an intersection of two grid lines. 2 Estimate π using the formula π ≈ 4 · x_ . 20 Try This 1. How close is your result to π? Average the results of the entire class to get a more accurate estimate. 2. In order for a penny to touch or cover an intersection, the center of the penny can land anywhere in the shaded area. a. Find the area of the shaded region. (Hint: Each corner part is one fourth of the circle. Put the four corner parts together to form a circle with radius r.) b. Find the area of the square. c. Write the expressions as a ratio and simplify to determine the probability of the center of the penny landing in the shaded area. 3. Explain why the formula in the activity can be used to estimate π. 9-6 Geometry Lab 637 637 � SECTION 9B Applying Geometric Formulas Step Right Up! A booster club organizes a carnival to raise money for sports uniforms. The carnival features several games that give visitors chances to win prizes. �� 1. The balloon game consists of 15 balloons attached to a vertical rectangular board with the dimensions shown. Each balloon has a diameter of 4 in. Each player throws a dart at the board and wins a prize if the dart pops a balloon. Assuming that all darts hit the board at random, what is the probability of winning a prize? �� 2. The organizers decide to make the game easier, so they double the diameter of the balloons. How does this affect the probability of winning? 3. The bean toss consists of a horizontal rectangular board that is divided into a grid. The board has coordinates (0, 0) , (100, 0) , (100, 60) , and (0, 60) . A quadrilateral on the board has coordinates A (60,0) , B (100, 30) , C (40, 60) , and D (0, 40) . Each player tosses a bean onto the board and wins a prize if the bean lands inside quadrilateral ABCD. Find the probability of winning a prize. 4. Of the three games described in Problems 1, 2, and 3, which one gives players the best chance of winning a prize? 638 638 Chapter 9 Extending Perimeter, Circumference, and Area SECTION 9B Quiz for Lessons 9-4 Through 9-6 9-4 Perimeter and Area in the Coordinate Plane Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 1. A (-2, 2) , B (2, 4) , C (2, -4) , D (-2, -2) 2. E (-1, 5) , F (3, 5) , G (3, -3) , H (-1, -3) Find the area of each polygon with the given vertices. 3. J (-3, 3) , K (2, 2) , L (-1, -3) , M (-4, -1) 4. N (-3, 1) , P (3, 3) , Q (5, 1) , R (2, -4) 9-5 Effects of Changing Dimensions Proportionally Describe the effect of each change on the perimeter and area of the given figure. 5. The side length of the square is tripled. 6. The diagonals of a rhombus in which d 1 = 3 ft and d 2 = 9 ft are both multiplied by 1 __ 3 . 7. The base and height of the rectangle are both doubled. 8. The base and height of a right triangle with base 15 in. and height 8 in. are multiplied by 1 __ 3 . 9. A square has vertices (-1, 2) , (3, 2) , (3, -2) , and (-1, -2) . If you quadruple the area, what happens to the side length? 10. A restaurant sells pancakes in two sizes, silver dollar and regular. The silver-dollar pancakes have a 4-inch diameter and require 1 __ 8 cup of batter per pancake. The diameter of a regular pancake is 2.5 times the diameter of a silver-dollar pancake. About how much batter is required to make a regular pancake? 9-6 Geometric Probability Use the spinner to find the probability of each event. 11. the pointer landing on red 12. the pointer landing on red or yellow 13. the pointer not landing on green 14. the pointer landing on yellow or blue 15. A radio station plays 12 commercials per hour. Each commercial is 1 minute long. If you turn on the radio at a random time, find the probability that a commercial will be playing. Ready to Go On? 639 639 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary apothem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 center of a circle . . . . . . . . . . . . . . . . . . . . . . 600 composite figure . . . . . . . . . . . . . . . . . . . . . . 606 center of a regular polygon . . . . . . . . . . . . 601 geometric probability . . . . . . . . . . . . . . . . . 630 central angle of a regular polygon . . . . . . 601 Complete the sentences below with vocabulary words from the list above. 1. A(n) ? is the length of a segment perpendicular to a side of a regular polygon. ̶̶̶̶ 2. The point that is equidistant from every point on a circle is the ? . ̶̶̶̶ 3. ? is based on a ratio of geometric measures. ̶̶̶̶ 9-1 Developing Formulas for Triangles and Quadrilaterals (pp. 589–597) E X A M P L E S EXERCISES TEKS G.1.B, G.3.C, G.3.E, G.5.A, G.8.C Find each measurement. ■ the perimeter of a square in which A = 36 in 2 Find each measurement. 4. the area of a square in which P = 36 in. A = s 2 = 36 in 2 S = √  36 = 6 in. P = 4s = 4 ⋅ 6 = 24 in. Use the Area Formula to find the side length. ■ the area of the triangle By the Pythagorean Theorem, 8 2 + b 2 = 17 2 64 + b 2 = 289 b 2 = 225, so b = 15 ft. bh = 1 _ A = 1 _ (15) (8) = 60 ft 2 2 2 ■ the diagonal d 2 of a rhombus in which A = 6 x 3 y 3 m and 4 x 2 y) d 2 2 d 2 = 3x y 2 Substitute the given values. Solve for d 2 . 5. the perimeter of a rectangle in which b = 4 cm and A = 28 cm 2 6. the height of a triangle in which A = 6 x 3 y in 2 and b = 4xy in. 7. the height of the trapezoid, in which A = 48xy ft 2 8. the area of a rhombus in which d 1 = 21 yd and d 2 = 24 yd 9. the diagonal d 2 of the rhombus, in which A = 630 x 3 y 7 in 2 10. the area of a kite in which d 1 = 32 m and d 2 = 18 m 640 640 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-2 Developing Formulas for Circles and Regular Polygons (pp. 600–605) E X A M P L E S Find each measurement. ■ the circumference and area of ⊙B in terms of π C = 2πr = 2π (5xy) = 10xyπ m A = π r 2 = π (5xy) 2 = 25 x 2 y 2 π m 2 EXERCISES TEKS G.5.A, G.8.A, G.8.C Find each measurement. Round to the nearest tenth, if necessary. 11. the circumference of ⊙G 12. the area of ⊙J
in which C = 14π yd ■ the area, to the nearest tenth, of a regular 13. the diameter of ⊙K in which A = 64 x 2 π m 2 hexagon with apothem 9 yd By the 30°-60°-90° Triangle Theorem, x = 9 √  3 ____ 3 = 3 √3 . So s = 2x = 6 √  3 , and P = 6 (6 √  3 ) = 36 √  3 . aP = 1 _ A = 1 _ (9) (36 √  3 ) = 162 √  3 ≈ 280.6 yd 2 2 2 14. the area of a regular pentagon with side length 10 ft 15. the area of an equilateral triangle with side length 4 in. 16. the area of a regular octagon with side length 8 cm 17. the area of the square 9-3 Composite Figures (pp. 606–612) TEKS G.8.A E X A M P L E EXERCISES ■ Find the shaded area. Round to the nearest tenth, if necessary. Find the shaded area. Round to the nearest tenth, if necessary. 18. 19. The area of the triangle is A = 1 _ (18) (20) = 180 cm 2 . 2 The area of the parallelogram is A = bh = 20 (10) = 200 cm 2 . The area of the figure is the sum of the two areas. 180 + 200 = 380 cm 2 20. Study Guide: Review 641 641 �� 9-4 Perimeter and Area in the Coordinate Plane (pp. 616–621) TEKS G.7.A, G.7.B EXERCISES G.8.A Estimate the area of each irregular shape. 21. 22. Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 23. H (0, 3) , J (3, 0) , K (0, -3) , L (-3, 0) 24. M (-2, 5) , N (3, -2) , P (-2, -2) 25. A (-2, 3) , B (2, 3) , C (4, -1) , D (-4, -1) 26. E (-1, 3) , F (3, 3) , G (1, 0) , H (-3, 0) Find the area of the polygon with the given vertices. 27. Q (1, 4) , R (4, 3) , S (2, -4) , T (-3, -2) 28. V (-2, 2) , W (4, 0) , X (2, -3) , Y (-3, 0) 29. A (1, 4) , B (2, 3) , C (0, -3) , D (-2, -1) 30. E (-1, 2) , F (2, 0) , G (1, -3) , H (-4, -1) E X A M P L E S ■ Estimate the area of the irregular shape. The shape has 28 approximately whole squares and 17 approximately half squares. The total area is approximately 28 + 1 _ (17) = 36.5 units 2 . 2 ■ Draw and classify the polygons with vertices R (2, 4) , S (3, 1) , T (2, -2) , and U (1, 1) . Find the perimeter and area of the polygon. RSTU appears to be a rhombus. Verify this by showing that the four sides are congruent. By the Distance Formula, UR = RS = ST = TU = √  10 units. The perimeter is 4 √  10 units. US ⋅ RT = 1 _ d 1 d 2 = 1 _ The area is A = 1 _ (2 ⋅ 6) 2 2 2 = 6 units 2 . ■ Find the area of the polygon with vertices A (-3, 4) , B (2, 3) , C (0, -2) , and D (-5, -1) . area of rectangle: 7 (6) = 42 units 2 area of triangles: a: A = 1 _ (2) (5) 2 = 5 units 2 b: A = 1 _ (5) (1) 2 = 2.5 units 2 c: A = 1 _ (2) (5) = 5 units 2 2 d: A = 1 _ (5) (1) = 2.5 units 2 2 area of polygon: A = 42 - 5 - 2.5 - 5 - 2.5 = 27 units 2 642 642 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-5 Effects of Changing Dimensions Proportionally (pp. 622–627) E X A M P L E EXERCISES TEKS G.5.A, G.5.B, G.11.D ■ The base and height of a rectangle with base 10 cm and height 15 cm are both doubled. Describe the effect on the area and perimeter of the figure. original: P = 2b + 2h = 2 (10) + 2 (15) = 50 cm A = bh = 10 (15) = 150 cm 2 doubled: P = 2b + 2h = 2 (20) + 2 (30) = 100 cm A = bh = 20 (30) = 600 cm 2 The perimeter increases by a factor of 2. The area increases by a factor of 4. Describe the effect of each change on the perimeter or circumference and area of the given figure. 31. The base and height of the triangle with vertices X (-1, 3) , Y (-3, -2) , and Z (2, -2) are tripled. 32. The side length of the square with vertices P (-1, 1) , Q (3, 1) , R (3, -3) , and S (-1, -3) is doubled. 33. The radius of ⊙A with radius 11 m is multiplied by 1 _ . 2 34. The base and height of a triangle with base 8 ft and height 20 ft are both multiplied by 4. 9-6 Geometric Probability (pp. 630–636) TEKS G.8.A E X A M P L E S EXERCISES A point is chosen randomly on probability of each event. ̶̶ WZ . Find the A point is chosen randomly on probability of each event. ̶̶ AD . Find the ■ The point is on P (XZ) = XZ _ WZ ■ The point is on = 5 _ 6 ̶̶ XZ . = 15 _ 18 ̶̶̶ ̶̶ WX or YZ . ̶̶̶ WX ) + P ( ̶̶ YZ ) = P ( P ( = 10 _ 18 ̶̶̶ WX or = 5 _ 9 ̶̶ YZ ) = 3 _ 18 + 7 _ 18 ■ Find the probability that a point chosen randomly inside the rectangle is inside the equilateral triangle. area of rectangle A = bh = 20 (10) = 200 ft 2 area of triangle ( 5 √  3 aP = = 43.3 _ 200 ≈ 0.22 ) (30) = 25 √  3 ≈ 43.3 ft 2 35. The point is on ̶̶ AB . 36. The point is not on 37. The point is on 38. The point is on ̶̶ CD . ̶̶ CD . ̶̶ CD . ̶̶ AB or ̶̶ BC or Find the probability that a point chosen randomly inside the 40 m by 24 m rectangle is in each shape. Round to the nearest hundredth. 39. the regular hexagon 40. the triangle 41. the circle or the triangle 42. inside the rectangle but not inside the hexagon, triangle, or circle Study Guide: Review 643 643 �� Find each measurement. 1. the height h of a triangle in which A = 12 x 2 y ft 2 and b = 3x ft 2. the base b 1 of a trapezoid in which A = 161.5 cm 2 , h = 17 cm, and b 2 = 13 cm 3. the area A of a kite in which d 1 = 25 in. and d 2 = 12 in. 4. Find the circumference and area of ⊙A with diameter 12 in. Give your answers in terms of π. 5. Find the area of a regular hexagon with a side length of 14 m. Round to the nearest tenth. Find the shaded area. Round to the nearest tenth, if necessary. 6. 7. 8. The diagram shows a plan for a pond. Use a composite figure to estimate the pond’s area. The grid has squares with side lengths of 1 yd. 9. Draw and classify the polygon with vertices A (1, 5) , B (2, 3) , C (-2, 1) , and D (-3, 3) . Find the perimeter and area of the polygon. Find the area of each polygon with the given vertices. 10. E (-3, 4) , F (1, 1) , G (0, -4) , H (-4, 1) 11. J (3, 4) , K (4, -1) , L (-2, -4) , M (-3, 3) Describe the effect of each change on the perimeter or circumference and area of the given figure. 12. The base and height of a triangle with base 10 cm and height 12 cm are multiplied by 3. 13. The radius of a circle with radius 12 m is multiplied by 1 _ . 2 14. A circular garden plot has a diameter of 21 ft. Janelle is planning a new circular plot with an area 1 __ 9 as large. How will the circumference of the new plot compare to the circumference of the old plot? A point is chosen randomly on probability of each event. ̶̶ NS . Find the 15. The point is on ̶̶̶ NQ . 16. The point is not on ̶̶̶ NQ or 17. The point is on ̶̶ RS . ̶̶ QR . 18. A shuttle bus for a festival stops at the parking lot every 18 minutes and stays at the lot for 2 minutes. If you go to the festival at a random time, what is the probability that the shuttle bus will be at the parking lot when you arrive? 644 644 Chapter 9 Extending Perimeter, Circumference, and Area �� FOCUS ON SAT STUDENT-PRODUCED RESPONSES There are two types of questions in the mathematics sections of the SAT: multiple-choice questions, where you select the correct answer from five choices, and student-produced response questions, for which you enter the correct answer in a special grid. On the SAT, the student-produced response items do not have a penalty for incorrect answers. If you are uncertain of your answer and do not have time to rework the problem, you should still grid in the answer you have. You may want to time yourself as you take this practice test. It should take you about 9 minutes to complete. 1. A triangle has two angles with a measure of 60° and one side with a length of 12. What is the perimeter of the triangle? 4. Three overlapping squares and the coordinates of a corner of each square are shown above. What is the y-intercept of line ℓ? 2. The figure above is composed of four congruent trapezoids arranged around a shaded square. What is the area of the shaded square? 3. If △PQR ∼ △STU, m∠P = 22°, m∠Q = 57°, and m∠U = x°, what is the value of x? 5. In the figure above, what is the value of y? 6. The three angles of a triangle have measures 12x°, 3x°, and 7y°, where 7y > 60. If x and y are integers, what is the value of x? College Entrance Exam Practice 645 645 �� Any Question Type: Use a Formula Sheet When you take a standardized mathematics test, you may be given a formula sheet or a mathematics chart that accompanies the test. Although many common formulas are given on these sheets, you still need to know when the formulas are applicable, and what the variables in the formulas represent. Mathematics Chart Perimeter rectangle P = 2ℓ + 2w or P = 2 (ℓ + w) Circumference circle C = 2πr or C = πd Area rectangle A = ℓw or A = bh triangle bh or A = bh _ A = 1 _ 2 2 trapezoid or circle A = π r 2 Multiple Choice In the figure, a rectangle is inscribed in a circle. Which best represents the shaded area to the nearest tenth of a square meter? 3.4 m 2 7.6 m 2 12.6 m 2 17.2 m 2 Which formula(s) do I need? area of a circle, area of a rectangle What do I substitute for each variable in the formulas? To use the formula for the area of a circle, I need to know the radius. The diameter of the circle is 5 m, so the radius is 2.5 m. I should substitute 2.5 for r and 3.14 for π. To use the formula for the area of a rectangle, I need to know its base and height. The base b is 4 m. To find the height, I can use the Pythagorean Theorem. 4 2 + h 2 = 5 2 16 + h 2 = 25 h 2 = 9 h = 3 What are the areas of the shapes? circle: A = π r 2 = π (2.5) 2 = 6.25π m 2 rectangle: A = bh = 4 (3) = 12 m 2 What do I do with the areas to find the answer? shaded area = area of circle - area of rectangle = 6.25π - 12 ≈ 7.6 m 2 Choice B is the correct answer. 646 646 Chapter 9 Extending Perimeter, Circumference, and Area �� Read each test problem and answer the questions that follow. Use the formula sheet below, if applicable. �� Before you begin a test, quickly review the formulas included on your formula sheet. Perimeter rectangle P
= 2ℓ + 2w or P = 2 (ℓ + w) Circumference circle Area rectangle triangle trapezoid C = 2πr or C = πd A = ℓw or A = bh bh or A = bh or circle A = π r 2 Pi π π ≈ 3.14 or π ≈ 22 _ 7 Item A The circumference of a circle is 48π meters. What is the radius in meters? 6.9 meters 24 meters 12 meters 36 meters 1. Which formula would you use to solve this problem? 2. After substituting the variables in the formula, what would you need to do to find the correct answer? Item B Gridded Response The area of a trapezoid is 171 square meters. The height is 9 meters, and one base length is 23 meters. What is the other base length of the trapezoid in meters? Item C Gridded Response The area of the rectangle is 48 square miles. What is the perimeter in miles? 5. What formula(s) would you use to solve this problem? 6. What would you substitute for each variable in the formula? 7. After substituting the variables in the formula, what would you need to do to find the correct answer? Item D Gridded Response A point is chosen randomly inside the rectangle. What is the probability that the point does not lie inside the triangle or the trapezoid? Round to the nearest hundredth. 8. Which formulas would you use to solve this problem? 9. What would you substitute for each variable 3. What formula(s) would you use to solve this in the formula(s)? problem? 4. What would you substitute for each variable in the formula? 10. After substituting the variables in the formula, what would you need to do to find the correct answer? TAKS Tackler 647 647 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–9 Multiple Choice 1. The floor of a tent is a regular hexagon. If the side length of the tent floor is 5 feet, what is the area of the floor? Round to the nearest tenth. 5. If ABCD is a rhombus in which m∠1 = (x + 15) ° and m∠2 = (2x + 12) °, what is the value of x? 32.5 square feet 65.0 square feet 75.0 square feet 129.9 square feet 2. If J is on the perpendicular bisector of ̶̶ KL , what is 3 9 18 21 the length of KL? 6. What is the area of the shaded portion of the rectangle? 34 square centimeters 36 square centimeters 38 square centimeters 50 square centimeters 7. If △XYZ is isosceles and m∠Y > 100°, which of the following must be true? m∠X < 40° m∠X > 40° ̶̶ XZ ≅ ̶̶ XY ≅ ̶̶ YZ ̶̶ XZ 8. The Eiffel Tower in Paris, France, is 300 meters tall. The first level of the tower has a height of 57 meters. A scale model of the Eiffel Tower in Shenzhen, China, is 108 meters tall. What is the height of the first level of the model? Round to the nearest tenth. 15.8 meters 20.5 meters 56.8 meters 61.6 meters 12 18 24 36 3. What is the length of ̶̶ VY ? 1.6 2 2.5 4 4. A sailor on a ship sights the light of a lighthouse at an angle of elevation of 15°. If the light in the lighthouse is 189 feet higher than the sailor’s line of sight, what is the horizontal distance between the ship and the lighthouse? Round to the nearest foot. 49 feet 51 feet 705 feet 730 feet 648 648 Chapter 9 Extending Perimeter, Circumference, and Area �� There is often more than one way to find a missing side length or angle measure in a figure. For example, you might be able to find a side length of a right triangle by using either the Pythagorean Theorem or a trigonometric ratio. Check your answer by using a different method than the one you originally used. 9. The lengths of both bases of a trapezoid are tripled. What is the effect of the change on the area of the trapezoid? The area remains the same. The area is tripled. The area increases by a factor of 6. The area increases by a factor of 9. 10. If ∠1 and ∠2 form a linear pair, which of the following must also be true about these angles? They are adjacent. They are complementary. They are congruent. They are vertical. 11. In △ABC, AB = 8, BC = 17, and AC = 2x + 1. Which of the following is a possible value of x? 3 4 9 12 12. Which line is parallel to the line with the equation y = -3x + 4? y - 3x = 8 4y - 12x = 1 3y - x = 3 2y + 6x = 5 Gridded Response 13. What is the radius of a circle in inches if the ratio of its area to its circumference is 2.5 square inches : 1 inch? 14. △JLM ∼ △RST. If JL = 5, LM = 4, RS = 3x - 1, and ST = x + 2, what is the value of x? 15. If the two diagonals of a kite measure 16 centimeters and 10 centimeters, what is the area of the kite in square centimeters? STANDARDIZED TEST PREP Short Response 16. Two gas stations on a straight highway are 8 miles apart. If a car runs out of gas at a random point between the two gas stations, what is the probability that the car will be at least 2 miles from either gas station? Draw a diagram or write and explanation to show how you determined your answer. 17. Use the figure below to find each measure. Show your work or explain in words how you found your answers. Round the angle measure to the nearest degree. a. m∠A b. AC 18. Given that ̶̶ DE , ̶̶ DF , and ̶̶ EF are midsegments of △ABC, determine m∠C to the nearest degree. Show your work or explain in words how you determined your answer. Extended Response 19. Quadrilateral LMNP has vertices at L (1, 4) , M (4, 4) , N (1, 0) and P (-2, 0) . a. Write a coordinate proof showing that LMNP is a parallelogram. b. Draw a rectangle with the same area as figure LMNP. Explain how you know that the figures have the same area. c. Does the rectangle you drew have the same perimeter as figure LMNP? Explain. Cumulative Assessment, Chapters 1–9 649 649 �� Spatial Reasoning 10A Three-Dimensional Figures 10-1 Solid Geometry 10-2 Representations of Three-Dimensional Figures Lab Use Nets to Create Polyhedrons 10-3 Formulas in Three Dimensions 10B Surface Area and Volume 10-4 Surface Area of Prisms and Cylinders Lab Model Right and Oblique Cylinders 10-5 Surface Area of Pyramids and Cones 10-6 Volume of Prisms and Cylinders 10-7 Volume of Pyramids and Cones 10-8 Spheres Lab Compare Surface Areas and Volumes Ext Spherical Geometry KEYWORD: MG7 ChProj Four Chromatic Gates by Herbert Bayer stands at Ervay and Federal Streets in downtown Dallas. 650 650 Chapter 10 Vocabulary Match each term on the left with a definition on the right. 1. equilateral A. the distance from the center of a regular polygon to a side of 2. parallelogram 3. apothem 4. composite figure the polygon B. a quadrilateral with four right angles C. a quadrilateral with two pairs of parallel sides D. having all sides congruent E. a figure made up of simple shapes, such as triangles, rectangles, trapezoids, and circles Find Area in the Coordinate Plane Find the area of each figure with the given vertices. 5. △ABC with A (0, 3) , B (5, 3) , and C (2, -1) 6. rectangle KLMN with K (-2, 3) , L (-2, 7) , M (6, 7) , and N (6, 3) 7. ⊙P with center P (2, 3) that passes through the point Q (-6, 3) Circumference and Area of Circles Find the circumference and area of each circle. Give your answers in terms of π. 8. 10. 9. Distance and Midpoint Formulas Find the length and midpoint of the segment with the given endpoints. 11. A (-3, 2) and B (5, 6) 13. E (0, 1) and F (-3, 4) 12. C (-4, -4) and D (2, -3) 14. G (2, -5) and H (-2, -2) Evaluate Expressions Evaluate each expression for the given values of the variables. 15. √A_ 16. 2A_ P π for A = 121π cm 2 for A = 128 ft 2 and P = 32 ft 17. √ c 2 - a 2 for a = 8 m and c = 17 m 18. 2A_ h - b 1 for A = 60 in 2 , b 1 = 8 in., and h = 6 in. Spatial Reasoning 651 651 �� Key Vocabulary/Vocabulario cone cylinder net polyhedron prism pyramid sphere cono cilindro plantilla poliedro prisma pirámide esfera surface area área total volume volumen Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following questions. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The word polyhedron begins with the root poly-. List some other words that begin with poly-. What do all of these words have in common? 2. The word cone comes from the root ko-, which means “to sharpen.” Think of sharpening a pencil. How do you think this relates to a cone? 3. What does the word surface mean? What do you think the surface area of a three-dimensional figure is? Geometry TEKS Les. 10-1 Les. 10-2 10-3 Geo. Lab Les. 10-3 Les. 10-4 10-4 Geo. Lab Les. 10-5 Les. 10-6 Les. 10-7 Les. 10-8 G.1.C Geometric structure* compare and contrast … Euclidean and non-Euclidean geometries G.6.A Dimensionality and the geometry of location* describe and draw the intersection of a given plane with various three-dimensional … figures ★ ★ ★ G.6.B Dimensionality and the geometry of location* ★ ★ ★ ★ ★ 10-8 Tech. Lab Ext. ★ use nets to represent and construct threedimensional geometric figures G.6.C Dimensionality and the geometry of location* use orthographic and isometric views of threedimensional geometric figures … ★ G.7.C Dimensionality and the geometry of location* ★ … use formulas involving length, slope, and midpoint G.8.D Congruence and the geometry of size* find ★ ★ ★ ★ ★ surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites … G.9.D Congruence and the geometry of size* ★ ★ ★ ★ ★ analyze the characteristics of polyhedra … G.11.D Similarity and the geometry of shape* ★ ★ ★ ★ ★ ★ describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed … * Knowledge and skills are written out completely on pages TX28–TX35. 652 652 Chapter 10 Writing Strategy: Draw Three-Dimensional Figures When you encounter a three-dimensional figure such as a cylinder, cone, sphere, prism, or pyramid, it may help you to make a quick sketch so that you can visualize its shape. Use these tips to help you draw quick sketches of three-dimensional figures. ��
�� Try This 1. Explain and show how to draw a cube, a prism with equal length, width, and height. �� 2. Draw a prism, starting with two hexagons. (Hint: Draw the �� hexagons as if you were viewing them at an angle.) 3. Draw a pyramid, starting with a triangle and a point above the triangle. Spatial Reasoning 653 653 10-1 Solid Geometry TEKS G.6.A Dimensionality and the geometry of location: describe and draw the intersection of a given plane with various three-dimensional geometric figures. Objectives Classify three-dimensional figures according to their properties. Use nets and cross sections to analyze threedimensional figures. Vocabulary face edge vertex prism cylinder pyramid cone cube net cross section Why learn this? Some farmers in Japan grow cube-shaped watermelons to save space in small refrigerators. Each fruit costs about the equivalent of U.S. $80. (See Example 4.) Three-dimensional figures, or solids, can be made up of flat or curved surfaces. Each flat surface is called a face . An edge is the segment that is the intersection of two faces. A vertex is the point that is the intersection of three or more faces. Three-Dimensional Figures TERM EXAMPLE A prism is formed by two parallel congruent polygonal faces called bases connected by faces that are parallelograms. A cylinder is formed by two parallel congruent circular bases and a curved surface that connects the bases. Also G.2.B, G.6.B, G.9.D A pyramid is formed by a polygonal base and triangular faces that meet at a common vertex. A cone is formed by a circular base and a curved surface that connects the base to a vertex. A cube is a prism with six square faces. Other prisms and pyramids are named for the shape of their bases. Triangular prism Rectangular prism Pentagonal prism Hexagonal prism Triangular pyramid Rectangular pyramid Pentagonal pyramid Hexagonal pyramid 654 654 Chapter 10 Spatial Reasoning �� E X A M P L E 1 Classifying Three-Dimensional Figures Classify each figure. Name the vertices, edges, and bases. A � B � � � � rectangular pyramid vertices: A, B, C, D, E edges: ̶̶ AB , ̶̶ BE , ̶̶ BC , ̶̶ CE , ̶̶ CD , ̶̶ DE ̶̶ AD , ̶̶ AE , � � cylinder vertices: none edges: none base: rectangle ABCD bases: ⊙P and ⊙Q Classify each figure. Name the vertices, edges, and bases. 1a. � 1b. � � � � � � A net is a diagram of the surfaces of a three-dimensional figure that can be folded to form the three-dimensional figure. To identify a three-dimensional figure from a net, look at the number of faces and the shape of each face. E X A M P L E 2 Identifying a Three-Dimensional Figure From a Net Describe the three-dimensional figure that can be made from the given net. A B The net has two congruent triangular faces. The remaining faces are parallelograms, so the net forms a triangular prism. The net has one square face. The remaining faces are triangles, so the net forms a square pyramid. Describe the three-dimensional figure that can be made from the given net. 2a. 2b. 10- 1 Solid Geometry 655 655 A cross section is the intersection of a three-dimensional figure and a plane. E X A M P L E 3 Describing Cross Sections of Three-Dimensional Figures Describe each cross section. A B The cross section is a triangle. The cross section is a circle. Describe each cross section. 3a. 3b. E X A M P L E 4 Food Application A chef is slicing a cube-shaped watermelon for a buffet. How can the chef cut the watermelon to make a slice of each shape? A a square B a hexagon Cut parallel to the bases. Cut through the midpoints of the edges. 4. How can a chef cut a cube-shaped watermelon to make slices that are triangles? THINK AND DISCUSS 1. Compare prisms and cylinders. 2. GET ORGANIZED Copy and complete the graphic organizer. 656 656 Chapter 10 Spatial Reasoning �� 10-1 Exercises Exercises KEYWORD: MG7 10-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary A ? has two circular bases. (prism, cylinder, or cone) ̶̶̶̶ Classify each figure. Name the vertices, edges, and bases. p. 655 2. � 3. � � � 4. 4 Describe the three-dimensional figure that can be made from the given net. p. 655 5 Describe each cross section. p. 656 8. 6. 9. 7. 10. 656 Art A sculptor has a cylindrical piece of clay. How can the sculptor slice the clay to make a slice of each given shape? 11. a circle 12. a rectangle Independent Practice For See Exercises Example 13–15 16–18 19–21 22–23 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S22 Application Practice p. S37 PRACTICE AND PROBLEM SOLVING Classify each figure. Name the vertices, edges, and bases. 13. 14. 15. Describe the three-dimensional figure that can be made from the given net. 16. 17. 18. 10- 1 Solid Geometry 657 657 �� Describe each cross section. 19. 20. 21. Architecture An architect is drawing plans for a building that is a hexagonal prism. How could the architect draw a cutaway of the building that shows a cross section in the shape of each given figure? 22. a hexagon 23. a rectangle Name a three-dimensional figure from which a cross section in the given shape can be made. 24. square 25. rectangle 26. circle 27. hexagon Write a verbal description of each figure. 28. 29. 30. Draw and label a figure that meets each description. 31. rectangular prism with length 3 cm, width 2 cm, and height 5 cm 32. regular pentagonal prism with side length 6 in. and height 8 in. 33. cylinder with radius 4 m and height 7 m Draw a net for each three-dimensional figure. 34. 35. 36. 37. This problem will prepare you for the Multi-Step TAKS Prep on page 678. A manufacturer of camping gear makes a wall tent in the shape shown in the diagram. a. Classify the three-dimensional figure that the wall tent forms. b. What shapes make up the faces of the tent? How many of each shape are there? c. Draw a net for the wall tent. 658 658 Chapter 10 Spatial Reasoning �� 38. /////ERROR ANALYSIS///// A regular hexagonal prism is intersected by a plane as shown. Which cross section is incorrect? Explain. 39. Critical Thinking A three-dimensional figure has 5 faces. One face is adjacent to every other face. Four of the faces are congruent. Draw a figure that meets these conditions. 40. Write About It Which of the following figures is not a net for a cube? Explain. a. b. c. d. 41. Which three-dimensional figure does the net represent? 42. Which shape CANNOT be a face of a hexagonal prism? triangle hexagon parallelogram rectangle 43. What shape is the cross section formed by a cone and a plane that is perpendicular to the base and that passes through the vertex of the cone? circle triangle trapezoid rectangle 44. Which shape best represents a hexagonal prism viewed from the top? 10- 1 Solid Geometry 659 659 �� CHALLENGE AND EXTEND A double cone is formed by two cones that share the same vertex. Sketch each cross section formed by a double cone and a plane. 45. 46. 47. Crafts Elena is designing patterns for gift boxes. Draw a pattern that she can use to create each box. Be sure to include tabs for gluing the sides together. 48. a box that is a square pyramid where each triangular face is an isosceles triangle with a height equal to three times the width 49. a box that is a cylinder with the diameter equal to the height 50. a box that is a rectangular prism with a base that is twice as long as it is wide, and with a rectangular pyramid on the top base 51. A net of a prism is shown. The bases of the prism are regular hexagons, and the rectangular faces are all congruent. a. List all pairs of parallel faces in the prism. b. Draw a net of a prism with bases that are regular pentagons. How many pairs of parallel faces does the prism have? � � � � � � � � SPIRAL REVIEW Write the equation that fits the description. (Previous course) 52. the equation of the graph that is the reflection of the graph of y = x 2 over the x-axis 53. the equation of the graph of y = x 2 after a vertical translation of 6 units upward 54. the quadratic equation of a graph that opens upward and is wider than y = x 2 Name the largest and smallest angles of each triangle. (Lesson 5-5) 55. � 56. � � � � � � �� 57. � �� Determine whether the two polygons are similar. If so, give the similarity ratio. (Lesson 7-2) 58. � � � � � �� 59. �� 660 660 Chapter 10 Spatial Reasoning 10-2 Representations of Three-Dimensional Figures TEKS G.9.D Congruence and the geometry of size: analyze the characteristics of polyhedra and other three-dimensional figures .... Also G.6.C Objectives Draw representations of three-dimensional figures. Recognize a threedimensional figure from a given representation. Vocabulary orthographic drawing isometric drawing perspective drawing vanishing point horizon Who uses this? Architects make many different kinds of drawings to represent three-dimensional figures. (See Exercise 34.) There are many ways to represent a threedimensional object. An orthographic drawing shows six different views of an object: top, bottom, front, back, left side, and right side. First National Bank, San Angelo E X A M P L E 1 Drawing Orthographic Views of an Object Draw all six orthographic views of
the given object. Assume there are no hidden cubes. Top: Bottom: Front: Back: Left: Right: 1. Draw all six orthographic views of the given object. Assume there are no hidden cubes. 10- 2 Representations of Three-Dimensional Figures 661 661 �� Isometric drawing is a way to show three sides of a figure from a corner view. You can use isometric dot paper to make an isometric drawing. This paper has diagonal rows of dots that are equally spaced in a repeating triangular pattern. E X A M P L E 2 Drawing an Isometric View of an Object Draw an isometric view of the given object. Assume there are no hidden cubes. 2. Draw an isometric view of the given object. Assume there are no hidden cubes. In a perspective drawing , nonvertical parallel lines are drawn so that they meet at a point called a vanishing point . Vanishing points are located on a horizontal line called the horizon . A one-point perspective drawing contains one vanishing point. A two-point perspective drawing contains two vanishing points. Perspective Drawing When making a perspective drawing, it helps me to remember that all vertical lines on the object will be vertical in the drawing. Jacob Martin MacArthur High School Three-dimensional figure One-point perspective Two-point perspective 662 662 Chapter 10 Spatial Reasoning �� E X A M P L E 3 Drawing an Object in Perspective A Draw a cube in one-point perspective. Draw a horizontal line to represent the horizon. Mark a vanishing point on the horizon. Then draw a square below the horizon. This is the front of the cube. From each corner of the square, lightly draw dashed segments to the vanishing point. Lightly draw a smaller square with vertices on the dashed segments. This is the back of the cube. Draw the edges of the cube, using dashed segments for hidden edges. Erase any segments that are not part of the cube. B Draw a rectangular prism in two-point perspective. In a one-point perspective drawing of a cube, you are looking at a face. In a two-point perspective drawing, you are looking at a corner. Draw a horizontal line to represent the horizon. Mark two vanishing points on the horizon. Then draw a vertical segment below the horizon and between the vanishing points. This is the front edge of the prism. From each endpoint of the segment, lightly draw dashed segments to each vanishing point. Draw two vertical segments connecting the dashed lines. These are other vertical edges of the prism. Lightly draw dashed segments from each endpoint of the two vertical segments to the vanishing points. Draw the edges of the prism, using dashed lines for hidden edges. Erase any lines that are not part of the prism. 3a. Draw the block letter L in one-point perspective. 3b. Draw the block letter L in two-point perspective. 10- 2 Representations of Three-Dimensional Figures 663 663 E X A M P L E 4 Relating Different Representations of an Object Relating Different Representations of an Object Determine whether each drawing represents the given object. Assume there are no hidden cubes. B D A C Yes; the drawing is a one-point perspective view of the object. No; the cubes that share a face in the object do not share a face in the drawing. No; the figure in the drawing is made up of four cubes, and the object is made up of only three cubes. Yes; the drawing shows the six orthographic views of the object. 4. Determine whether the drawing represents the given object. Assume there are no hidden cubes. THINK AND DISCUSS 1. Describe the six orthographic views of a cube. 2. In a perspective drawing, are all parallel lines drawn so that they meet at a vanishing point? Why or why not? 3. GET ORGANIZED Copy and complete the graphic organizer. 664 664 Chapter 10 Spatial Reasoning �� 10-2 Exercises Exercises KEYWORD: MG7 10-2 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In a(n) ? drawing, the vanishing points are located on the ̶̶̶̶ horizon. (orthographic, isometric, or perspective Draw all six orthographic views of each object. Assume there are no hidden cubes. p. 661 2. 3. 662 Draw an isometric view of each object. Assume there are no hidden cubes. 5. 6. 4. 7. 663 Draw each object in one-point and two-point perspectives. Assume there are no hidden cubes. 8. rectangular prism 9. block letter . 664 Determine whether each drawing represents the given object. Assume there are no hidden cubes. 11. 10. 12. 13. PRACTICE AND PROBLEM SOLVING Draw all six orthographic views of each object. Assume there are no hidden cubes. 14. 15. 16. 10- 2 Representations of Three-Dimensional Figures 665 665 �� Independent Practice Draw an isometric view of each object. Assume there are no hidden cubes. For See Exercises Example 17. 18. 19. 14–16 17–19 20–21 22–25 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S22 Application Practice p. S37 Draw each object in one-point and two-point perspective. Assume there are no hidden cubes. 20. right triangular prism 21. block letter Determine whether each drawing represents the given object. Assume there are no hidden cubes. 22. 23. 24. 25. 26. Use the top, front, side, and isometric views to build the three-dimensional figure out of unit cubes. Then draw the figure in one-point perspective. Use the top, side, and front views to draw an isometric view of each figure. 27. 28. 29. This problem will prepare you for the Multi-Step TAKS Prep on page 678. A camping gear catalog shows the three given views of a tent. a. Draw a bottom view of the tent. b. Make a sketch of the tent. c. Each edge of the three-dimensional figure from part b represents one pole of the tent. How many poles does this tent have? 666 666 Chapter 10 Spatial Reasoning �� Draw all six orthographic views of each object. 30. 31. 32. 33. Critical Thinking Describe or draw two figures that have the same left, right, front, and back orthographic views but have different top and bottom views. 34. Architecture Perspective drawings are used by architects to show what a finished room will look like. a. Is the architect’s sketch in one-point or two-point perspective? b. Write About It How would you locate the vanishing point(s) in the architect’s sketch? 35. Which three-dimensional figure has these three views? 36. Which drawing best represents the top view of the three-dimensional figure? 37. Short Response Draw a one-point perspective view and an isometric view of a triangular prism. Explain how the two drawings are different. 10- 2 Representations of Three-Dimensional Figures 667 667 �� CHALLENGE AND EXTEND Draw each figure using one-point perspective. (Hint: First lightly draw a rectangular prism. Enclose the figure in the prism.) 38. an octagonal prism 39. a cylinder 40. a cone 41. A frustum of a cone is a part of a cone with two parallel bases. Copy the diagram of the frustum of a cone. a. Draw the entire cone. b. Draw all six orthographic views of the frustum. c. Draw a net for the frustum. 42. Art Draw a one-point or two-point perspective drawing of the inside of a room. Include at least two pieces of furniture drawn in perspective. SPIRAL REVIEW Find the two numbers. (Previous course) 43. The sum of two numbers is 30. The difference between 2 times the first number and 2 times the second number is 20. 44. The difference between the first number and the second number is 7. When the second number is added to 4 times the first number, the result is 38. 45. The second number is 5 more than the first number. Their sum is 5. For A (4, 2) , B (6, 1) , C (3, 0) , and D (2, 0) , find the slope of each line. (Lesson 3-5) 46.  AB 48.  AD 47.  AC Describe the faces of each figure. (Lesson 10-1) 49. pentagonal prism 50. cube 51. triangular pyramid Using Technology You can use geometry software to draw figures in one- and two-point perspectives. 1. a. Draw a horizontal line to represent the horizon. Create a vanishing point on the horizon. Draw a rectangle with two sides parallel to the horizon. Draw a segment from each vertex to the vanishing point. b. Draw a smaller rectangle with vertices on the segments that intersect the horizon. Hide these segments and complete the figure. c. Drag the vanishing point to different locations on the horizon. Describe what happens to the figure. 2. Describe how you would use geometry software to draw a figure in two-point perspective. 668 668 Chapter 10 Spatial Reasoning 10-3 Use with Lesson 10-3 Activity Use Nets to Create Polyhedrons A polyhedron is formed by four or more polygons that intersect only at their edges. The faces of a regular polyhedron are all congruent regular polygons, and the same number of faces intersect at each vertex. Regular polyhedrons are also called Platonic solids. There are exactly five regular polyhedrons. TEKS G.6.B Dimensionality and the geometry of location: use nets to represent and construct three-dimensional geometric figures. Also G.9.D Use geometry software or a compass and straightedge to create a larger version of each net on heavy paper. Fold each net into a polyhedron. NAME FACES EXAMPLE NET REGULAR POLYHEDRONS Tetrahedron 4 triangles Octahedron 8 triangles Icosahedron 20 triangles Cube 6 squares Dodecahedron 12 pentagons Try This 1. Complete the table for the number of vertices V, edges E, and faces F for each of the polyhedrons you made in Activity 1. 2. Make a Conjecture What do you think is true about the relationship between the number of vertices, edges, and faces of a polyhedron? POLYHEDRON V E F V - E + F Tetrahedron Octahedron Icosahedron Cube Dodecahedron 10-3 Geometry Lab 669 669 10-3 Formulas in Three Dimensions TEKS G.7.C Dimensionality and the geome
try of location: develop and use formulas involving length, slope, and midpoint. Also G.5.A, G.8.C, G.9.D Objectives Apply Euler’s formula to find the number of vertices, edges, and faces of a polyhedron. Develop and apply the distance and midpoint formulas in three dimensions. Vocabulary polyhedron space Why learn this? Divers can use a three-dimensional coordinate system to find distances between two points under water. (See Example 5.) A polyhedron is formed by four or more polygons that intersect only at their edges. Prisms and pyramids are polyhedrons, but cylinders and cones are not. Polyhedrons Not polyhedrons In the lab before this lesson, you made a conjecture about the relationship between the vertices, edges, and faces of a polyhedron. One way to state this relationship is given below. Euler’s Formula For any polyhedron with V vertices, E edges, and F faces, V - E + F = 2. E X A M P L E 1 Using Euler’s Formula Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. A B Euler is pronounced “Oiler.” V = 4, E = 6 = 10, E = 15, F = 7 Use Euler’s formula. Simplify. 10 - 15 + 7 ≟ 2 2 = 2 Find the number of vertices, edges, and faces of the polyhedron. Use your results to verify Euler’s formula. 1a. 1b. 670 670 Chapter 10 Spatial Reasoning A diagonal of a three-dimensional figure connects two vertices of two different faces. Diagonal d of a rectangular prism is shown in the diagram. By the Pythagorean Theorem, ℓ 2 + w 2 = x 2 , and x 2 + h 2 = d 2 . Using substitution . Diagonal of a Right Rectangular Prism The length of a diagonal d of a right rectangular prism with length ℓ, width w, and height h is d = √  Using the Pythagorean Theorem in Three Dimensions Find the unknown dimension in each figure. A the length of the diagonal of a 3 in. by 4 in. by 5 in. rectangular prism 3 2 + 4 2 + 5 2 d = √  = √  9 + 16 + 25 = √  50 ≈ 7.1 in. Substitute 3 for ℓ, 4 for w, and 5 for h. Simplify. B the height of a rectangular prism with an 8 ft by 12 ft base and an 18 ft diagonal 18 = √  8 2 + 12 2 + h 2 18 2 = ( √  8 2 + 12 2 + h 2 ) 324 = 64 + 144 + h 2 h 2 = 116 h = √  116 ≈ 10.8 ft 2 Substitute 18 for d, 8 for ℓ, and 12 for w. Square both sides of the equation. Simplify. Solve for h 2 . Take the square root of both sides. 2. Find the length of the diagonal of a cube with edge length 5 cm. Space is the set of all points in three dimensions. Three coordinates are needed to locate a point in space. A three-dimensional coordinate system has 3 perpendicular axes: the x-axis, the y-axis, and the z-axis. An ordered triple (x, y, z) is used to locate a point. To locate the point (3, 2, 4) , start at (0, 0, 0) . From there move 3 units forward, 2 units right, and then 4 units up. E X A M P L E 3 Graphing Figures in Three Dimensions Graph each figure. A a cube with edge length 4 units and one vertex at (0, 0, 0) The cube has 8 vertices: (0, 0, 0) , (0, 4, 0) , (0, 0, 4) , (4, 0, 0) , (4, 4, 0) , (4, 0, 4) , (0, 4, 4) , (4, 4, 4) . 10- 3 Formulas in Three Dimensions 671 671 �� Graph each figure. B a cylinder with radius 3 units, height 5 units, and one base centered at (0, 0, 0) Graph the center of the bottom base at (0, 0, 0) . Since the height is 5, graph the center of the top base at (0, 0, 5) . The radius is 3, so the bottom base will cross the x-axis at (3, 0 ,0) and the y-axis at (0, 3, 0) . Draw the top base parallel to the bottom base and connect the bases. 3. Graph a cone with radius 5 units, height 7 units, and the base centered at (0, 0, 0) . You can find the distance between the two points ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) by drawing a rectangular prism with the given points as endpoints of a diagonal. Then use the formula for the length of the diagonal. You can also use a formula related to the Distance Formula. (See Lesson 1-6.) The formula for the midpoint between ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) is related to the Midpoint Formula. (See Lesson 1-6.) Distance and Midpoint Formulas in Three Dimensions The distance between the points ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) is d = √  ( . The midpoint of the segment with endpoints ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) is Finding Distances and Midpoints in Three Dimensions Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. A (0, 0, 0) and (3, 4, 12) distance: d = √  ( = √  (3 - 0) 2 + (4 - 0) 2 + (12 - 0) 2 = √  9 + 16 + 144 = √  169 = 13 units midpoint + 12 1.5, 2, 6) 672 672 Chapter 10 Spatial Reasoning �� Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. B (3, 8, 10) and (7, 12, 15) distance: d = √  midpoint ( ) 10 + 15 8 + 12 3 + 7 _ _ _ (7 - 3) 2 + (12 - 8) 2 + (15 - 10) 2 , , 2 2 2 M (5, 10, 12.5) = √  = √  16 + 16 + 25 = √  57 ≈ 7.5 units Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 4a. (0, 9, 5) and (6, 0, 12) 4b. (5, 8, 16) and (12, 16, 20) E X A M P L E 5 Recreation Application Two divers swam from a boat to the locations shown in the diagram. How far apart are the divers? Recreation Divers in the Comal Springs in New Braunfels can see a variety of plant and animal life. The springs, which are fed by the Edwards Aquifer, have a constant temperature of 68°F and are home to many endangered species. The location of the boat can be represented by the ordered triple (0, 0, 0) , and the locations of the divers can be represented by the ordered triples (18, 9, -8) and (-15, -6, -12) . d = √  ( = √  = √  (-15 - 18) 2 + (-6 - 9) 2 + (-12 + 8) 2 1330 ≈ 36.5 ft c10l03002a Use the Distance Formula to find the distance between the divers. 5. What if…? If both divers swam straight up to the surface, how far apart would they be? THINK AND DISCUSS 1. Explain how to find the distance between two points in a three-dimensional coordinate system. 2. GET ORGANIZED Copy and complete the graphic organizer. 10- 3 Formulas in Three Dimensions 673 673 Depth: 8 ft9 ft18 ft15 ft6 ftDepth: 12 ftge07se_c10l03002aAB�� 10-3 Exercises Exercises KEYWORD: MG7 10-3 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Explain why a cylinder is not a polyhedron. 670 Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. 2. 3. 4 Find the unknown dimension in each figure. Round to the nearest tenth, if necessary. p. 671 5. the length of the diagonal of a 4 ft by 8 ft by 12 ft rectangular prism 6. the height of a rectangular prism with a 6 in. by 10 in. base and a 13 in. diagonal 7. the length of the diagonal of a square prism with a base edge length of 12 in. and a height of 1 in Graph each figure. p. 671 8. a cone with radius 8 units, height 4 units, and the base centered at (0, 0, 0) 9. a cylinder with radius 3 units, height 4 units, and one base centered at (0, 0, 0) 10. a cube with edge length 7 units and one vertex at (0, 0, 0. 672 . 673 Independent Practice For See Exercises Example 15–17 18–20 21–23 24–26 27 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S22 Application Practice p. S37 Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 11. (0, 0, 0) and (5, 9, 10) 12. (0, 3, 8) and (7, 0, 14) 13. (4, 6, 10) and (9, 12, 15) 14. Recreation After a day hike, a group of hikers set up a camp 3 km east and 7 km north of the starting point. The elevation of the camp is 0.6 km higher than the starting point. What is the distance from the camp to the starting point? PRACTICE AND PROBLEM SOLVING Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. 15. 16. 17. Find the unknown dimension in each figure. Round to the nearest tenth, if necessary. 18. the length of the diagonal of a 7 yd by 8 yd by 16 yd rectangular prism 19. the height of a rectangular prism with a 15 m by 6 m base and a 17 m diagonal 20. the edge length of a cube with an 8 cm diagonal 674 674 Chapter 10 Spatial Reasoning Meteorology A typical cumulus cloud weighs about 1.4 billion pounds, which is more than 100,000 elephants. Source: usgs.gov Graph each figure. 21. a cylinder with radius 5 units, height 3 units, and one base centered at (0, 0, 0) 22. a cone with radius 2 units, height 4 units, and the base centered at (0, 0, 0) 23. a square prism with base edge length 5 units, height 3 units, and one vertex at (0, 0, 0) Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 24. (0, 0, 0) and (4, 4, 4) 25. (2, 3, 7) and (9, 10, 10) 26. (2, 5, 3) and (8, 8, 10) 27. Meteorology A cloud has an elevation of 6500 feet. A raindrop falling from the cloud was blown 700 feet south and 500 feet east before it hit the ground. How far did the raindrop travel from the cloud to the ground? 28. Multi-Step Find the length of a diagonal of the rectangular prism at right. If the length, width, and height are doubled, what happens to the length of the diagonal? For each three-dimensional figure, find the missing value and draw a figure with the correct number of vertices, edges, and faces. Vertices V Edges E Faces F Diagram 5 8 7 29. 30. 31. 8 12 9 5 5 7 32. Algebra Each base of a prism is a polygon with n sides. Write an expression for the number of vertices V, the number of edges E, and the number of faces F in terms of n. Use your results
to show that Euler’s formula is true for all prisms. 33. Algebra The base of a pyramid is a polygon with n sides. Write an expression for the number of vertices V, the number of edges E, and the number of faces F in terms of n. Use your results to show that Euler’s formula is true for all pyramids. 34. This problem will prepare you for the Multi-Step TAKS Prep on page 678. ̶̶̶ NM ≅ ̶̶ NP The tent at right is a triangular prism where ̶̶ KJ ≅ ̶̶ KL and has the given dimensions. and a. The tent manufacturer sets up the tent on a coordinate system so that J is at the origin and M has coordinates (7, 0, 0) . Find the coordinates of the other vertices. b. The manufacturer wants to know the distance from K to P in order to make an extra support pole for the tent. Find KP to the nearest tenth. � �� 10- 3 Formulas in Three Dimensions 675 675 �� Find the missing dimension of each rectangular prism. Give your answers in simplest radical form. Length ℓ Width w Height h Diagonal d 6 in. 24 12 35. 36. 37. 38. 6 in. 18 2 6 in. 60 3 65 24 4 Graph each figure. 39. a cylinder with radius 4 units, height 5 units, and one base centered at (1, 2, 5) 40. a cone with radius 3 units, height 7 units, and the base centered at (3, 2, 6) 41. a cube with edge length 6 units and one vertex at (4, 2, 3) 42. a rectangular prism with vertices at (4, 2, 5) , (4, 6, 5) , (4, 6, 8) , (8, 6, 5) , (8, 2, 5) , (8, 6, 8) , (4, 2, 8) , and (8, 2, 8) 43. a cone with radius 4 units, the vertex at (4, 7, 8) , and the base centered at (4, 7, 1) 44. a cylinder with a radius of 5 units and bases centered at (2, 3, 7) and (2, 3, 15) Graph each segment with the given endpoints in a three-dimensional coordinate system. Find the length and midpoint of each segment. 46. (4, 3, 3) and (7, 4, 4) 45. (1, 2, 3) and (3, 2, 1) 47. (4, 7, 8) and (3, 1, 5) 48. (0, 0, 0) and (8, 3, 6) 49. (6, 1, 8) and (2, 2, 6) 50. (2, 8, 5) and (3, 6, 3) 51. Multi-Step Find z if the distance between R (6, -1, -3) and S (3, 3, z) is 13. 52. Draw a figure with 6 vertices and 6 faces. 53. Estimation Measure the net for a rectangular prism and estimate the length of a diagonal. 54. Make a Conjecture What do you think is the longest segment joining two points on a rectangular prism? Test your conjecture using at least three segments whose endpoints are on the prism with vertices A (0, 0, 0) , B (1, 0, 0) , C (1, 2, 0) , D (0, 2, 0) , E (0, 0, 2) , F (1, 0, 2) , G (1, 2, 2) , and H (0, 2, 2) . 55. Critical Thinking The points A (3, 2, -3) , B (5, 8, 6) , and C (-3, -5, 3) form a triangle. Classify the triangle by sides and angles. 56. Write About It A cylinder has a radius of 4 and a height of 6. What is the length of the longest segment with both endpoints on the cylinder? Describe the location of the endpoints and explain why it is the longest possible segment. 676 676 Chapter 10 Spatial Reasoning 57. How many faces, edges, and vertices does a hexagonal pyramid have? 6 faces, 10 edges, 6 vertices 7 faces, 12 edges, 7 vertices 7 faces, 10 edges, 7 vertices 8 faces, 18 edges, 12 vertices 58. Which is closest to the length of the diagonal of the rectangular prism with length 12 m, width 8 m, and height 6 m? 6.6 m 44 m 15.6 m 244.0 m 59. What is the distance between the points (7, 14, 8) and (9, 3, 12) to the nearest tenth? 10.9 11.9 119.0 141.0 CHALLENGE AND EXTEND 60. Multi-Step The bases of the right hexagonal prism are regular hexagons with side length a, and the height of the prism is h. Find the length of the indicated diagonal in terms of a and h. 61. Determine if the points A (-1, 2, 4) , B (1, -2, 6) , and C (3, -6, 8) are collinear. 62. Algebra Write a coordinate proof of the Midpoint Formula using the Distance Formula. Given: points ) , and M ( Prove: A, B, and M are collinear, and AM = MB _____ _____ _____ 2 , 2 , 2 63. Algebra Write a coordinate proof that the diagonals of a rectangular prism are congruent and bisect each other. Given: a rectangular prism with vertices A (0, 0, 0) , B (a, 0, 0) , C (a, b, 0) , D (0, b, 0) , E (0, 0, c) , F (a, 0, c) , G (a, b, c) , and H (0, b, c) ̶̶ AG and ̶̶ BH are congruent and bisect each other. Prove: SPIRAL REVIEW The histogram shows the number of people by age group who attended a natural history museum opening. Find the following. (Previous course) 64. the number of people between 10 and 29 years of age that were in attendance 65. the age group that had the greatest number of people in attendance Write a formula for the area of each figure after the given change. (Lesson 9-5) 66. A parallelogram with base b and height h has its height doubled. 67. A trapezoid with height h and bases b 1 and b 2 has its base b 1 multiplied by 1 __ 2 . 68. A circle with radius r has its radius tripled. Use the diagram for Exercises 69–71. (Lesson 10-1) 69. Classify the figure. 70. Name the edges. 71. Name the base. 10- 3 Formulas in Three Dimensions 677 677 �� SECTION 10A Three-Dimensional Figures Your Two Tents A manufacturer of camping gear offers two types of tents: an A-frame tent and a pyramid tent. A-frame tent Pyramid tent 1. The manufacturer’s catalog shows the top, front, and side views of each tent. It shows a two-dimensional shape for each that can be folded to form the three-dimensional shape of the tent. Draw the catalog display for each tent. The manufacturer uses a three-dimensional coordinate system to represent the vertices of each tent. Each unit of the coordinate system represents one foot. 2. Which tent offers a greater sleeping area? 3. Compare the heights of the tents. Which tent offers more headroom? 4. A camper wants to purchase the tent that has shorter support poles so that she can fit the folded tent in ̶̶ EF in her car more easily. Find the length of pole ̶̶ TR in the the A-frame tent and the length of pole pyramid tent. Which tent should the camper buy? A-Frame Tent Vertex Coordinates A B C D E F (0, 0, 0) (0, 7, 0) (0, 3.5, 7) (8, 0, 0) (8, 7, 0) (8, 3.5, 7) Pyramid Tent Vertex Coordinates P Q R S T (0, 0, 0) (8, 0, 0) (8, 8, 0) (0, 8, 0) (4, 4, 8) 678 678 Chapter 10 Spatial Reasoning Camping at Big Bend National Park �� SECTION 10A Quiz for Lessons 10-1 Through 10-3 10-1 Solid Geometry Classify each figure. Name the vertices, edges, and bases. 1. 2. 3. Describe the three-dimensional figure that can be made from the given net. 4. Describe each cross section. 7. 5. 8. 6. 9. 10-2 Representations of Three-Dimensional Figures Use the figure made of unit cubes for Problems 10 and 11. Assume there are no hidden cubes. 10. Draw all six orthographic views. 11. Draw an isometric view. 12. Draw the block letter T in one-point perspective. 13. Draw the block letter T in two-point perspective. 10-3 Formulas in Three Dimensions Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. 14. a square prism 15. a hexagonal pyramid 16. a triangular pyramid 17. A bird flies from its nest to a point that is 6 feet north, 7 feet west, and 6 feet higher in the tree than the nest. How far is the bird from the nest? Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 19. (3, 1, -2) and (5, -5, 7) 18. (0, 0, 0) and (4, 6, 12) 20. (3, 5, 9) and (7, 2, 0) Ready to Go On? 679 679 �� 10-4 Surface Area of Prisms and Cylinders TEKS G.8.D Congruence and the geometry of size: find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites …. Objectives Learn and apply the formula for the surface area of a prism. Learn and apply the formula for the surface area of a cylinder. Vocabulary lateral face lateral edge right prism oblique prism altitude surface area lateral surface axis of a cylinder right cylinder oblique cylinder Also G.5.A, G.5.B, G.6.B, G.11.D Why learn this? The surface area of ice affects how fast it will melt. If the surface exposed to the air is increased, the ice will melt faster. (See Example 5.) Prisms and cylinders have 2 congruent parallel bases. A lateral face is not a base. The edges of the base are called base edges. A lateral edge is not an edge of a base. The lateral faces of a right prism are all rectangles. An oblique prism has at least one nonrectangular lateral face. An altitude of a prism or cylinder is a perpendicular segment joining the planes of the bases. The height of a three-dimensional figure is the length of an altitude. �� Surface area is the total area of all faces and curved surfaces of a three-dimensional figure. The lateral area of a prism is the sum of the areas of the lateral faces. The net of a right prism can be drawn so that the lateral faces form a rectangle with the same height as the prism. The base of the rectangle is equal to the perimeter of the base of the prism. Lateral Area and Surface Area of Right Prisms The lateral area of a right prism with base perimeter P and height h is L = Ph. The surface area of a right prism with lateral area L and base area B is S = L + 2B, or S = Ph + 2B. The surface area of a cube with edge length s is S = 6 s 2 . The surface area of a right rectangular prism with length ℓ, width w, and height h can be written as S = 2ℓw + 2wh + 2ℓh. 680 680 Chapter 10 Spatial Reasoning �� E X A M P L E 1 Finding Lateral Areas and Surface Areas of Prisms Find the lateral area and surface area of each right prism. Round to the nearest tenth, if necessary. A the rectangular prism L = Ph = (28) 12 = 336 cm 2 S = Ph + 2B = 336 + 2 (6) (8) = 432 cm 2 P = 2 (8) + 2 (6) = 28 cm B the regular hexagonal prism The surface area formula is only true for right prisms. To find the surface area of an oblique prism, add the areas of the faces. L = Ph = 36 (10) =
360 m 2 S = Ph + 2B = 360 + 2 (54 √  3 ) ≈ 547.1 m 2 P = 6 (6) = 36 m The base area is B = 1 __ aP = 54 √  3 m. 2 1. Find the lateral area and surface area of a cube with edge length 8 cm. The lateral surface of a cylinder is the curved surface that connects the two bases. The axis of a cylinder is the segment with endpoints at the centers of the bases. The axis of a right cylinder is perpendicular to its bases. The axis of an oblique cylinder is not perpendicular to its bases. The altitude of a right cylinder is the same length as the axis. Lateral Area and Surface Area of Right Cylinders The lateral area of a right cylinder with radius r and height h is L = 2πrh. The surface area of a right cylinder with lateral area L and base area B is S = L + 2B, or S = 2πrh + 2π r 2 . 10- 4 Surface Area of Prisms and Cylinders 681 681 �� E X A M P L E 2 Finding Lateral Areas and Surface Areas of Right Cylinders Find the lateral area and surface area of each right cylinder. Give your answers in terms of π. A L = 2πrh = 2π (1) (5) = 10π m 2 S = L + 2π r 2 = 10π + 2π (1) 2 = 12π m 2 The radius is half the diameter, or 1 m. B a cylinder with a circumference of 10π cm and a height equal to 3 times the radius Step 1 Use the circumference to find the radius. C = 2πr 10π = 2πr r = 5 Circumference of a circle Substitute 10π for C. Divide both sides by 2π. Step 2 Use the radius to find the lateral area and surface area. The height is 3 times the radius, or 15 cm. L = 2π rh = 2π (5) (15) = 150π cm 2 S = 2π rh + 2π r 2 = 150π + 2π (5) 2 = 200π cm 2 Lateral area Surface area 2. Find the lateral area and surface area of a cylinder with a base area of 49π and a height that is 2 times the radius. E X A M P L E 3 Finding Surface Areas of Composite Three-Dimensional Figures Always round at the last step of the problem. Use the value of π given by the π key on your calculator. Find the surface area of the composite figure. Round to the nearest tenth. The surface area of the right rectangular prism is S = Ph + 2B = 80 (20) + 2 (24) (16) = 2368 ft 2 . A right cylinder is removed from the rectangular prism. The lateral area is L = 2π rh = 2π (4) (20) = 160π ft 2 . The area of each base is B = π r 2 = π (4) 2 = 16π ft 2 . The surface area of the composite figure is the sum of the areas of all surfaces on the exterior of the figure. S = (prism surface area) + (cylinder lateral area) - (cylinder base area) = 2368 + 160π -2 (16π) = 2368 + 128π ≈ 2770.1 ft 2 3. Find the surface area of the composite figure. Round to the nearest tenth. 682 682 Chapter 10 Spatial Reasoning �� E X A M P L E 4 Exploring Effects of Changing Dimensions The length, width, and height of the right rectangular prism are doubled. Describe the effect on the surface area. original dimensions: length, width, and height doubled: S = Ph + 2B = 16 (3) + 2 (6) (2) = 72 in 2 S = Ph + 2B = 32 (6) + 2 (12) (4) = 288 in 2 Notice that 288 = 4 (72) . If the length, width, and height are doubled, the surface area is multiplied by 2 2 , or 4. 4. The height and diameter of the cylinder are multiplied by 1 __ 2 . Describe the effect on the surface area. E X A M P L E 5 Chemistry Application Entertainment Ice sculptures are carved from blocks of ice that weigh hundreds of pounds. A typical ice sculpture can last 6–8 hours at room temperature (70°F). If two pieces of ice have the same volume, the one with the greater surface area will melt faster because more of it is exposed to the air. One piece of ice shown is a rectangular prism, and the other is half a cylinder. Given that the volumes are approximately equal, which will melt faster? �� rectangular prism: S = Ph + 2B = 12 (3) + 2 (8) = 52 cm 2 �� half cylinder: S = πrh + π r 2 + 2rh = π (4) (1) + π (4) 2 + 8 (1) �� = 20π + 8 ≈ 70.8 cm 2 �� The half cylinder of ice will melt faster. Use the information above to answer the following. 5. A piece of ice shaped like a 5 cm by 5 cm by 1 cm rectangular prism has approximately the same volume as the pieces above. Compare the surface areas. Which will melt faster? THINK AND DISCUSS 1. Explain how to find the surface area of a cylinder if you know the �� lateral area and the radius of the base. 2. Describe the difference between an oblique prism and a right prism. 3. GET ORGANIZED Copy and complete the graphic organizer. Write the formulas in each box. 10- 4 Surface Area of Prisms and Cylinders 683 683 �� 10-4 Exercises Exercises KEYWORD: MG7 10-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary How many lateral faces does a pentagonal prism have Find the lateral area and surface area of each right prism. p. 681 2. 3. 4. a cube with edge length 9 inches . 682 Find the lateral area and surface area of each right cylinder. Give your answers in terms of π. 5. 6. 7. a cylinder with base area 64π m 2 and a height 3 meters less than the radius . 682 Multi-Step Find the surface area of each composite figure. Round to the nearest tenth. 8. 9 Describe the effect of each change on the surface area of the given figure. p. 683 10. The dimensions are cut in half. 11. The dimensions are multiplied by 2 __ . 683 12. Consumer Application The greater the lateral area of a florescent light bulb, the more light the bulb produces. One cylindrical light bulb is 16 inches long with a 1-inch radius. Another cylindrical light bulb is 23 inches long with a 3 __ 4 -inch radius. Which bulb will produce more light? 684 684 Chapter 10 Spatial Reasoning �� Independent Practice For See Exercises Example 13–15 16–18 19–20 21–22 23 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S22 Application Practice p. S37 PRACTICE AND PROBLEM SOLVING Find the lateral area and surface area of each right prism. Round to the nearest tenth, if necessary. 13. 14. 15. a right equilateral triangular prism with base edge length 8 ft and height 14 ft Find the lateral area and surface area of each right cylinder. Give your answers in terms of π. 16. 17. 18. a cylinder with base circumference 16π yd 2 and a height equal to 3 times the radius Multi-Step Find the surface area of each composite figure. Round to the nearest tenth. 19. 20. Describe the effect of each change on the surface area of the given figure. 21. The dimensions are tripled. 22. The dimensions are doubled. 23. Biology Plant cells are shaped approximately like a right rectangular prism. Each cell absorbs oxygen and nutrients through its surface. Which cell can be expected to absorb at a greater rate? (Hint: 1 µm = 1 micrometer = 0.000001 meter) 10- 4 Surface Area of Prisms and Cylinders 685 685 �� 24. Find the height of a right cylinder with surface area 160π ft 2 and radius 5 ft. 25. Find the height of a right rectangular prism with surface area 286 m 2 , length 10 m, and width 8 m. 26. Find the height of a right regular hexagonal prism with lateral area 1368 m 2 and base edge length 12 m. 27. Find the surface area of the right triangular prism with vertices at (0, 0, 0) , (5, 0, 0) , (0, 2, 0) , (0, 0, 9) , (5, 0, 9) , and (0, 2, 9) . The dimensions of various coins are given in the table. Find the surface area of each coin. Round to the nearest hundredth. Coin Diameter (mm) Thickness (mm) Surface Area ( mm 2 ) 28. Penny 29. Nickel 30. Dime 31. Quarter 19.05 21.21 17.91 24.26 1.55 1.95 1.35 1.75 32. How can the edge lengths of a rectangular prism be changed so that the surface area is multiplied by 9? 33. How can the radius and height of a cylinder be changed so that the surface area is multiplied by 1 __ 4 ? 34. Landscaping Ingrid is building a shelter to protect her plants from freezing. She is planning to stretch plastic sheeting over the top and the ends of a frame. Which of the frames shown will require more plastic? 35. Critical Thinking If the length of the measurements of the net are correct to the nearest tenth of a centimeter, what is the maximum error in the surface area? 36. Write About It Explain how to use the net of a three-dimensional figure to find its surface area. 37. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A juice container is a square prism with base edge length 4 in. When an 8 in. straw is inserted into the container as shown, exactly 1 in. remains outside the container. a. Find AB and BC. b. What is the height AC of the container to the nearest tenth? c. Use your result from part b to find how much material is required to manufacture the container. Round to the nearest tenth. 686 686 Chapter 10 Spatial Reasoning 10 ftge07sec10l04004aa1st pass4/23/5cmurphy10 ft10 ftge07sec10l04005a1st pass4/12/5cmurphy10 ft10 ft�� 38. Measure the dimensions of the net of a cylinder to the nearest millimeter. Which is closest to the surface area of the cylinder? 35.8 cm 2 18.8 cm 2 16.0 cm 2 13.2 cm 2 39. The base of a triangular prism is an equilateral triangle with a perimeter of 24 inches. If the height of the prism is 5 inches, find the lateral area. 120 in 2 60 in 2 40 in 2 360 in 2 40. Gridded Response Find the surface area in square inches of a cylinder with a radius of 6 inches and a height of 5 inches. Use 3.14 for π and round your answer to the nearest tenth. CHALLENGE AND EXTEND 41. A cylinder has a radius of 8 cm and a height of 3 cm. Find the height of another cylinder that has a radius of 4 cm and the same surface area as the first cylinder. 42. If one gallon of paint covers 250 square feet, how many gallons of paint will be needed to cover the shed, not including the roof? If a ga
llon of paint costs $25, about how much will it cost to paint the walls of the shed? 43. The lateral area of a right rectangular prism is 144 cm 2 . Its length is three times its width, and its height is twice its width. Find its surface area. SPIRAL REVIEW 44. Rebecca’s car can travel 250 miles on one tank of gas. Rebecca has traveled 154 miles. Write an inequality that models m, the number of miles farther Rebecca can travel on the tank of gas. (Previous course) 45. Blood sugar is a measure of the number of milligrams of glucose in a deciliter of blood (mg/dL). Normal fasting blood sugar levels are above 70 mg/dL and below 110 mg/dL. Write an inequality that models s, the blood sugar level of a normal patient. (Previous course) Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. (Lesson 8-5) 46. BC 47. m∠ABC Draw the top, left, and right views of each object. Assume there are no hidden cubes. (Lesson 10-2) 48. 49. 50. 10- 4 Surface Area of Prisms and Cylinders 687 687 �� 10-4 Model Right and Oblique Cylinders In Lesson 10-4, you learned the difference between right and oblique cylinders. In this lab, you will make models of right and oblique cylinders. Use with Lesson 10-4 TEKS G.9.D Congruence and the geometry of size: analyze the characteristics of polyhedra and other three-dimensional figures and their component parts …. Activity 1 1 Use a compass to draw at least 10 circles with a radius of 3 cm each on cardboard and then cut them out. 2 Poke a hole through the center of each circle. 3 Unbend a paper clip part way and push it through the center of each circle to model a cylinder. The stack of cardboard circles can be held straight to model a right cylinder or tilted to model an oblique cylinder. Try This 1. On each cardboard model, use string or a rubber band to outline a cross section that is parallel to the base of the cylinder. What shape is each cross section? 2. Use string or a rubber band to outline a cross section of the cardboard model of the oblique cylinder that is perpendicular to the lateral surface. What shape is the cross section? Activity 2 1 Roll a piece of paper to make a right cylinder. Tape the edges. 2 Cut along the bottom and top to approximate an oblique cylinder. 3 Untape the edge and unroll the paper. What does the net for an oblique cylinder look like? Try This 3. Cut off the curved part of the net you created in Activity 2 and translate it to the opposite side to form a rectangle. How do the side lengths of the rectangle relate to the dimensions of the cylinder? Estimate the lateral area and surface area of the oblique cylinder. 688 688 Chapter 10 Spatial Reasoning 10-5 Surface Area of Pyramids and Cones TEKS G.8.D Congruence and the geometry of size: find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites …. Objectives Learn and apply the formula for the surface area of a pyramid. Learn and apply the formula for the surface area of a cone. Vocabulary vertex of a pyramid regular pyramid slant height of a regular pyramid altitude of a pyramid vertex of a cone axis of a cone right cone oblique cone slant height of a right cone altitude of a cone Also G.5.A, G.5.B, G.6.B, G.11.D Why learn this? A speaker uses part of the lateral surface of a cone to produce sound. Speaker cones are usually made of paper, plastic, or metal. (See Example 5.) �� The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base. The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid. The width of the rectangle is equal to the base perimeter of the pyramid. �� Lateral and Surface Area of a Regular Pyramid The lateral area of a regular pyramid with perimeter P and slant height ℓ is L = 1 __ Pℓ. 2 The surface area of a regular pyramid with lateral area L and base area B is S = L + B, or S = 1 __ Pℓ + B Finding Lateral Area and Surface Area of Pyramids Find the lateral area and surface area of each pyramid. A a regular square pyramid with base edge length 5 in. Pℓ and slant height 9 in20) (9) = 90 in 2 2 S = 1 _ 2 = 90 + 25 = 115 in 2 Pℓ + B Lateral area of a regular pyramid P = 4(5) = 20 in. Surface area of a regular pyramid B = 5 2 = 25 in 2 10- 5 Surface Area of Pyramids and Cones 689 689 �� Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth. B Step 1 Find the base perimeter and apothem. The base perimeter is 6 (4) = 24 m 2 . The apothem is 2 √  3 m, so the base area is 1 __ 2 aP = 1 __ 2 (2 √  3 ) (24) = 24 √  3 m 2 . Step 2 Find the lateral area. Pℓ L = 1 _ 2 = 1 _ (24) (7) = 84 m 2 2 Step 3 Find the surface area. S = 1 _ 2 Pℓ + B Lateral area of a regular pyramid Substitute 24 for P and 7 for ℓ. Surface area of a regular pyramid = 84 + 24 √  3 ≈ 125.6 cm 2 Substitute 24 √  3 for B. 1. Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft. The vertex of a cone is the point opposite the base. The axis of a cone is the segment with endpoints at the vertex and the center of the base. The axis of a right cone is perpendicular to the base. The axis of an oblique cone is not perpendicular to the base. The slant height of a right cone is the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a cone is a perpendicular segment from the vertex of the cone to the plane of the base. Lateral and Surface Area of a Right Cone The lateral area of a right cone with radius r and slant height ℓ is L = πrℓ. The surface area of a right cone with lateral area L and base area B is S = L + B, or S = πrℓ + π r 2 . 690 690 Chapter 10 Spatial Reasoning �� E X A M P L E 2 Finding Lateral Area and Surface Area of Right Cones Find the lateral area and surface area of each cone. Give your answers in terms of π. A a right cone with radius 2 m and slant height 3 m L = π rℓ = π (2) (3) = 6π m 2 S = π rℓ + π r 2 = 6π + π (2) 2 = 10π m 2 Lateral area of a cone Substitute 2 for r and 3 for ℓ. Surface area of a cone Substitute 2 for r and 3 for ℓ. B Step 1 Use the Pythagorean Theorem to find ℓ. ℓ = √  5 2 + 12 2 = 13 ft Step 2 Find the lateral area and surface area. L = πrℓ = π (5) (13) = 65π ft 2 S = πrℓ + π r 2 = 65π + π (5) 2 = 90π ft 2 Lateral area of a right cone Substitute 5 for r and 13 for ℓ. Surface area of a right cone Substitute 5 for r and 13 for ℓ. 2. Find the lateral area and surface area of the right cone. E X A M P L E 3 Exploring Effects of Changing Dimensions The radius and slant height of the right cone are tripled. Describe the effect on the surface area. original dimensions: S = π rℓ + π r 2 = π (3) (5) + π (3) 2 = 24π cm 2 radius and slant height tripled: S = π rℓ + π r 2 = π (9) (15) + π (9) 2 = 216π cm 2 Notice that 216π = 9 (24π) . If the length, width, and height are tripled, the surface area is multiplied by 3 2 , or 9. 3. The base edge length and slant height of the regular square pyramid are both multiplied by 2 __ 3 . Describe the effect on the surface area. 10- 5 Surface Area of Pyramids and Cones 691 691 �� E X A M P L E 4 Finding Surface Area of Composite Three-Dimensional Figures Find the surface area of the composite figure. The height of the cone is 90 - 45 = 45 cm. By the Pythagorean Theorem, ℓ = √  the cone is 28 2 + 45 2 = 53 cm. The lateral area of L = πrℓ = π (28) (53) = 1484π cm 2 . The lateral area of the cylinder is L = 2πrh = 2π (28) (45) = 2520π cm 2 . The base area is B = π r 2 = π (28) 2 = 784π cm 2 . S = (cone lateral area) + (cylinder lateral area) + (base area) = 2520π + 784π + 1484π = 4788π cm 2 4. Find the surface area of the composite figure. E X A M P L E 5 Electronics Application Electronics The paper cones of The paper cones of antique speakers were both functional and decorative. Some had elaborate patterns or shapes. Tim is replacing the paper cone of an antique speaker. He measured the existing cone and created the pattern for the lateral surface from a large circle. What is the diameter of the cone? The radius of the large circle used to create the pattern is the slant height of the cone. The area of the pattern is the lateral area of the cone. The area of the pattern is also 3 __ 4 of the area of the large circle, so πrℓ = 3 __ 4 π r 2 . πr (10) = 3 _ π (10) 2 4 Substitute 10 for ℓ, the slant height of the cone and the radius of the large circle. r = 7.5 in. Solve for r. The diameter of the cone is 2 (7.5) = 15 in. 5. What if…? If the radius of the large circle were 12 in., what would be the radius of the cone? THINK AND DISCUSS 1. Explain why the lateral area of a regular pyramid is 1 __ 2 the base perimeter times the slant height. 2. In a right cone, which is greater, the height or the slant height? Explain. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write the name of the part of the cone. 692 692 Chapter 10 Spatial Reasoning �� 10-5 Exercises Exercises GUIDED PRACTICE 1. Vocabulary Describe the endpoints of an axis of a cone Find the lateral area and surface area of each regular pyramid. p. 689 2. 3. KEYWORD: MG7 10-5 KEYWORD: MG7 Parent 4. a regular triangular pyramid with base edge length 15 in. and slant height 20 in. 691 Find the lateral area and surface area of each right cone
. Give your answers in terms of π. 5. 6. 7. a cone with base area 36π ft 2 and slant height 8 ft Describe the effect of each change on the surface area of the given figure. p. 691 8. The dimensions are cut in half. 9. The dimensions are tripled Find the surface area of each composite figure. p. 692 10. 11. 692 12. Crafts Anna is making a birthday hat from a pattern that is 3 __ 4 of a circle of colored paper. If Anna’s head is 7 inches in diameter, will the hat fit her? Explain. 10- 5 Surface Area of Pyramids and Cones 693 693 �� Independent Practice For See Exercises Example 13–15 16–18 19–20 21–22 23 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S23 Application Practice p. S37 PRACTICE AND PROBLEM SOLVING Find the lateral area and surface area of each regular pyramid. 13. 14. 15. a regular hexagonal pyramid with base edge length 7 ft and slant height 15 ft Find the lateral area and surface area of each right cone. Give your answers in terms of π. 16. 17. 18. a cone with radius 8 m and height that is 1 m less than twice the radius Describe the effect of each change on the surface area of the given figure. 19. The dimensions are divided by 3. 20. The dimensions are doubled. Find the surface area of each composite figure. 21. 22. 23. It is a tradition in England to celebrate May 1st by hanging cone-shaped baskets of flowers on neighbors’ door handles. Addy is making a basket from a piece of paper that is a semicircle with diameter 12 in. What is the diameter of the basket? �� Find the surface area of each figure. Shape Base Area Slant Height Surface Area 24. 25. 26. 27. Regular square pyramid Regular triangular pyramid Right cone Right cone 36 cm 2 √  3 m 2 16π in 2 π ft 2 5 cm √  3 m 7 in. 2 ft 694 694 Chapter 10 Spatial Reasoning �� 28. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A juice container is a regular square pyramid with the dimensions shown. a. Find the surface area of the container to the nearest tenth. b. The manufacturer decides to make a container in the shape of a right cone that requires the same amount of material. The base diameter must be 9 cm. Find the slant height of the container to the nearest tenth. 29. Find the radius of a right cone with slant height 21 m and surface area 168π m 2 . 30. Find the slant height of a regular square pyramid with base perimeter 32 ft and surface area 256 ft 2 . 31. Find the base perimeter of a regular hexagonal pyramid with slant height 10 cm and lateral area 120 cm 2 . 32. Find the surface area of a right cone with a slant height of 25 units that has its base centered at (0, 0, 0) and its vertex at (0, 0, 7) . Architecture Find the surface area of each composite figure. 33. 34. The Pyramid Arena seats 21,000 people. The base of the pyramid is larger than six football fields. 35. Architecture The Pyramid Arena in Memphis, Tennessee, is a square pyramid with base edge lengths of 200 yd and a height of 32 stories. Estimate the area of the glass on the sides of the pyramid. (Hint: 1 story ≈ 10 ft) 36. Critical Thinking Explain why the slant height of a regular square pyramid must be greater than half the base edge length. 37. Write About It Explain why slant height is not defined for an oblique cone. 38. Which expressions represent the surface area of the regular square pyramid? I. t 2 _ 16 II. t 2 _ 16 + tℓ _ 2 + ts _ 2 III. t _ 2 ( t _ + ℓ) 8 I only II only I and II II and III 39. A regular square pyramid has a slant height of 18 cm and a lateral area of 216 cm 2 . What is the surface area? 252 cm 2 234 cm 2 225 cm 2 240 cm 2 40. What is the lateral area of the cone? 450π cm 2 1640π cm 2 360π cm 2 369π cm 2 10- 5 Surface Area of Pyramids and Cones 695 695 �� CHALLENGE AND EXTEND 41. A frustum of a cone is a part of the cone with two parallel bases. The height of the frustum of the cone is half the height of the original cone. a. Find the surface area of the original cone. b. Find the lateral area of the top of the cone. c. Find the area of the top base of the frustum. d. Use your results from parts a, b, and c to find the �� surface area of the frustum of the cone. �� 42. A frustum of a pyramid is a part of the pyramid with two parallel bases. The lateral faces of the frustum are trapezoids. Use the area formula for a trapezoid to derive a formula for the lateral area of a frustum of a regular square pyramid with base edge lengths b 1 and b 2 and slant height ℓ. 43. Use the net to derive the formula for the lateral area of a right cone with radius r and slant height ℓ. a. The length of the curved edge of the lateral surface must equal the circumference of the base. Find the circumference c of the base in terms of r. b. The lateral surface is part of a larger circle. Find the circumference C of the larger circle. c. The lateral surface area is c __ times the area of C the larger circle. Use your results from parts a and b to find c __ . C �� d. Find the area of the larger circle. Use your result and the result from part c to find the lateral area L. SPIRAL REVIEW State whether the following can be described by a linear function. (Previous course) 44. the surface area of a right circular cone with height h and radius r 45. the perimeter of a rectangle with a height h that is twice as large as its width w 46. the area of a circle with radius r A point is chosen randomly in ACEF. Find the probability of each event. Round to the nearest hundredth. (Lesson 9-6) 47. The point is in △BDG. 48. The point is in ⊙H. 49. The point is in the shaded region. Find the surface area of each right prism or right cylinder. Round your answer to the nearest tenth. (Lesson 10-4) � � � � �� 50. 51. �� 52. �� 696 696 Chapter 10 Spatial Reasoning 10-6 Volume of Prisms and Cylinders TEKS G.8.D Congruence and the geometry of size: find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites …. Objectives Learn and apply the formula for the volume of a prism. Learn and apply the formula for the volume of a cylinder. Vocabulary volume Also G.1.B, G.5.A, G.5.B, G.11.D Who uses this? Marine biologists must ensure that aquariums are large enough to accommodate the number of fish inside them. (See Example 2.) The volume of a threedimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill the interior. A cube built out of 27 unit cubes has a volume of 27 cubic units. Cavalieri’s principle says that if two three-dimensional figures have the same height and have the same cross-sectional area at every level, they have the same volume. A right prism and an oblique prism with the same base and height have the same volume. Volume of a Prism The volume of a prism with base area B and height h is V = Bh. The volume of a right rectangular prism with length ℓ, width w, and height h is V = ℓwh. The volume of a cube with edge length s is Finding Volumes of Prisms Find the volume of each prism. Round to the nearest tenth, if necessary. A V = ℓwh = (10) (12) (8) = 960 cm 3 Volume of a right rectangular prism Substitute 10 for ℓ, 12 for w, and 8 for h. B a cube with edge length 10 cm V = s 3 = 10 3 = 1000 cm 3 Volume of a cube Substitute 10 for s. 10-6 Volume of Prisms and Cylinders 697 697 �� To review the area of a regular polygon, see page 601. To review tangent ratios, see page 525. Find the volume of each prism. Round to the nearest tenth, if necessary. C a right regular pentagonal prism with base edge length 5 m and height 7 m Step 1 Find the apothem a of the base. First draw a right triangle on one base as shown. The measure of the angle with its vertex at = 36°. 10 the center is 360° _ tan 36° = 2.5 _ a a = 2.5 _ tan 36° The leg of the triangle is half the side length, or 2.5 m. Solve for a. Step 2 Use the value of a to find the base area. aP = .5 _ tan 36° ) (25) = 31.25 _ tan 36° P = 5 (5) = 25 m Step 3 Use the base area to find the volume. V = Bh = 31.25 _ · 7 ≈ 301.1 m 3 tan 36° 1. Find the volume of a triangular prism with a height of 9 yd whose base is a right triangle with legs 7 yd and 5 yd long. E X A M P L E 2 Marine Biology Application The aquarium at the right is a rectangular prism. Estimate the volume of the water in the aquarium in gallons. The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds. (Hint: 1 gallon ≈ 0.134 ft 3 ) Step 1 Find the volume of the aquarium in cubic feet. V = ℓwh = (120) (60) (8) = 57,600 ft 3 Step 2 Use the conversion factor to estimate the volume in gallons. 57, 600 ft 3 · 1 gallon _ 0.134 ft 3 ≈ 429,851 gallons 1 gallon _ = 1 0.134 ft 3 1 gallon _ 0.134 ft 3 Step 3 Use the conversion factor of the water. 8.33 pounds __ 1 gallon to estimate the weight 429,851 gallons · ≈ 3,580,659 pounds 8.33 pounds __ 1 gallon 8.33 pounds __ = 1 1 gallon Marine Biology The Islands of Steel habitat at the Texas State Aquarium in Corpus Christi has a volume of 132,000 gallons. Over 150 animals live in the habitat, including a sand tiger shark and nurse sharks. The aquarium holds about 429,851 gallons. The water in the aquarium weighs about 3,580,659 pounds. 2. What if…? Estimate the volume in gallons and the weight of the water in the aquarium above if the height were doubled. 698 698 Chapter 10 Spatial Reasoning 120 ft60 ft8 ftge07se_c10l06003aAB �� Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they have the same volume. Volume of a Cylinder The volume of a cylinder with base area B, radius r, and height h is V = Bh, or V = π r 2h. E X A M P L E 3 Finding Volumes of Cylinders Find the v
olume of each cylinder. Give your answers both in terms of π and rounded to the nearest tenth. A V = π r 2h = π (8)2(12) Volume of a cylinder Substitute 8 for r and 12 for h. = 768π cm 3 ≈ 2412.7 cm 3 B a cylinder with a base area of 36π in 2 and a height equal to twice the radius Step 1 Use the base area to find the radius. π r 2 = 36π r = 6 Substitute 36π for the base area. Solve for r. Step 2 Use the radius to find the height. The height is equal to twice the radius. h = 2r = 2 (6) = 12 cm Step 3 Use the radius and height to find the volume. Volume of a cylinder V = π r 2h = π (6)2(12)= 432π in 3 ≈ 1357.2 in 3 Substitute 6 for r and 12 for h. 3. Find the volume of a cylinder with a diameter of 16 in. and a height of 17 in. Give your answer both in terms of π and rounded to the nearest tenth. 10-6 Volume of Prisms and Cylinders 699 699 �� E X A M P L E 4 Exploring Effects of Changing Dimensions The radius and height of the cylinder are multiplied by 1 __ . Describe the effect on the volume. 2 original dimensions6) 2 (12) = 432π m 3 radius and height multiplied by 1 __ 3) 2 (6) = 54π m 3 Notice that 54π = 1 __ 8 (432π) . If the radius and height are multiplied by 1 __ 2 , the volume is multiplied by ( 1 __ 2 ) , or 1 __ 8 . 3 4. The length, width, and height of the prism are doubled. Describe the effect on the volume. E X A M P L E 5 Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. The base area of the prism is B = 1 __ 2 (6) (8) = 24 m 2 . The volume of the prism is V = Bh = 24 (9) = 216 m 3 . �� The cylinder’s diameter equals the hypotenuse of the prism’s base, 10 m. So the radius is 5 m. The volume of the cylinder is V = π r 2 h = π (5) 2 (5) = 125π m 3 . �� The total volume of the figure is the sum of the volumes. V = 216 + 125π ≈ 608.7 m 3 �� 5. Find the volume of the composite figure. Round to the nearest tenth. THINK AND DISCUSS 1. Compare the formula for the volume of a prism with the formula for the volume of a cylinder. 2. Explain how Cavalieri’s principle relates to the formula for the volume of an oblique prism. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write the formula for the volume. 700 700 Chapter 10 Spatial Reasoning �� 10-6 Exercises Exercises KEYWORD: MG7 10-6 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In a right cylinder, the altitude is ? the axis. (longer than, shorter ̶̶̶̶ than, or the same length as Find the volume of each prism. p. 697 2. 3. 4. a cube with edge length 8 ft . Food The world’s largest ice cream cake, built in p. 698 New York City on May 25, 2004, was approximately a 19 ft by 9 ft by 2 ft rectangular prism. Estimate the volume of the ice cream cake in gallons. If the density of the ice cream was 4.73 pounds per gallon, estimate the weight of the cake. (Hint: 1 gallon ≈ 0.134 cubic feet. 699 Find the volume of each cylinder. Give your answers both in terms of π and rounded to the nearest tenth. 6. 7. 8. a cylinder with base area 25π cm 2 and height 3 cm more than the radius . 700 Describe the effect of each change on the volume of the given figure. 9. The dimensions are multiplied by 1 _ . 4 10. The dimensions are tripled Find the volume of each composite figure. Round to the nearest tenth. p. 700 11. 12. 10-6 Volume of Prisms and Cylinders 701 701 �� PRACTICE AND PROBLEM SOLVING Find the volume of each prism. 13. 14. Independent Practice For See Exercises Example 13–15 16 17–19 20–21 22–23 1 2 3 4 5 TEKS TEKS TAKS TAKS 15. a square prism with a base area of 49 ft 2 and a height 2 ft less than the base Skills Practice p. S23 Application Practice p. S37 edge length 16. Landscaping Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle. If he wants to fill it to a depth of 4 in., how many cubic yards of dirt does he need? If dirt costs $25 per yd 3 , how much will the project cost? (Hint: 1 yd 3 = 27 ft 3 ) Find the volume of each cylinder. Give your answers both in terms of π and rounded to the nearest tenth. 17. 18. 19. a cylinder with base area 24π cm 2 and height 16 cm Describe the effect of each change on the volume of the given figure. 20. The dimensions are multiplied by 5. 21. The dimensions are multiplied by 3 _ . 5 Find the volume of each composite figure. 22. 23. 24. One cup is equal to 14.4375 in 3 . If a 1 c cylindrical measuring cup has a radius of 2 in., what is its height? If the radius is 1.5 in., what is its height? 25. Food A cake is a cylinder with a diameter of 10 in. and a height of 3 in. For a party, a coin has been mixed into the batter and baked inside the cake. The person who gets the piece with the coin wins a prize. a. Find the volume of the cake. Round to the nearest tenth. b. Probability Keka gets a piece of cake that is a right rectangular prism with a 3 in. by 1 in. base. What is the probability that the coin is in her piece? Round to the nearest tenth. 702 702 Chapter 10 Spatial Reasoning �� 26. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A cylindrical juice container with a 3 in. diameter has a hole for a straw that is 1 in. from the side. Up to 5 in. of a straw can be inserted. a. Find the height h of the container to the nearest tenth. b. Find the volume of the container to the nearest tenth. c. How many ounces of juice does the container hold? (Hint: 1 in 3 ≈ 0.55 oz) Math History 27. Find the height of a rectangular prism with length 5 ft, width 9 ft, and volume 495 ft 3 . 28. Find the area of the base of a rectangular prism with volume 360 in 3 and height 9 in. 29. Find the volume of a cylinder with surface area 210π m 2 and height 8 m. 30. Find the volume of a rectangular prism with vertices (0, 0, 0) , (0, 3, 0) , (7, 0, 0) , (7, 3, 0) , (0, 0, 6) , (0, 3, 6) , (7, 0, 6) , and (7, 3, 6) . 31. You can use displacement to find the volume of an irregular object, such as a stone. Suppose the tank shown is filled with water to a depth of 8 in. A stone is placed in the tank so that it is completely covered, causing the water level to rise by 2 in. Find the volume of the stone. �� Archimedes (287–212 B.C.E.) used displacement to find the volume of a gold crown. He discovered that the goldsmith had cheated the king by substituting an equal weight of silver for part of the gold. 32. Food A 1 in. cube of cheese is one serving. How many servings are in a 4 in. by 4 in. by 1 __ 4 in. slice? 33. History In 1919, a cylindrical tank containing molasses burst and flooded the city of Boston, Massachusetts. The tank had a 90 ft diameter and a height of 52 ft. How many gallons of molasses were in the tank? (Hint: 1 gal ≈ 0.134 ft 3 ) 34. Meteorology If 3 in. of rain fall on the property shown, what is the volume in cubic feet? In gallons? The density of water is 8.33 pounds per gallon. What is the weight of the rain in pounds? (Hint: 1 gal ≈ 0.134 ft 3 ) �� 35. Critical Thinking The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism. Make a conjecture about the ratio of the surface area of the new prism to its volume. Test your conjecture using a cube with an edge length of 1 and a scale factor of 2. 36. Write About It How can you change the edge length of a cube so that its volume is doubled? 37. Abigail has a cylindrical candle mold with the dimensions shown. If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm, about how many candles can she make after melting the block of wax? 14 31 35 76 10-6 Volume of Prisms and Cylinders 703 703 �� 38. A 96-inch piece of wire was cut into equal segments that were then connected to form the edges of a cube. What is the volume of the cube? 576 in 3 512 in 3 729 in 3 1728 in 3 39. One juice container is a rectangular prism with a height of 9 in. and a 3 in. by 3 in. square base. Another juice container is a cylinder with a radius of 1.75 in. and a height of 9 in. Which best describes the relationship between the two containers? The prism has the greater volume. The cylinder has the greater volume. The volumes are equivalent. The volumes cannot be determined. 40. What is the volume of the three-dimensional object with the dimensions shown in the three views below? �� 160 cm 3 240 cm 3 840 cm 3 1000 cm 3 CHALLENGE AND EXTEND Algebra Find the volume of each three-dimensional figure in terms of x. 41. 42. �� 43. � �� 44. The volume in cubic units of a cylinder is equal to its surface area in square units. Prove that the radius and height must both be greater than 2. SPIRAL REVIEW 45. Marcy, Rachel, and Tina went bowling. Marcy bowled 100 less than twice Rachel’s score. Tina bowled 40 more than Rachel’s score. Rachel bowled a higher score than Marcy. What is the greatest score that Tina could have bowled? (Previous course) 46. Max can type 40 words per minute. He estimates that his term paper contains about 5000 words, and he takes a 15-minute break for every 45 minutes of typing. About how much time will it take Max to type his term paper? (Previous course) ABCD is a parallelogram. Find each measure. (Lesson 6-2) 47. m∠ABC 48. BC 49. AB Find the surface area of each figure. Round to the nearest tenth. (Lesson 10-5) 50. a square pyramid with slant height 10 in. and base edge length 8 in. �� 51. a regular pentagonal pyramid with slant height 8 cm and base edge length 6 cm 52. a right cone with slant height 2 ft and a base with circumference of π ft 704 704 Chapter 10 Spatial Reasoning 10-7 Volume of Pyramids and Cones
TEKS G.8.D Congruence and the geometry of size: find surface areas and volumes of ..., pyramids, ..., cones, .... Objectives Learn and apply the formula for the volume of a pyramid. Learn and apply the formula for the volume of a cone. Also G.5.A, G.5.B, G.11.D Who uses this? The builders of the Rainforest Pyramid in Galveston, Texas, needed to calculate the volume of the pyramid to plan the climate control system. (See Example 2.) The volume of a pyramid is related to the volume of a prism with the same base and height. The relationship can be verified by dividing a cube into three congruent square pyramids, as shown. The square pyramids are congruent, so they have the same volume. The volume of each pyramid is one third the volume of the cube. Volume of a Pyramid The volume of a pyramid with base area B and height h is V = 1 _ 3 Bh. E X A M P L E 1 Finding Volumes of Pyramids Find the volume of each pyramid. A a rectangular pyramid with length 7 ft, width 9 ft, and height 12 ft Bh = 1 _ V = 1 _ (7 · 9) (12) = 252 ft 3 3 3 B the square pyramid The base is a square with a side length of 4 in., and the height is 6 in. Bh = 6) = 32 in 3 3 3 10-7 Volume of Pyramids and Cones 705 705 �� Find the volume of the pyramid. C the trapezoidal pyramid with base ABCD, where and ̶̶ AE ⊥ plane ABC ̶̶ AB ǁ ̶̶ CD Step 1 Find the area of the base9 + 18) 6 2 = 81 m 2 Area of a trapezoid Substitute 9 for b 1 , 18 for b 2 , and 6 for h. Simplify. Step 2 Use the base area and the height to find the volume. ̶̶ AE is the altitude, so the height Because ̶̶ AE ⊥ plane ABC, Bh is equal to AE81) (10) 3 = 270 m 3 Volume of a pyramid Substitute 81 for B and 10 for h. 1. Find the volume of a regular hexagonal pyramid with a base edge length of 2 cm and a height equal to the area of the base. E X A M P L E 2 Architecture Application The Rainforest Pyramid in Galveston, Texas, is a square pyramid with a base area of about 1 acre and a height of 10 stories. Estimate the volume in cubic yards and in cubic feet. (Hint: 1 acre = 4840 yd 2 , 1 story ≈ 10 ft) The base is a square with an area of about 4840 yd 2 . The base edge length is √  is about 10 (10) = 100 ft, or about 33 yd. 4840 ≈ 70 yd. The height First find the volume in cubic yards. Bh V = 1 _ 3 = 1 _ ( 70 2 ) (33) = 53,900 yd 3 3 Volume of a regular pyramid Substitute 70 2 for B and 33 for h. Then convert your answer to find the volume in cubic feet. The volume of one cubic yard is (3 ft) (3 ft) (3 ft) = 27 ft 3 . Use the conversion factor 27 ft 3 ____ 1 yd 3 to find the volume in cubic feet. 53,900 yd 3 · 27 ft 3 _ 1 yd 3 ≈ 1,455,300 ft 3 2. What if…? What would be the volume of the Rainforest Pyramid if the height were doubled? 706 706 Chapter 10 Spatial Reasoning �� Volume of Cones The volume of a cone with base area B, radius r, and height h is V = 1 _ 3 or V = 1 _ π r 2 h. 3 Bh, E X A M P L E 3 Finding Volumes of Cones Find the volume of each cone. Give your answers both in terms of π and rounded to the nearest tenth. A a cone with radius 5 cm and height 12 cm 5) 2 (12) 3 = 100π cm 3 ≈ 314.2 cm 3 Volume of a cone Substitute 5 for r and 12 for h. Simplify. B a cone with a base circumference of 21π cm and a height 3 cm less than twice the radius Step 1 Use the circumference to find the radius. 2πr = 21π Substitute 21π for C. r = 10.5 cm Divide both sides by 2π. Step 2 Use the radius to find the height. 2 (10.5) - 3 = 18 cm The height is 3 cm less than twice the radius. Step 3 Use the radius and height to find the volume10.5) 2 (18) 3 Volume of a cone Substitute 10.5 for r and 18 for h. = 661.5π cm 3 ≈ 2078.2 cm 3 Simplify. C Step 1 Use the Pythagorean Theorem to find the height. 7 2 + h 2 = 25 2 h 2 = 576 h = 24 Pythagorean Theorem Subtract 7 2 from both sides. Take the square root of both sides. Step 2 Use the radius and height to find the volume7) 2 (24) 3 Volume of a cone Substitute 7 for r and 24 for h. = 392π ft 3 ≈ 1231.5 ft 3 Simplify. 3. Find the volume of the cone. 10-7 Volume of Pyramids and Cones 707 707 �� E X A M P L E 4 Exploring Effects of Changing Dimensions The length, width, and height of the rectangular pyramid are multiplied by 1 __ . 4 Describe the effect on the volume. original dimensions: Bh V = 1 _ 3 = 1 _ (24 · 20) (20) 3 = 3200 ft 3 length, width, and height multiplied by 1 _ : 4 Bh V = 1 _ 3 = 1 _ (6 · 5) (5) 3 = 50 ft 3 Notice that 50 = 1 __ 64 (3200) . If the length, width, and height are multiplied by 1 __ 4 , the volume is multiplied by ( 1 __ 4 ) , or 1 __ 64 . 3 4. The radius and height of the cone are doubled. Describe the effect on the volume. E X A M P L E 5 Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. The volume of the cylinder is V = π r 2 h = π (2) 2 (2) = 8π in 3 . The volume of the cone is 2) 2 (3) = 4π in 3 . 3 3 The volume of the composite figure is the sum of the volumes. V = 8π + 4π = 12π in 3 ≈ 37.7 in 3 5. Find the volume of the composite figure. THINK AND DISCUSS 1. Explain how the volume of a pyramid is related to the volume of a prism with the same base and height. 2. GET ORGANIZED Copy and complete the graphic organizer. 708 708 Chapter 10 Spatial Reasoning �� 10-7 Exercises Exercises KEYWORD: MG7 10-7 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary The altitude of a pyramid is or oblique) ? to the base. (perpendicular, parallel, ̶̶̶̶ Find the volume of each pyramid. Round to the nearest tenth, if necessary. p. 705 2. 3. 4. a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft . 706 5. Geology A crystal is cut into the shape formed by two square pyramids joined at the base. Each pyramid has a base edge length of 5.7 mm and a height of 3 mm. What is the volume to the nearest cubic millimeter of the crystal. 707 Find the volume of each cone. Give your answers both in terms of π and rounded to the nearest tenth. 6. 7. 8. a cone with radius 12 m and height 20 Describe the effect of each change on the volume of the given figure. p. 708 9. The dimensions are tripled. 10. The dimensions are multiplied by Find the volume of each composite figure. Round to the nearest tenth, if necessary. p. 708 11. 12. 10-7 Volume of Pyramids and Cones 709 709 ge07sec10l07003aAB3 mm5.7 mm�� Independent Practice For See Exercises Example 13–15 16 17–19 20–21 22–23 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S23 Application Practice p. S37 PRACTICE AND PROBLEM SOLVING Find the volume of each pyramid. Round to the nearest tenth, if necessary. 13. 14. 15. 15. a regular square pyramid with base edge length 12 ft and slant height 10 ft 16. Carpentry A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards. What is the volume of the attic in cubic feet? In cubic yards? �� Find the volume of each cone. Give your answers both in terms of π and rounded to the nearest tenth. 17. 18. 19. a cone with base area 36π ft 2 and a height equal to twice the radius �� Describe the effect of each change on the volume of the given figure. 20. The dimensions are multiplied by 1 _ . 3 21. The dimensions are multiplied by 6. Find the volume of each composite figure. Round to the nearest tenth, if necessary. 22. 23. Find the volume of each right cone with the given dimensions. Give your answers in terms of π. 24. radius 3 in. height 7 in. 25. diameter 5 m height 2 m 26. radius 28 ft 27. diameter 24 cm slant height 53 ft slant height 13 cm 710 710 Chapter 10 Spatial Reasoning �� Find the volume of each regular pyramid with the given dimensions. Round to the nearest tenth, if necessary. Number of sides of base Base edge length Height Volume 28. 29. 30. 31. 3 4 5 6 10 ft 15 m 9 in. 8 cm 6 ft 18 m 12 in. 3 cm 32. Find the height of a rectangular pyramid with length 3 m, width 8 m, and volume 112 m 3 . 33. Find the base circumference of a cone with height 5 cm and volume 125π cm 3 . 34. Find the volume of a cone with slant height 10 ft and height 8 ft. 35. Find the volume of a square pyramid with slant height 17 in. and surface area 800 in 2 . 36. Find the surface area of a cone with height 20 yd and volume 1500π yd 3 . 37. Find the volume of a triangular pyramid with vertices (0, 0, 0) , (5, 0, 0) , (0, 3, 0) , and (0, 0, 7) . 38. /////ERROR ANALYSIS///// Which volume is incorrect? Explain the error. 39. Critical Thinking Write a ratio comparing the volume of the prism to the volume of the composite figure. Explain your answer. 40. Write About It Explain how you would find the volume of a cone, given the radius and the surface area. 41. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A juice stand sells smoothies in cone-shaped cups that are 8 in. tall. The regular size has a 4 in. diameter. The jumbo size has an 8 in. diameter. a. Find the volume of the regular size to the nearest tenth. b. Find the volume of the jumbo size to the nearest tenth. c. The regular size costs $1.25. What would be a reasonable price for the jumbo size? Explain your reasoning. �� 10-7 Volume of Pyramids and Cones 711 711 �� 42. Find the volume of the cone. 432π cm 3 720π cm 3 1296π cm 3 2160π cm 3 43. A square pyramid has a slant height of 25 m and a lateral area of 350 m 2 . Which is closest to the volume? 392 m 3 1176 m 3 404 m 3 1225 m 3 44. A cone has a volume of 18π in 3 . Which are possible dimensions of the cone? Diameter 3 in., height 6 in. Diameter 1 in., height 18 in. Diameter 6 in., height 6 in. Diameter 6 in., height 3 in. 45. Gridde
d Response Find the height in centimeters of a square pyramid with a volume of 243 cm 3 and a base edge length equal to the height. CHALLENGE AND EXTEND Each cone is inscribed in a regular pyramid with a base edge length of 2 ft and a height of 2 ft. Find the volume of each cone. 46. 47. 48. 49. A regular octahedron has 8 faces that are equilateral triangles. Find the volume of a regular octahedron with a side length of 10 cm. 50. A cylinder has a radius of 5 in. and a height of 3 in. Without calculating the volumes, find the height of a cone with the same base and the same volume as the cylinder. Explain your reasoning. SPIRAL REVIEW Find the unknown numbers. (Previous course) 51. The difference of two numbers is 24. The larger number is 4 less than 3 times the smaller number. 52. Three times the first number plus the second number is 88. The first number times 10 is equal to 4 times the second. 53. The sum of two numbers is 197. The first number is 20 more than 1 __ 2 of the second number. Explain why the triangles are similar, then find each length. (Lesson 7-3) 54. AB 55. PQ Find AB and the coordinates of the midpoint of if necessary. (Lesson 10-3) 56. A (1, 1, 2) , B (8, 9, 10) 57. A (-4, -1, 0) , B (5, 1, -4) ̶̶ AB . Round to the nearest tenth, 58. A (2, -2, 4) , B (-2, 2, -4) 59. A (-3, -1, 2) , B (-1, 5, 5) 712 712 Chapter 10 Spatial Reasoning �� Functional Relationships in Formulas Algebra You have studied formulas for several solid figures. Here you will see how a change in one dimension affects the measurements of the other dimensions. See Skills Bank page S63 Example A square prism has a volume of 21 cubic units. Write an equation that describes the base edge length s in terms of the height h. Graph the relationship in a coordinate plane with h on the horizontal axis and s on the vertical axis. What happens to the base edge length as the height increases? First use the volume formula to write an equation. V = Bh 100 = s 2h Volume of a prism Substitute 100 for V and s 2 for B. Then solve for s to get an equation for s in terms of h. s 2 = 100_ h s = √100_ h s = 10_ √  h Divide both sides by h. Take the square root of both sides. √  100 = 10 Graph the equation. First make a table of h- and s-values. Then plot the points and draw a smooth curve through the points. Notice that the function is not defined for h = 0. h 1 4 9 16 25 s 10 5 ̶ 3 3. 2.5 2 As the height of the prism increases, the base edge length decreases. Try This TAKS Grades 9–11 Obj. 8 1. A right cone has a radius of 10 units. Write an equation that describes the slant height ℓ in terms of the surface area S. Graph the relationship in a coordinate plane with S on the horizontal axis and ℓ on the vertical axis. What happens to the slant height as the surface area increases? 2. A cylinder has a height of 5 units. Write an equation that describes the radius r in terms of the volume V. Graph the relationship in a coordinate plane with V on the horizontal axis and r on the vertical axis. What happens to the radius as the volume increases? On Track for TAKS 713 713 �� 10-8 Spheres TEKS G.8.D Congruence and the geometry of size: find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites …. Objectives Learn and apply the formula for the volume of a sphere. Learn and apply the formula for the surface area of a sphere. Vocabulary sphere center of a sphere radius of a sphere hemisphere great circle Who uses this? Biologists study the eyes of deep-sea predators such as the giant squid to learn about their behavior. (See Example 2.) A sphere is the locus of points in space that are a fixed distance from a given point called the center of a sphere . A radius of a sphere connects the center of the sphere to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres. Also G.5.A, G.5.B, G.11.D The figure shows a hemisphere and a cylinder with a cone removed from its interior. The cross sections have the same area at every level, so the volumes are equal by Cavalieri’s Principle. You will prove that the cross sections have equal areas in Exercise 39. V (hemisphere) = V (cylinder) - V (cone = 2_ 3 = 2_ 3 = 2_ 3 π r 2(r) π r 2h π r 3 The height of the hemisphere is equal to the radius. The volume of a sphere with radius r is twice the volume of the hemisphere, or V = 4__ 3 π r 3 . Volume of a Sphere The volume of a sphere with radius r is Finding Volumes of Spheres Find each measurement. Give your answer in terms of π. A the volume of the sphere 9) 3 3 Substitute 9 for r. = 972π cm 3 Simplify. 714 714 Chapter 10 Spatial Reasoning �� Find each measurement. Give your answer in terms of π. B the diameter of a sphere with volume 972π in 3 972π = 4 _ π r 3 3 729 = r 3 r = 9 d = 18 in. Substitute 972π for V. Divide both sides by 4 _ π. 3 Take the cube root of both sides. d = 2r C the volume of the hemisphere 4) 3 = 128π _ = 2 _ 3 3 m 3 Volume of a hemisphere Substitute 4 for r. 1. Find the radius of a sphere with volume 2304π ft Biology Application Giant squid need large eyes to see their prey in low light. The eyeball of a giant squid is approximately a sphere with a diameter of 25 cm, which is bigger than a soccer ball. A human eyeball is approximately a sphere with a diameter of 2.5 cm. How many times as great is the volume of a giant squid eyeball as the volume of a human eyeball? human eyeball: giant squid eyeball1.25) 3 ≈ 8.18 cm 12.5) 3 ≈ 8181.23 cm 3 3 A giant squid eyeball is about 1000 times as great in volume as a human eyeball. 2. A hummingbird eyeball has a diameter of approximately 0.6 cm. How many times as great is the volume of a human eyeball as the volume of a hummingbird eyeball? In the figure, the vertex of the pyramid is at the center of the sphere. The height of the pyramid is approximately the radius r of the sphere. Suppose the entire sphere is filled with n pyramids that each have base area B and height r. Br + 1 _ V (sphere) ≈ 1 _ Br + … + Br) 4 _ 3 4π r 2 ≈ nB 3 Br The sphere’s volume is close to the sum of the volumes of the pyramids. Divide both sides by 1 _ 3 πr. If the pyramids fill the sphere, the total area of the bases is approximately equal to the surface area of the sphere S, so 4π r 2 ≈ S. As the number of pyramids increases, the approximation gets closer to the actual surface area. 10 - 8 Spheres 715 715 �� Surface Area of a Sphere The surface area of a sphere with radius r is S = 4π Finding Surface Area of Spheres Find each measurement. Give your answers in terms of π. A the surface area of a sphere with diameter 10 ft S = 4π r 2 S = 4π (5)2 = 200π ft 2 Substitute 5 for r. B the volume of a sphere with surface area 144π m 2 S = 4π r 2 144π = 4π 6) 3 = 288π m 3 3 Substitute 144π for S. Solve for r. Substitute 6 for r. The volume of the sphere is 288π m 3 . C the surface area of a sphere with a great circle that has an area of 4π in 2 π r 2 = 4π Substitute 4π for A in the formula for the area of a circle. Solve for r. r = 2 S = 4π r 2 = 4π (2) 2 = 16π in 2 Substitute 2 for r in the surface area formula. 3. Find the surface area of the sphere. E X A M P L E 4 Exploring Effects of Changing Dimensions The radius of the sphere is tripled. Describe the effect on the volume. original dimensions3) 3 3 = 36π m 3 radius tripled9) 3 3 = 972π m 3 Notice that 972π = 27 (36π) . If the radius is tripled, the volume is multiplied by 27. 4. The radius of the sphere above is divided by 3. Describe the effect on the surface area. 716 716 Chapter 10 Spatial Reasoning �� E X A M P L E 5 Finding Surface Areas and Volumes of Composite Figures Find the surface area and volume of the composite figure. Give your answers in terms of π. Step 1 Find the surface area of the composite figure. The surface area of the composite figure is the sum of the surface area of the hemisphere and the lateral area of the cone. S (hemisphere) = 1 _ (4π r 2 ) = 2π (7) 2 = 98π cm 2 2 L (cone) = πrℓ = π (7) (25) = 175π cm 2 The surface area of the composite figure is 98π + 175π = 273π cm 2 . Step 2 Find the volume of the composite figure. First find the height of the cone. h = √  25 2 - 7 2 Pythagorean Theorem = √  576 = 24 cm Simplify. The volume of the composite figure is the sum of the volume of the hemisphere and the volume of the conehemispherecone) = 1 _ π (7) 2 (24) = 392π cm 3 3 3 π (7) 3 = 686π _ 3 3 cm 3 The volume of the composite figure is 686π _ + 392π = 1862π _ cm 3 . 3 3 5. Find the surface area and volume of the composite figure. THINK AND DISCUSS 1. Explain how to find the surface area of a sphere when you know the area of a great circle. 2. Compare the volume of the sphere with the volume of the composite figure. 3. GET ORGANIZED Copy and complete the graphic organizer. 10 - 8 Spheres 717 717 �� 10-8 Exercises Exercises KEYWORD: MG7 10-8 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Describe the endpoints of a radius of a sphere Find each measurement. Give your answers in terms of π. p. 714 2. the volume of the hemisphere 3. the volume of the sphere 4. the radius of a sphere with volume 288π cm . Food Approximately how many times as great is the volume of the grapefruit p. 715 as the volume of the lime? �� Find each measurement. Give your answers in terms of π. p. 716 6. the surface area of the sphere 7. the surface area of the sphere 8. the volume of a sphere with surface area 6724π ft Describe the effect of each change on the given measurement of the figure. p. 716 9. surface area 10. volume The dimensions are doubled. The dimensions are multiplied by Find the surface area and volume of each composite figure. p. 717 11. 12. 718 718 Chapter 10 Spatial Reasoning ��
�� Independent Practice For See Exercises Example 13–15 16 17–19 20–21 22–23 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S23 Application Practice p. S37 PRACTICE AND PROBLEM SOLVING Find each measurement. Give your answers in terms of π. 13. the volume of the sphere 14. the volume of the hemisphere 15. the diameter of a sphere with volume 7776π in 3 16. Jewelry The size of a cultured pearl is typically indicated by its diameter in mm. How many times as great is the volume of the 9 mm pearl as the volume of the 6 mm pearl? �� Find each measurement. Give your answers in terms of π. �� 17. the surface area of the sphere 18. the surface area of the sphere 19. the volume of a sphere with surface area 625π m 2 Describe the effect of each change on the given measurement of the figure. 20. surface area The dimensions are multiplied by 1 _ . 5 21. volume The dimensions are multiplied by 6. Find the surface area and volume of each composite figure. 22. 23. 24. Find the radius of a hemisphere with a volume of 144π cm 3 . 25. Find the circumference of a sphere with a surface area of 60π in 2 . 26. Find the volume of a sphere with a circumference of 36π ft. 27. Find the surface area and volume of a sphere centered at (0, 0, 0) that passes through the point (2, 3, 6) . 28. Estimation A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter. Estimate the surface area and volume of the bead. 10 - 8 Spheres 719 719 �� Marine Biology In 1934, the bathysphere reached a record depth of 3028 feet. The pressure on the hull was about half a ton per square inch. Sports Find the unknown dimensions of the ball for each sport. Sport Ball Diameter Circumference Surface Area Volume 29. Golf 30. Cricket 31. Tennis 32. Petanque 9 in. 1.68 in. 2.5 in. 74 mm 33. Marine Biology The bathysphere was an early version of a submarine, invented in the 1930s. The inside diameter of the bathysphere was 54 inches, and the steel used to make the sphere was 1.5 inches thick. It had three 8-inch diameter windows. Estimate the volume of steel used to make the bathysphere. 34. Geography Earth’s radius is approximately 4000 mi. About two-thirds of Earth’s surface is covered by water. Estimate the land area on Earth. Astronomy Use the table for Exercises 35–38. 35. How many times as great is the volume of Jupiter as the volume of Earth? 36. The sum of the volumes of Venus and Mars is about equal to the volume of which planet? 37. Which is greater, the sum of the surface areas of Uranus and Neptune or the surface area of Saturn? 38. How many times as great is the surface area of Mercury as the surface area of Pluto? Planet Diameter (mi) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto 3,032 7,521 7,926 4,222 88,846 74,898 31,763 30,775 1,485 39. Critical Thinking In the figure, the hemisphere and the cylinder both have radius and height r. Prove that the shaded cross sections have equal areas. 40. Write About It Suppose a sphere and a cube have equal surface areas. Using r for the radius of the sphere and s for the side of a cube, write an equation to show the relationship between r and s. 41. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A company sells orange juice in spherical containers that look like oranges. Each container has a surface area of approximately 50.3 in 2 . a. What is the volume of the container? Round to the nearest tenth. b. The company decides to increase the radius of the container by 10%. What is the volume of the new container? 720 720 Chapter 10 Spatial Reasoning �� 42. A sphere with radius 8 cm is inscribed in a cube. Find the ratio of the volume of the cube to the volume of the sphere. 2 : 1 _ π 3 2 : 3π 43. What is the surface area of a sphere with volume 10 2 _ π in 3 ? 3 8π in 2 10 2 _ π in 2 3 16π in 2 32π in 2 44. Which expression represents the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r? 3 π + 82π + 12 CHALLENGE AND EXTEND 45. Food The top of a gumball machine is an 18 in. sphere. The machine holds a maximum of 3300 gumballs, which leaves about 43% of the space in the machine empty. Estimate the diameter of each gumball. 46. The surface area of a sphere can be used to determine its volume. a. Solve the surface area formula of a sphere to get an expression for r in terms of S. b. Substitute your result from part a into the volume formula to find the volume V of a sphere in terms of its surface area S. c. Graph the relationship between volume and surface area with S on the horizontal axis and V on the vertical axis. What shape is the graph? Use the diagram of a sphere inscribed in a cylinder for Exercises 47 and 48. 47. What is the relationship between the volume of the sphere and the volume of the cylinder? 48. What is the relationship between the surface area of the sphere and the lateral area of the cylinder? SPIRAL REVIEW Write an equation that describes the functional relationship for each set of ordered pairs. (Previous course)   (0, 1) , (1, 2) , (-1, 2) , (2, 5) , (-2, 5) 49. ⎬ ⎨     (-1, 9) , (0, 10) , (1, 11) , (2, 12) , (3, 13) 50. ⎬ ⎨   Find the shaded area. Round to the nearest tenth, if necessary. (Lesson 9-3) 51. 52. Describe the effect on the volume that results from the given change. (Lesson 10-6) 53. The side lengths of a cube are multiplied by 3 __ 4 . 54. The height and the base area of a prism are multiplied by 5. 10 - 8 Spheres 721 721 �� 10-8 Use with Lesson 10-8 Activity 1 Compare Surface Areas and Volumes In some situations you may need to find the minimum surface area for a given volume. In others you may need to find the maximum volume for a given surface area. Spreadsheet software can help you analyze these problems. TEKS G.11.D Similarity and the geometry of shape: describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed …. 1 Create a spreadsheet to compare surface areas and volumes of rectangular prisms. Create columns for length L, width W, height H, surface area SA, volume V, and ratio of surface area to volume SA/V. In the column for SA, use the formula shown. 2 Create a formula for the V column and a formula for the SA/V column. 3 Fill in the measurements L = 8, W = 2, and H = 4 for the first rectangular prism. 4 Choose several values for L, W, and H to create rectangular prisms that each have the same volume as the first one. Which has the least surface area? Sketch the prism and describe its shape in words. (Is it tall or short, skinny or wide, flat or cubical?) Make a conjecture about what type of shape has the minimum surface area for a given volume. 722 722 Chapter 10 Spatial Reasoning �� Try This 1. Repeat Activity 1 for cylinders. Create columns for radius R, height H, surface area SA, volume V, and ratio of surface area to volume SA/V. What shape cylinder has the minimum surface area for a given volume? (Hint: To use π in a formula, input “PI( )” into your spreadsheet.) 2. Investigate packages such as cereal boxes and soda cans. Do the manufacturers appear to be using shapes with the minimum surface areas for their volume? What other factors might influence a company’s choice of packaging? Activity 2 1 Create a new spreadsheet with the same column headings used in Activity 1. Fill in the measurements L = 8, W = 2, and H = 4 for the first rectangular prism. To create a new prism with the same surface area, choose new values for L and W, and use the formula shown to calculate H. 2 Choose several more values for L and W, and calculate H so that SA = 112. Examine the V and SA/V columns. Which prism has the greatest volume? Sketch the prism and describe it in words. Make a conjecture about what type of shape has the maximum volume for a given surface area. Try This 3. Repeat Activity 2 for cylinders. Create columns for radius R, height H, surface area SA, volume V, and the ratio of surface area to volume SA/V. What shape cylinder has the maximum volume for a given surface area? 4. Solve the formula SA = 2LW + 2LH + 2WH for H. Use your result to explain the formula that was used to find H in Activity 2. 5. If a rectangular prism, a pyramid, a cylinder, a cone, and a sphere all had the same volume, which do you think would have the least surface area? Which would have the greatest surface area? Explain. 6. Use a spreadsheet to analyze what happens to the ratio of surface area to volume of a rectangular prism when the dimensions are doubled. Explain how you set up the spreadsheet and describe your results. 10-8 Technology Lab 723 723 �� SECTION 10B Surface Area and Volume Juice for Fun You are in charge of designing containers for a new brand of juice. Your company wants you to compare several different container shapes. The container must be able to hold a 6-inch straw so that exactly 1 inch remains outside the container when the straw is inserted as far as possible. 1. One possible container is a cylinder with a base diameter of 4 in., as shown. How much material is needed to make this container? Round to the nearest tenth. 2. Estimate the volume of juice in ounces that the cylinder will hold. Round to the nearest tenth. (Hint: 1 in 3 ≈ 0.55 oz) 3. Another option is a square prism with a 3 in. by 3 in. base, as shown. How much material is needed to make this container? 4. Estimate the volume of juice in ounces that the prism will hold. (Hint: 1 in 3 ≈ 0.55 oz) 5. Which container would you recommend to your company? Justify your answer. �� 724 724 Chapter 10 Spatial Reasoning SECTION 10B Quiz for Lessons 10-4 Through 10-8 10-4 Surface Area of Prisms and Cylinders Find the surface area of each figure. Round to the nearest tenth, if necessary. 1. 2. 3. 4. The dimensions of a 12 mm by 8 mm by 24 mm right rectangular prism are multiplied by 3
__ 4 . Describe the effect on the surface area. 10-5 Surface Area of Pyramids and Cones Find the surface area of each figure. Round to the nearest tenth, if necessary. 5. a regular pentagonal pyramid with base edge length 18 yd and slant height 20 yd 6. a right cone with diameter 30 in. and height 8 in. 7. the composite figure formed by two cones 10-6 Volume of Prisms and Cylinders Find the volume of each figure. Round to the nearest tenth, if necessary. 8. a regular hexagonal prism with base area 23 in 2 and height 9 in. 9. a cylinder with radius 8 yd and height 14 yd 10. A brick patio measures 10 ft by 12 ft by 4 in. Find the volume of the bricks. If the density of brick is 130 pounds per cubic foot, what is the weight of the patio in pounds? 11. The dimensions of a cylinder with diameter 2 ft and height 1 ft are doubled. Describe the effect on the volume. 10-7 Volume of Pyramids and Cones Find the volume of each figure. Round to the nearest tenth, if necessary. 12. 13. 14. 10-8 Spheres Find the surface area and volume of each figure. 15. a sphere with diameter 20 in. 16. a hemisphere with radius 12 in. 17. A baseball has a diameter of approximately 3 in., and a softball has a diameter of approximately 5 in. About how many times as great is the volume of a softball as the volume of a baseball? Ready to Go On? 725 725 �� EXTENSION Spherical Geometry EXTENSION TEKS G.1.C Geometric structure: compare and contrast the structures and implications of Euclidean and non-Euclidean geometries. Objective Understand spherical geometry as an example of non-Euclidean geometry. Vocabulary non-Euclidean geometry spherical geometry Euclidean geometry is based on figures in a plane. Non-Euclidean geometry is based on figures in a curved surface. In a non-Euclidean geometry system, the Parallel Postulate is not true. One type of non-Euclidean geometry is spherical geometry , which is the study of figures on the surface of a sphere. A line in Euclidean geometry is the shortest path between two points. On a sphere, the shortest path between two points is along a great circle, so “lines” in spherical geometry are defined as great circles. In spherical geometry, there are no parallel lines. Any two lines intersect at two points. Pilots usually fly along great circles because a great circle is the shortest route between two points on Earth. Spherical Geometry Parallel Postulate Through a point not on a line, there is no line parallel to a given line. E X A M P L E 1 Classifying Figures in Spherical Geometry Name a line, a segment, and a triangle on the sphere. The two points used to name a line cannot be exactly opposite each other on the sphere. In Example 1,   AB could refer to more than one line.  AC is a line. ̶̶ AC is a segment. △ACD is a triangle. 1. Name another line, segment, and triangle on the sphere above. In Example 1, the lines   AC and   AD are both perpendicular to   CD . This means that △ACD has two right angles. So the sum of its angle measures must be greater than 180°. Imagine cutting an orange in half and then cutting each half in quarters using two perpendicular cuts. Each of the resulting triangles has three right angles. Spherical Triangle Sum Theorem The sum of the angle measures of a spherical triangle is greater than 180°. 726 726 Chapter 10 Spatial Reasoning �� E X A M P L E 2 Classifying Spherical Triangles Classify each spherical triangle by its angle measures and by its side lengths. A △ABC △ABC is an obtuse scalene triangle. B △NPQ on Earth has vertex N at the North Pole and vertices P and Q on the equator. PQ is equal to 1 __ the circumference of Earth. 3 NP and NQ are both equal to 1 __ 4 the circumference of Earth. The equator is perpendicular to both of the other two sides of the triangle. Thus △NPQ is an isosceles right triangle. 2. Classify △VWX by its angle measures and by its side lengths. The area of a spherical triangle is part of the surface area of the sphere. For the piece of orange on page 726, the area is 1 __ 8 of the surface area of the orange, or 1 __ 8 (4π r 2 ) = π r 2 ___ 2 . If you know the radius of a sphere and the measure of each angle, you can find the area of the triangle. Area of a Spherical Triangle The area of spherical △ABC on a sphere with radius r is A = π r 2 _ 180° (m∠A + m∠B + m∠C - 180°) . E X A M P L E 3 Finding the Area of Spherical Triangles Find the area of each spherical triangle. Round to the nearest tenth, if necessary. A △ABC (m∠A + m∠B + m∠C - 180°) (100 + 106 + 114 - 180) ≈ 152.4 cm 2 B △DEF on Earth’s surface with m∠D = 75°, m∠E = 80°, and m∠F = 30°. (Hint: average radius of Earth = 3959 miles) (m∠D + m∠E + m∠F - 180°) (75 + 80 + 30 - 180) ≈ 1,367,786.7 mi 2 A = π r 2 _ 180° π (14) 2 _ 180° A = A = π r 2 _ 180° π (3959) 2 _ 180° = 3. Find the area of △KLM on a sphere with diameter 20 ft, where m∠K = 90°, m∠L = 90°, and m∠M = 30°. Round to the nearest tenth. Chapter 10 Extension 727 727 �� EXTENSION Exercises Exercises Use the figure for Exercises 1–3. 1. Name all lines on the sphere. 2. Name three segments on the sphere. 3. Name a triangle on the sphere. Determine whether each figure is a line in spherical geometry. 4. m 5. n 6. p Classify each spherical triangle by its angle measures and by its side lengths. 7. 9. Find the area of each spherical triangle. 11. 13. 8. 10. 12. 14. 15. △ABC on the Moon’s surface with m∠A = 35°, m∠B = 48°, and m∠C = 100° (Hint: average radius of the Moon ≈ 1079 miles) 16. △RST on a scale model of Earth with radius 6 m, m∠R = 80°, m∠S = 130°, and m∠T = 150° 728 728 Chapter 10 Spatial Reasoning �� 17. △ABC is an acute triangle. a. Write an inequality for the sum of the angle measures of △ABC, based on the fact that △ABC is acute. b. Use your result from part a to write an inequality for the area of △ABC. c. Use your result from part b to compare the area of an acute spherical triangle to the total surface area of the sphere. 18. Draw a quadrilateral on a sphere. Include one diagonal in your drawing. Use the sum of the angle measures of the quadrilateral to write an inequality. Geography Compare each length to the length of a great circle on Earth. 19. the distance between the North 20. the distance between the North Pole and the South Pole Pole and any point on the equator 21. Geography If the area of a triangle on Earth’s surface is 100,000 mi 2 , what is the sum of its angle measures? (Hint: average radius of Earth ≈ 3959 miles) 22. Sports Describe the curves on the basketball that are lines in spherical geometry. 23. Navigation Pilots navigating long distances often travel along the lines of spherical geometry. Using a globe and string, determine the shortest route for a plane traveling from Washington, D.C., to London, England. What do you notice? 24. Write About It Can a spherical triangle be right and obtuse at the same time? Explain. 25. Write About It A 2-gon is a polygon with two edges. Draw two lines on a sphere. How many 2-gons are formed? What can you say about the positions of the vertices of the 2-gons on the sphere? 26. Challenge Another type of non-Euclidean geometry, called hyperbolic geometry, is defined on a surface that is curved like the bell of a trumpet. What do you think is true about the sum of the angle measures of the triangle shown at right? Compare the sum of the angle measures of a triangle in Euclidean, spherical, and hyperbolic geometry. Chapter 10 Extension 729 729 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary altitude . . . . . . . . . . . . . . . . . . . . 680 isometric drawing . . . . . . . . . . 662 right cone . . . . . . . . . . . . . . . . . . 690 altitude of a cone . . . . . . . . . . . 690 lateral edge . . . . . . . . . . . . . . . . . 680 right cylinder . . . . . . . . . . . . . . . 681 altitude of a pyramid . . . . . . . . 689 lateral face . . . . . . . . . . . . . . . . . 680 right prism . . . . . . . . . . . . . . . . . 680 axis of a cone . . . . . . . . . . . . . . . 690 lateral surface . . . . . . . . . . . . . . 681 slant height of a axis of a cylinder . . . . . . . . . . . . 681 net . . . . . . . . . . . . . . . . . . . . . . . . 655 center of a sphere . . . . . . . . . . . 714 oblique cone . . . . . . . . . . . . . . . 690 cone . . . . . . . . . . . . . . . . . . . . . . . 654 oblique cylinder . . . . . . . . . . . . 681 cross section . . . . . . . . . . . . . . . 656 oblique prism . . . . . . . . . . . . . . 680 cube . . . . . . . . . . . . . . . . . . . . . . . 654 orthographic drawing . . . . . . . 661 cylinder . . . . . . . . . . . . . . . . . . . . 654 perspective drawing . . . . . . . . 662 edge . . . . . . . . . . . . . . . . . . . . . . . 654 polyhedron . . . . . . . . . . . . . . . . . 670 face . . . . . . . . . . . . . . . . . . . . . . . . 654 prism . . . . . . . . . . . . . . . . . . . . . . 654 great circle . . . . . . . . . . . . . . . . . 714 pyramid . . . . . . . . . . . . . . . . . . . 654 hemisphere . . . . . . . . . . . . . . . . 714 radius of a sphere . . . . . . . . . . . 714 horizon . . . . . . . . . . . . . . . . . . . . 662 regular pyramid . . . . . . . . . . . . 689 regular pyramid . . . . . . . . . . 689 slant height of a right cone . . . . . . . . . . . . . . . 690 space . . . . . . . . . . . . . . . . . . . . . . 671 sphere . . . . . . . . . . . . . . . . . . . . . 714 surface area . . . . . . . . . . . . . . . . 680 vanishing point . . . . . . . . . . . . . 662 vertex . . . . . . . . . . . . . . . . . . . . . . 654 vertex of a cone . . . . . . . . . . . . . 690 vertex of a pyramid . . . . . . . . . 689 volume . . . . . . . . . . . . . . . . . . . . 697 Complete the sentences below with vocabulary words from the list above. 1. A(n) ? has at least one nonrectangular lateral face. ̶̶̶̶ 2. A n
ame given to the intersection of a three-dimensional figure and a plane is ? . ̶̶̶̶ 10-1 Solid Geometry (pp. 654–660) TEKS G.2.B, G.6.A, G.6.B, G.9.D E X A M P L E S ■ Classify the figure. Name the vertices, edges, and bases. pentagonal prism vertices: A, B, C, D, E, F, EXERCISES Classify each figure. Name the vertices, edges, and bases. 3. 4. G, H, J, K ̶̶ AB , ̶̶ EK , edges: ̶̶ ̶̶ AF , KF , ̶̶ BC , ̶̶ DJ , ̶̶ CD , ̶̶ CH , ̶̶ DE , ̶̶ BG ̶̶ AE , ̶̶ FG , ̶̶̶ GH , ̶̶ HJ , ̶̶ JK , bases: ABCDE, FGHJK ■ Describe the three-dimensional figure that can be made from the given net. The net forms a rectangular prism. 730 730 Chapter 10 Spatial Reasoning Describe the three-dimensional figure that can be made from the given net. 5. 6. �� 10-2 Representations of Three-Dimensional Figures (pp. 661–668) TEKS G.6.C, G.9.D E X A M P L E S ■ Draw all six orthographic views of the given object. Assume there are no hidden cubes. EXERCISES Use the figure made of unit cubes for Exercises 7–10. Assume there are no hidden cubes. 7. Draw all six orthographic views. Top: Bottom: 8. Draw an isometric view. Front: Back: Left: Right: ■ Draw an isometric view of the given object. Assume there are no hidden cubes. 9. Draw the object in one-point perspective. 10. Draw the object in two-point perspective. Determine whether each drawing represents the given object. Assume there are no hidden cubes. 11. 12. 10-3 Formulas in Three Dimensions (pp. 670–677) TEKS G.5.A, G.7.C, G.8.C, G.9.D E X A M P L E S ■ Find the number of vertices, edges, and faces of the given polyhedron. Use your results to verify Euler’s formula. V = 12, E = 18, F = 8 12 - 18 + 8 = 2 ■ Find the distance between the points (6, 3, 4) and (2, 7, 9) . Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. distance: d = √  (2 - 6) 2 + (7 - 3) 2 + (9 - 4) 2 = √  57 ≈ 7.5 midpoint4, 5, 6.5) EXERCISES Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. 13. 14. Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 15. (2, 6, 4) and (7, 1, 1) 16. (0, 3, 0) and (5, 7, 8) 17. (7, 2, 6) and (9, 1, 5) 18. (6, 2, 8) and (2, 7, 4) Study Guide: Review 731 731 10-4 Surface Area of Prisms and Cylinders (pp. 680–687) E X A M P L E S EXERCISES TEKS G.5.A, G.5.B, G.6.B, G.8.D, G.11.D Find the lateral area and surface area of each right prism or cylinder. ■ Find the lateral area and surface area of each right prism or cylinder. Round to the nearest tenth, if necessary. 19. L = Ph = 28 (10) = 280 in 2 S = Ph + 2B = 280 + 2 (49) = 378 in 2 20. a cube with side length 5 ft ■ a cylinder with radius 8 m and height 12 m L = 2πrh = 2π (8)(12) = 192π ≈ 603.2 m 2 S = L + 2B = 192π + 2π (8)2 = 320π ≈ 1005.3 m 2 21. an equilateral triangular prism with height 7 m and base edge lengths 6 m 22. a regular pentagonal prism with height 8 cm and base edge length 4 cm 10-5 Surface Area of Pyramids and Cones (pp. 689–696) E X A M P L E S EXERCISES TEKS G.5.A, G.5.B, G.6.B, G.8.D, G.11.D Find the lateral area and surface area of each right pyramid or cone. ■ Find the lateral area and surface area of each right pyramid or cone. 23. a square pyramid with side length 15 ft and slant height 21 ft 24. a cone with radius 7 m and height 24 m The radius is 8 m, so the slant height is 25. a cone with diameter 20 in. and slant height 15 in. √ 8 2 + 15 2 = 17 m. L = πrℓ = π (8)(17) = 136π m 2 S = πrℓ + π r 2 = 136π + (8)2π = 200π m 2 ■ a regular hexagonal pyramid with base edge length 8 in. and slant height 20 in. L = 1 _ Pℓ = 1 _ (48) (20) = 480 in 2 2 2 S = L + B = 480 + 1 _ (4 √  3 ) (48) ≈ 646.3 in 2 2 Find the surface area of each composite figure. 26. 27. 10-6 Volume of Prisms and Cylinders (pp. 697–704) E X A M P L E S ■ Find the volume of the prism. 2 V = Bh = ( 1 _ aP) h = 1 _ (4 √  3 ) (48) (12) 2 = 1152 √  3 ≈ 1995.3 cm 3 732 732 Chapter 10 Spatial Reasoning EXERCISES Find the volume of each prism. 28. 29. TEKS G.1.B, G.5.A, G.5.B, G.8.D, G.11.D �� ■ Find the volume of the cylinder6) 2 (14) = 504π ≈ 1583.4 ft 3 Find the volume of each cylinder. 30. 31. 10-7 Volume of Pyramids and Cones (pp. 705–712) TEKS G.5.A, G.5.B, G.8.D, G.11.D E X A M P L E S ■ Find the volume of the pyramid. Bh = 1 _ V = 1 _ (8 · 3) (14) 3 3 = 112 in 3 ■ Find the volume of the cone9) 2 (16) 3 3 = 432π ft 3 ≈ 1357.2 ft 3 EXERCISES Find the volume of each pyramid or cone. 32. a hexagonal pyramid with base area 42 m 2 and height 8 m 33. an equilateral triangular pyramid with base edge 3 cm and height 8 cm 34. a cone with diameter 12 cm and height 10 cm 35. a cone with base area 16π ft 2 and height 9 ft Find the volume of each composite figure. 36. 37. 10-8 Spheres (pp. 714–721) TEKS G.5.A, G.5.B, G.8.D, G.11.D E X A M P L E EXERCISES ■ Find the volume and surface area of the sphere. Give your answers in terms of π9) 3 = 972π m 2 3 3 S = 4π r 2 = 4π (9) 2 = 324π m 2 Find each measurement. Give your answers in terms of π. 38. the volume of a sphere with surface area 100π m 2 39. the surface area of a sphere with volume 288π in 3 40. the diameter of a sphere with surface area 256π ft 2 Find the surface area and volume of each composite figure. 41. 42. Study Guide: Review 733 733 �� Use the diagram for Items 1–3. 1. Classify the figure. Name the vertices, edges, and bases. 2. Describe a cross section made by a plane parallel to the base. 3. Find the number of vertices, edges, and faces of the polyhedron. Use your results to verify Euler’s formula. Use the figure made of unit cubes for Items 4–6. Assume there are no hidden cubes. 4. Draw all six orthographic views. 5. Draw an isometric view. 6. Draw the object in one-point perspective. Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 7. (0, 0, 0) and (5, 5, 5) 8. (6, 0, 9) and (7, 1, 4) 9. (-1, 4, 3) and (2, -5, 7) Find the surface area of each figure. Round to the nearest tenth, if necessary. 10. 13. 11. 14. 12. 15. Find the volume of each figure. Round to the nearest tenth, if necessary. 16. 19. 17. 20. 18. 21. 22. Earth’s diameter is approximately 7930 miles. The Moon’s diameter is approximately 2160 miles. About how many times as great is the volume of Earth as the volume of the Moon? 734 734 Chapter 10 Spatial Reasoning �� FOCUS ON SAT MATHEMATICS SUBJECT TEST SAT Mathematics Subject Test results include scaled scores and percentiles. Your scaled score is a number from 200 to 800, calculated using a formula that varies from year to year. The percentile indicates the percentage of people who took the same test and scored lower than you did. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. The questions are written so that you should not need to do any lengthy calculations. If you find yourself getting involved in a long calculation, think again about all of the information in the problem to see if you might have missed something helpful. 1. A line intersects a cube at two points, A and B. If each edge of the cube is 4 cm, what is the greatest possible distance between A and B? 4. If triangle ABC is rotated about the x-axis, what is the volume of the resulting cone? (A) 2 √  3 cm (B) 4 cm (C) 4 √  2 cm (D) 4 √  3 cm (E) 16 √  3 cm 2. The lateral area of a right cylinder is 3 times the area of its base. What is the height h of the cylinder in terms of its radius r? (A) 1 _ r 2 (B) 2 _ r 3 (C) 3 _ r 2 (D) 3r (E) 3 r 2 3. What is the lateral area of a right cone with radius 6 ft and height 8 ft? (A) 30π ft 2 (B) 48π ft 2 (C) 60π ft 2 (D) 180π ft 2 (E) 360π ft 2 (A) 100π cubic units (B) 144π cubic units (C) 240π cubic units (D) 300π cubic units (E) 720π cubic units in 3 5. An oxygen tank is the shape of a cylinder with a hemisphere at each end. If the radius of the tank is 5 inches and the overall length is 32 inches, what is the volume of the tank? (A) 500 _ 3π (B) 2275 _ 12 (C) 1900 _ 3 (D) 2150 _ 3 (E) 2900 _ 3 π in 3 π in 3 π in 3 π in 3 College Entrance Exam Practice 735 735 �� Any Question Type: Measure to Solve Problems On some tests, you may have to measure a figure in order to answer a question. Pay close attention to the units of measure asked for in the question. Some questions ask you to measure to the nearest centimeter, and some ask you to measure to the nearest inch. Multiple Choice: The net of a square pyramid is shown below. Use a ruler to measure the dimensions of the pyramid to the nearest centimeter. Which of the following best represents the total surface area of the square pyramid? 9 square centimeters 21 square centimeters 33 square centimeters 36 square centimeters Use a centimeter ruler to measure one side of the square base. The measurement to the nearest centimeter is 3 cm. The base is a square, so all four side lengths are 3 cm. Measure the altitude of a triangular face, which is the slant height of the pyramid. The altitude is 2 cm. Label the drawing with the measurements. To find the total surface area of the pyramid, find the base area and the lateral area. The base of the pyramid is a square. The base area of the pyramid is A = s 2 = (3) 2 = 9 cm 2 . The area of one triangular face is A = 1 _ 2 bh = 1 _ (3) (2) = 3 cm 2 . 2 The pyramid has 4 faces, so the lateral area is 4 (3) = 12 cm 2 . The total surface area is 9 + 12 = 21 cm 2 . The correct answer choice is B. 736 736 Chapter 10 Spatial Reasoning ��
�� Read each test item and answer the questions that follow. �� Measure carefully and make sure you are using the correct units to measure the figure. Item A The net of a cube is shown below. Use a ruler to measure the dimensions of the cube to the nearest 1 __ Which best represents the volume of the cube to the nearest cubic inch? inch. 4 1 cubic inch 2 cubic inches 5 cubic inches 9 cubic inches 1. Measure one edge of the net for the cube. What is the length to the nearest 1__ 4 inch? 2. How would you use the measurement to find the volume of the cube? Item B The net of a cylinder is shown below. Use a ruler to measure the dimensions of the cylinder to the nearest tenth of a centimeter. Which best represents the total surface area of the cylinder to the nearest square centimeter? 6 square centimeters 16 square centimeters 19 square centimeters 42 square centimeters 3. Which part of the net do you need to measure in order to find the height of the cylinder? Find the height of the cylinder to the nearest tenth of a centimeter. 4. What other measurement(s) do you need in order to find the surface area of the cylinder? Find the measurement(s) to the nearest tenth of a centimeter. 5. How would you use the measurements to find the surface area of the cylinder? TAKS Tackler 737 737 KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–10 Multiple Choice 1. If a point (x, y) is chosen at random in the coordinate plane such that -1 ≤ x ≤ 1 and -5 ≤ y ≤ 3, what is the probability that x ≥ 0 and y ≥ 0? 0.1875 0.25 0.375 0.8125 2. △ABC ∼ △DEF, and △DEF ∼ △GHI . If the similarity ratio of △ABC to △DEF is 1 __ and the 2 similarity ratio of △DEF to △GHI is 3 __ , what is the 4 similarity ratio of △ABC to △GHI. Which expression represents the number of faces of a prism with bases that are n-gons? n + 1 n + 2 2n 3n 4. Parallelogram ABCD has a diagonal ̶̶ AC with endpoints A (-1, 3) and C (-3, -3) . If B has coordinates (x, y) , which of the following represents the coordinates for D? D (-3x, -y) D (-x, -y) D (-x - 4, -y) D (x - 2, y) 5. Right △ABC with legs AB = 9 millimeters and BC = 12 millimeters is the base of a right prism that has a surface area of 450 square millimeters. What is the height of the prism? 4.75 millimeters 9.5 millimeters 6 millimeters 11 millimeters 6. The radius of a sphere is doubled. What happens to the ratio of the volume of the sphere to the surface area of the sphere? It remains the same. It is doubled. It is increased by a factor of 4. It is increased by a factor of 8. 738 738 Chapter 10 Spatial Reasoning 7. ̶̶ AB has endpoints A (x, y, z) and B (-2, 6, 13) and midpoint M (2, -6, 3) . What are the coordinates of A? A (-6, 18, 23) A (0, 0, 8) A (2, -6, 19) A (6, -18, -7) 8. If ̶̶ DE bisects ∠CEF, which of the following additional statements would allow you to conclude that △DEF ≅ △ABC? ∠DEF ≅ ∠BAC ∠DEF ≅ ∠CDE ̶̶ ̶̶ CD EF ≅ ̶̶ ̶̶ EC EF ≅ 9. To the nearest tenth of a cubic centimeter, what is the volume of a right regular octagonal prism with base edge length 4 centimeters and height 7 centimeters? 180.3 cubic centimeters 224.0 cubic centimeters 270.4 cubic centimeters 540.8 cubic centimeters 10. Which of the following must be true about a conditional statement? If the inverse is false, then the converse is false. If the conditional is true, then the contrapositive is false. If the conditional is true, then the converse is false. If the hypothesis of the conditional is true, then the conditional is true. �� It may be helpful to include units in your calculations of measures of geometric figures. If your answer includes the proper units, you are less likely to have made an error. 11. A right cylinder has a height of 10 inches. The area of the base is 63.6 square inches. To the nearest tenth of a square inch, what is the lateral area for this cylinder? 53.6 square inches 282.7 square inches 409.9 square inches 634.6 square inches 12. The volume of the smaller sphere is 288 cubic centimeters. Find the volume of the larger sphere. 864 cubic centimeters 2,592 cubic centimeters 7,776 cubic centimeters 23,328 cubic centimeters Gridded Response 13.  u = 〈3, -7〉, and  v = 〈-6, 5〉. What is the magnitude of the resultant vector to the nearest tenth of  u and  v ? STANDARDIZED TEST PREP Short Response 17. The area of trapezoid GHIJ is 103.5 square centimeters. Find each of the following. Round answers to the nearest tenth. Show your work or explain in words how you found your answers. a. the height of trapezoid GHIJ b. m∠J 18. The figure shows the top view of a stack of cubes. The number on each cube represents the number of stacked cubes. Draw the bottom, back, and right views of the object. 19. △ABC has vertices A (1, -2) , B (-2, -3) , and C (-2, 2) . a. Graph △A'B'C', the image of △ABC, after a dilation with a scale factor of 3 __ . 2 b. Show that   AB ǁ   A'B' ,   BC ǁ   B'C' , and   CA ǁ   C'A' . Use slope to justify your answer. 14. If a polyhedron has 12 vertices and 8 faces, how many edges does the polyhedron have? Extended Response 15. If Y is the circumcenter of △PQR, what is the value of x? 20. A right cone has a lateral area of 30π square inches and a slant height of 6 inches. 16. How many cubes with edge length 3 centimeters will fit in a box that is a rectangular prism with length 12 centimeters, width 15 centimeters, and height 24 centimeters? a. Find the height of the cone. Show your work or explain in words how you determined your answer. Round your answer to the nearest tenth. b. Find the volume of this cone. Round your answer to the nearest tenth. c. Given a right cone with a lateral area of L and a slant height of ℓ, find an equation for the volume in terms of L and ℓ. Show your work. Cumulative Assessment, Chapters 1–10 739 739 �� T E X A S TAKS Grades 9–11 Obj. 10 Reliant Stadium When Houston’s Reliant Stadium opened in 2002, it was the first NFL stadium to have a retractable roof. In addition to football games, Reliant Stadium hosts rodeos, concerts, and other events. When configured for football, up to 72,000 fans can sit around 97,000 square feet of playing field. Choose one or more strategies to solve each problem. 1. Inside the playing field, the football field is 160 ft by 360 ft. Approximately how many acres of land surround the football field? (Hint: 1 acre = 43,560 ft 2 ) �� For 2 and 3, use the table. 2. There are two scoreboards in Reliant Stadium. Suppose the ratio of each scoreboard’s length to its width were 46 : 7. What would the dimensions of each scoreboard be? 3. Each scoreboard is equipped with a large video screen. A screen’s width is 72.5 ft greater than its height. What are the dimensions of the video screens on the scoreboards? Approximate Scoreboard Measurements Perimeter (ft) Scoreboard Video Screen 636 241 Area ( ft 2 ) 11,592 2,316 4. Reliant Stadium can be divided into 12 seating sections of equal size. When the roof is opened at a certain time of day, 4 of these sections are in the sunlight. Suppose you choose a seat in the arena at random when the roof is retracted. What is the probability that you are sitting in sunlight? What is the probability that you are sitting in the shade? 740 740 Chapter 10 Spatial Reasoning �� Texas Coins Although Texas does not have its own mint, Texas has a rich history in coins. Starting in the mid-1800s, many Texas businesses issued tokens that were redeemable for merchandise in place of U.S. money. In 1935, the U.S. Mint issued a coin to celebrate explorer Alvar Nunez Cabeza de Vaca’s travels across southern America and Texas. In 2004, the U.S. Mint issued the Texas state quarter. Choose one or more strategies and use the table to solve each problem. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List 1. Quarters are stamped out of a rectangular metal strip that is 13 in. wide by 1500 ft long. Given that the diameter of a quarter is just under an inch (0.955 in.), what is the minimum number of strips needed for 700,000 Texas state quarters? 2. A Spanish Trail memorial half dollar contains a small amount of copper, but most of the metal in the coin is silver. The volume of copper in a Spanish Trail memorial half dollar is about 6.58 mm 3 . What percent of the half dollar is copper? 3. Many Texas trade tokens were made from aluminum. About how many tokens could be made from a block of aluminum with a volume of 1 m 3 ? Texas State Quarter Spanish Trail Memorial Half Dollar Texas Trade Token Coin Specifications Diameter (mm) Thickness (mm) 24.26 1.75 30.60 2.15 44.00 1.42 4. 4. The regular octagonal token shown was issued by a 4. barber shop in Fort Worth. If the distance from the midpoint of one side to the midpoint of the opposite side is 25 mm, what is the area of the face of the coin? Problem Solving on Location 741 741 Circles 11A Lines and Arcs in Circles 11-1 Lines That Intersect Circles 11-2 Arcs and Chords 11-3 Sector Area and Arc Length 11B Angles and Segments in Circles 11-4 Inscribed Angles Lab Explore Angle Relationships in Circles 11-5 Angle Relationships in Circles Lab Explore Segment Relationships in Circles 11-6 Segment Relationships in Circles 11-7 Circles in the Coordinate Plane Ext Polar Coordinates KEYWORD: MG7 ChProj On the floor of the Capitol rotunda are six seals, commemorating the flags that have flown over Texas. 742 742 Chapter 11 Vocabulary Match each term on the left with a definition on the right. A. the distance around a circle 1. radius 2. pi 3. circle 4. circumference B. the locus of points in a plane that are a fixed distance from a given point C. a segment with one endpoint on a circle and one endpoint at the
center of the circle D. the point at the center of a circle E. the ratio of a circle’s circumference to its diameter Tables and Charts The table shows the number of students in each grade level at Middletown High School. Find each of the following. 5. the percentage of students who are freshman 6. the percentage of students who are juniors 7. the percentage of students who are sophomores or juniors Year Freshman Sophomore Junior Senior Number of Students 192 208 216 184 Circle Graphs The circle graph shows the age distribution of residents of Mesa, Arizona, according to the 2000 census. The population of the city is 400,000. 8. How many residents are between the ages of 18 and 24? 9. How many residents are under the age of 18? 10. What percentage of the residents are over the age of 45? 11. How many residents are over the age of 45? Solve Equations with Variables on Both Sides Solve each equation. 12. 11y - 8 = 8y + 1 14. z + 30 = 10z - 15 16. -2x - 16 = x + 6 13. 12x + 32 = 10 + x 15. 4y + 18 = 10y + 15 17. -2x - 11 = -3x - 1 Solve Quadratic Equations Solve each equation. 18. 17 = x 2 - 32 20. 4 x 2 + 12 = 7 x 2 19. 2 + y 2 = 18 21. 188 - 6 x 2 = 38 Circles 743 743 �� Key Vocabulary/Vocabulario arc arco arc length longitud de arco central angle ángulo central chord secant cuerda secante sector of a circle sector de un círculo segment of a circle segmento de un círculo semicircle semicírculo tangent of a circle tangente de un círculo Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, answer the following questions. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The word semicircle begins with the prefix semi-. List some other words that begin with semi-. What do all of these words have in common? 2. The word central means “located at the center.” How can you use this definition to understand the term central angle of a circle? 3. The word tangent comes from the Latin word tangere, which means “to touch.” What does this tell you about a line that is a tangent to a circle? Geometry TEKS Les. 11-1 Les. 11-2 Les. 11-3 Les. 11-4 G.1.A Geometric structure* develop an awareness of the ★ ★ ★ ★ 11-5 Tech. Lab 11-6 Tech. Lab Les. 11-5 ★ Les. 11-6 Les. 11-7 Ext ★ ★ ★ structure of a mathematical system, ... G.1.B Geometric structure* recognize the historical development of geometric systems ... ★ G.2.A Geometric structure* use constructions to explore ★ ★ ★ ★ ★ attributes of geometric figures and to make conjectures ... G.2.B Geometric structure* make conjectures about ... circles, ... and determine the validity of the conjectures, ... ★ ★ ★ ★ ★ ★ ★ ★ G.3.B Geometric structure* construct and justify statements ★ about geometric figures ... G.5.A Geometric patterns* use ... patterns to develop algebraic ★ ★ ★ ★ ★ expressions representing geometric properties G.5.B Geometric patterns* use numeric and geometric ★ ★ ★ patterns to make generalizations about geometric properties, including ... angle relationships in ... circles G.8.B Congruence and the geometry of size* find areas of ★ sectors and arc lengths of circles using proportional reasoning G.8.C Congruence and the geometry of size* … use the ★ Pythagorean Theorem G.9.C Congruence and the geometry of size* formulate and test conjectures about the properties and attributes of circles and the lines that intersect them ... ★ ★ ★ ★ ★ ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 744 744 Chapter 11 Reading Strategy: Read to Solve Problems A word problem may be overwhelming at first. Once you identify the important parts of the problem and translate the words into math language, you will find that the problem is similar to others you have solved. Reading Tips: ✔ Read each phrase slowly. Write down ✔ Translate the words or phrases what the words mean as you read them. into math language. ✔ Draw a diagram. Label the diagram so it ✔ Highlight what is being asked. makes sense to you. ✔ Read the problem again before finding your solution. From Lesson 10-3: Use the Reading Tips to help you understand this problem. 14. After a day hike, a group of hikers set up a camp 3 km east and 7 km north of the starting point. The elevation of the camp is 0.6 km higher than the starting point. What is the distance from the camp to the starting point? After a day hike, a group of hikers set up a camp 3 km east and 7 km north of the starting point. The starting point can be represented by the ordered triple (0, 0, 0) . The elevation of the camp is 0.6 km higher than the starting point. The camp can be represented by the ordered triple (3, 7, 0.6) . What is the distance from the camp to the starting point? Distance can be found using the Distance Formula. Use the Distance Formula to find the distance between the camp and the starting point = √  = √  (3 - 0) 2 + (7 - 0) 2 + (0.6 - 0) 2 ≈ 7.6 km Try This For the following problem, apply the following reading tips. Do not solve. • Identify key words. • Translate each phrase into math language. • Draw a diagram to represent the problem. 1. The height of a cylinder is 4 ft, and the diameter is 9 ft. What effect does doubling each measure have on the volume? Circles 745 745 �� 11-1 Lines That Intersect Circles TEKS G.9.C Congruence and the geometry of size: … test conjectures about the properties and attributes of circles and the lines that intersect …. Also G.1.A, Objectives Identify tangents, secants, and chords. Use properties of tangents to solve problems. Vocabulary interior of a circle exterior of a circle chord secant tangent of a circle point of tangency congruent circles concentric circles tangent circles common tangent Why learn this? You can use circle theorems to solve problems about Earth. (See Example 3.) This photograph was taken 216 miles above Earth. From this altitude, it is easy to see the curvature of the horizon. Facts about circles can help us understand details about Earth. Recall that a circle is the set of all points in a plane that are equidistant from a given point, called the center of the circle. A circle with center C is called circle C, or ⊙C. The interior of a circle is the set of all points inside the circle. The exterior of a circle is the set of all points outside the circle. �� Lines and Segments That Intersect Circles TERM DIAGRAM A chord is a segment whose endpoints lie on a circle. Also G.2.A, G.2.B A secant is a line that intersects a circle at two points. A tangent is a line in the same plane as a circle that intersects it at exactly one point. The point where the tangent and a circle intersect is called the point of tangency . E X A M P L E 1 Identifying Lines and Segments That Intersect Circles Identify each line or segment that intersects ⊙A. chords: ̶̶ EF and ̶̶ BC tangent: ℓ ̶̶ AC and radii: ̶̶ AB secant:   EF diameter: ̶̶ BC 746 746 Chapter 11 Circles �� 1. Identify each line or segment that intersects ⊙P. Remember that the terms radius and diameter may refer to line segments, or to the lengths of segments. Pairs of Circles TERM DIAGRAM Two circles are congruent circles if and only if they have congruent radii. Concentric circles are coplanar circles with the same center. Two coplanar circles that intersect at exactly one point are called tangent circles . ̶̶ ⊙A ≅ ⊙B if BD . ̶̶ ̶̶ BD if ⊙A ≅ ⊙B. AC ≅ ̶̶ AC ≅ Internally tangent circles Externally tangent circles E X A M P L E 2 Identifying Tangents of Circles Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. radius of ⊙A : 4 radius of ⊙B : 2 Center is (-1, 0) . Pt. on ⊙ is (3, 0) . Dist. between the 2 pts. is 4. Center is (1, 0) . Pt. on ⊙ is (3, 0) . Dist. between the 2 pts. is 2. point of tangency: (3, 0) Pt. where the ⊙s and tangent line intersect equation of tangent line: x = 3 Vert. line through (3, 0) 2. Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. 11- 1 Lines That Intersect Circles 747 747 �� A common tangent is a line that is tangent to two circles. Lines ℓ and m are common external tangents to ⊙A and ⊙B. Lines p and q are common internal tangents to ⊙A and ⊙B. Construction Tangent to a Circle at a Point    Draw ⊙P. Locate a point on the circle and label it Q. Draw  PQ . Construct the perpendicular ℓ to  PQ at Q. This line is tangent to ⊙P at Q. Notice that in the construction, the tangent line is perpendicular to the radius at the point of tangency. This fact is the basis for the following theorems. Theorems THEOREM HYPOTHESIS CONCLUSION 11-1-1 Theorem 11-1-2 is the converse of Theorem 11-1-1. 11-1-2 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. (line tangent to ⊙ → line ⊥ to radius) If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. (line ⊥ to radius → line tangent to ⊙) ℓ is tangent to ⊙A m is ⊥ to ̶̶ CD at D ℓ ⊥ ̶̶ AB m is tangent to ⊙C. You will prove Theorems 11-1-1 and 11-1-2 in Exercises 28 and 29. 748 748 Chapter 11 Circles �� E X A M P L E 3 Problem Solving Application The summit of Mount Everest is approximately 29,000 ft above sea level. What is the distance from the summit to the horizon to the nearest mile? Understand the Problem The answer will be the length of an imaginary segment from the summit of Mount Everest to Earth’s horizon. Make a Plan Draw a sketch. Let C be the center of Earth, E be the summit of Mount Everest, and H be a point on the horizon. You need to find the length of tangent to ⊙C at H. By Theorem 11-1-1, △CHE is a right triangle. ̶̶ EH , which is ̶̶ EH ⊥ ̶̶ CH . So 5280 ft = 1 mi Earth’s radius ≈ 4000 mi Solve ED = 29,000 ft Giv
en = 29,000 _ 5280 ≈ 5.49 mi EC = CD + ED Change ft to mi. Seg. Add. Post. = 4000 + 5.49 = 4005.49 mi Substitute 4000 for CD and 5.49 for ED. EC 2 = EH 2 + CH 2 4005.49 2 = EH 2 + 4000 2 43,950.14 ≈ EH 2 210 mi ≈ EH Look Back Pyth. Thm. Substitute the given values. Subtract 4000 2 from both sides. Take the square root of both sides. The problem asks for the distance to the nearest mile. Check if your answer is reasonable by using the Pythagorean Theorem. Is 210 2 + 4000 2 ≈ 4005 2 ? Yes, 16,044,100 ≈ 16,040,025. 3. Kilimanjaro, the tallest mountain in Africa, is 19,340 ft tall. What is the distance from the summit of Kilimanjaro to the horizon to the nearest mile? Theorem 11-1-3 THEOREM HYPOTHESIS CONCLUSION If two segments are tangent to a circle from the same external point, then the segments are congruent. (2 segs. tangent to ⊙ from same ext. pt. → segs. ≅) ̶̶ AB ≅ ̶̶ AC ̶̶ AB and ̶̶ AC are tangent to ⊙P. You will prove Theorem 11-1-3 in Exercise 30. 11- 1 Lines That Intersect Circles 749 749 12��34�� You can use Theorem 11-1-3 to find the length of segments drawn tangent to a circle from an exterior point. E X A M P L E 4 Using Properties of Tangents ̶̶ DE and ̶̶ DF are tangent to ⊙C. Find DF. DE = DF 2 segs. tangent to ⊙ from same ext. pt. → segs. ≅. 5y - 28 = 3y Substitute 5y - 28 for DE and 3y for DF. 2y - 28 = 0 Subtract 3y from both sides. 2y = 28 y = 14 DF = 3 (14) = 42 Add 28 to both sides. Divide both sides by 2. Substitute 14 for y. Simplify. ̶̶ RS and 4a. ̶̶ RT are tangent to ⊙Q. Find RS. 4b. THINK AND DISCUSS 1. Consider ⊙A and ⊙B. How many different lines are common tangents to both circles? Copy the circles and sketch the common external and common internal tangent lines. 2. Is it possible for a line to be tangent to two concentric circles? Explain your answer. 3. Given ⊙P, is the center P a part of the circle? Explain your answer. 4. In the figure, ̶̶ RQ is tangent to ⊙P at Q. Explain how you can find m∠PRQ. 5. GET ORGANIZED Copy and complete the graphic organizer below. In each box, write a definition and draw a sketch. 750 750 Chapter 11 Circles �� 11-1 Exercises Exercises KEYWORD: MG7 11-1 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. A ? is a line in the plane of a circle that intersects the circle at two points. ̶̶̶̶ (secant or tangent) 2. Coplanar circles that have the same center are called ? . ̶̶̶̶ (concentric or congruent) 3. ⊙Q and ⊙R both have a radius of 3 cm. Therefore the circles are ? . ̶̶̶̶ (concentric or congruent Identify each line or segment that intersects each circle. p. 746 4. 5. 747 Multi-Step Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. 6. 7. 749 8. Space Exploration The International Space Station orbits Earth at an altitude of 240 mi. What is the distance from the space station to Earth’s horizon to the nearest mile The segments in each figure are tangent to the circle. Find each length. p. 750 9. JK 10. ST 11- 1 Lines That Intersect Circles 751 751 �� PRACTICE AND PROBLEM SOLVING Identify each line or segment that intersects each circle. 11. 12. Multi-Step Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. 13. 14. Independent Practice For See Exercises Example 11–12 13–14 15 16–17 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S24 Application Practice p. S38 Astronomy 15. Astronomy Olympus Mons’s peak rises 25 km above the surface of the planet Mars. The diameter of Mars is approximately 6794 km. What is the distance from the peak of Olympus Mons to the horizon to the nearest kilometer? Olympus Mons, located on Mars, is the tallest known volcano in the solar system. The segments in each figure are tangent to the circle. Find each length. 16. AB 17. RT Tell whether each statement is sometimes, always, or never true. 18. Two circles with the same center are congruent. 19. A tangent to a circle intersects the circle at two points. 20. Tangent circles have the same center. 21. A tangent to a circle will form a right angle with a radius that is drawn to the point of tangency. 22. A chord of a circle is a diameter. Graphic Design Use the following diagram for Exercises 23–25. The blue topaz was adopted as the Texas state gemstone in 1969. Identify the following. � 23. diameter 24. radii 25. chord � � � � 752 752 Chapter 11 Circles �� In each diagram, ̶̶ PR and 26. m∠Q ̶̶ PS are tangent to ⊙Q. Find each angle measure. 27. m∠P 28. Complete this indirect proof of Theorem 11-1-1. Given: ℓ is tangent to ⊙A at point B. Prove: ℓ ⊥ ̶̶ AB ̶̶ AC such that ̶̶ AB . Then it is possible to ̶̶ AC ⊥ ℓ. If this is true, then △ACB is ? . Since ℓ is a ̶̶̶̶ Proof: Assume that ℓ is not ⊥ draw a right triangle. AC < AB because a. tangent line, it can only intersect ⊙A at b. ? , and C must be in the ̶̶̶̶ exterior of ⊙A. That means that AC > AB since contradicts the fact that AC < AB. Thus the assumption is false, and d. ? . This ̶̶̶̶ ̶̶ AB is a c. ? . ̶̶̶̶ 29. Prove Theorem 11-1-2. ̶̶ Given: m ⊥ CD Prove: m is tangent to ⊙C. (Hint: Choose a point on m. Then use the Pythagorean Theorem to prove that if the point is not D, then it is not on the circle.) 30. Prove Theorem 11-1-3. ̶̶ AB and ̶̶ ̶̶ AB ≅ AC Given: Prove: ̶̶ AC are tangent to ⊙P. Plan: Draw auxiliary segments the triangles formed are congruent. Then use CPCTC. ̶̶ PA , ̶̶ PB , and ̶̶ PC . Show that Algebra Assume the segments that appear to be tangent are tangent. Find each length. 31. ST 32. DE 33. JL 34. ⊙M has center M (2, 2) and radius 3. ⊙N has center N (-3, 2) and is tangent to ⊙M. Find the coordinates of the possible points of tangency of the two circles. 35. This problem will prepare you for the Multi-Step TAKS Prep on page 770. The diagram shows the gears of a bicycle. AD = 5 in., and BC = 3 in. CD, the length of the chain between the gears, is 17 in. a. What type of quadrilateral is BCDE? Why? b. Find BE and AE. c. What is AB to the nearest tenth of an inch? � � � � � 11- 1 Lines That Intersect Circles 753 753 �� 36. Critical Thinking Given a circle with diameter ̶̶ BC , is it possible to draw tangents to B and C from an external point X? If so, make a sketch. If not, explain why it is not possible. 37. Write About It ̶̶ PR and Explain why ∠P and ∠Q are supplementary. ̶̶ PS are tangent to ⊙Q. 38. ̶̶ AB and is closest to AD? ̶̶ AC are tangent to ⊙D. Which of these 9.5 cm 10 cm 10.4 cm 13 cm 39. ⊙P has center P (3, -2) and radius 2. Which of these lines is tangent to ⊙P? x = 4 y = -4 y = -2 x = 0 40. ⊙A has radius 5. ⊙B has radius 6. What is the ratio of the area of ⊙A to that of ⊙B? 125 _ 216 25 _ 36 5 _ 6 36 _ 25 CHALLENGE AND EXTEND ̶̶̶ 41. Given: ⊙G with GH ⊥ ̶̶ KH ̶̶ JH ≅ Prove: ̶̶ JK 42. Multi-Step ⊙A has radius 5, ⊙B has radius 2, ̶̶ CD is a common tangent. What is AB? and (Hint: Draw a perpendicular segment from B to E, a point on ̶̶ AC .) 43. Manufacturing A company builds metal stands for bicycle wheels. A new design calls for a V-shaped stand that will hold wheels with a 13 in. radius. The sides of the stand form a 70° angle. To the nearest tenth of an inch, what should be the length XY of a side so that it is tangent to the wheel? SPIRAL REVIEW 44. Andrea and Carlos both mow lawns. Andrea charges $14.00 plus $6.25 per hour. Carlos charges $12.50 plus $6.50 per hour. If they both mow h hours and Andrea earns more money than Carlos, what is the range of values of h? (Previous course) ̶̶ LR . A point is chosen randomly on Use the diagram to find the probability of each event. (Lesson 9-6) ̶̶̶ MP . 45. The point is not on ̶̶̶ MN or 47. The point is on ̶̶ PR . 754 754 Chapter 11 Circles 46. The point is on 48. The point is on ̶̶ LP . ̶̶ QR . �� Circle Graphs Data Analysis A circle graph compares data that are parts of a whole unit. When you make a circle graph, you find the measure of each central angle. A central angle is an angle whose vertex is the center of the circle. See Skills Bank page S80 Example Make a circle graph to represent the following data. Step 1 Add all the amounts. 110 + 40 + 300 + 150 = 600 Step 2 Write each part as a fraction of the whole. fiction: 110 _ ; nonfiction: 40 _ ; children’s: 300 _ ; audio books: 150 _ 600 600 600 600 Books in the Bookmobile Fiction Nonfiction Children’s 110 40 300 150 Step 3 Multiply each fraction by 360° to calculate the central Audio books angle measure. 110 _ 600 (360°) = 66°; 40 _ 600 (360°) = 24°; 300 _ (360°) = 180°; 150 _ (360°) = 90° 600 600 Step 4 Make a circle graph. Then color each section of the circle to match the data. The section with a central angle of 66° is green, 24° is orange, 180° is purple, and 90° is yellow. Try This TAKS Grades 9–11 Obj. 8, 9 Choose the circle graph that best represents the data. Show each step. 1. Books in Linda’s Library Novels Reference Textbooks 18 10 8 2. Vacation Expenses ($) 3. Puppy Expenses ($) Travel Meals Lodging Other 450 120 900 330 Food Health Training Other 190 375 120 50 On Track for TAKS 755 755 �� 11-2 Arcs and Chords TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system …. Also G.2.A, G.2.B, G.8.C, G.9.C Objectives Apply properties of arcs. Apply properties of chords. Who uses this? Market analysts use circle graphs to compare sales of different products. Vocabulary central angle arc minor arc major arc semicircle adjacent arcs congruent arcs Minor arcs may be named by two points. Major arcs and semicircles must be named by three points. A central angle is an angle whose vertex is the center of a circle. A
n arc is an unbroken part of a circle consisting of two points called the endpoints and all the points on the circle between them. Arcs and Their Measure ARC MEASURE DIAGRAM A minor arc is an arc whose points are on or in the interior of a central angle. The measure of a minor arc is equal to the measure of its central angle. m ⁀ AC = m∠ABC = x° A major arc is an arc whose points are on or in the exterior of a central angle. The measure of a major arc is equal to 360° minus the measure of its central angle. m ⁀ ADC = 360° - m∠ABC = 360° - x° If the endpoints of an arc lie on a diameter, the arc is a semicircle . The measure of a semicircle is equal to 180°. m ⁀ EFG = 180° E X A M P L E 1 Data Application The circle graph shows the types of music sold during one week at a music store. Find m ⁀ BC . m ⁀ BC = m∠BMC m of arc = m of central ∠. m∠BMC = 0.13 (360°) Central ∠ is 13% = 46.8° of the ⊙. Use the graph to find each of the following. 1a. m∠FMC 1b. m ⁀ AHB 1c. m∠EMD 756 756 Chapter 11 Circles �� Adjacent arcs are arcs of the same circle that intersect at exactly one point. ⁀ RS and ⁀ ST are adjacent arcs. Postulate 11-2-1 Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. m⁀ABC = m ⁀AB + m ⁀BC E X A M P L E 2 Using the Arc Addition Postulate Find m ⁀CDE m ⁀CD = 90° m∠DFE = 18° m ⁀DE = 18° m ⁀CE = m ⁀CD + m ⁀DE m∠CFD = 90° Vert.  Thm. m∠DFE = 18° Arc Add. Post. = 90° + 18° = 108° Substitute and simplify. Find each measure. 2a. m ⁀JKL 2b. m ⁀LJN Within a circle or congruent circles, congruent arcs are two arcs that have the same measure. In the figure, ⁀ ST ≅ ⁀ UV . Theorem 11-2-2 THEOREM HYPOTHESIS CONCLUSION In a circle or congruent circles: (1) Congruent central angles have congruent chords. (2) Congruent chords have congruent arcs. (3) Congruent arcs have congruent central angles. ∠EAD ≅ ∠BAC ̶̶ DE ≅ ̶̶ BC ̶̶ ED ≅ ̶̶ BC ⁀ DE ≅ ⁀ BC ⁀ ED ≅ ⁀ BC ∠DAE ≅ ∠BAC You will prove parts 2 and 3 of Theorem 11-2-2 in Exercises 40 and 41. 11-2 Arcs and Chords 757 757 �� The converses of the parts of Theorem 11-2-2 are also true. For example, with part 1, congruent chords have congruent central angles. PROOF PROOF Theorem 11-2-2 (Part 1) Given: ∠BAC ≅ ∠DAE ̶̶ BC ≅ Prove: ̶̶ DE Proof: Statements Reasons 1. ∠BAC ≅ ∠DAE ̶̶ AB ≅ ̶̶̶ AD , ̶̶ AC ≅ ̶̶ AE 2. 3. △BAC ≅ △DAE ̶̶ DE ̶̶ BC ≅ 4. 1. Given 2. All radii of a ⊙ are ≅. 3. SAS Steps 2, 1 4. CPCTC E X A M P L E 3 Applying Congruent Angles, Arcs, and Chords Find each measure. A ̶̶ RS ≅ ̶̶ TU . Find m ⁀ RS . ⁀ RS ≅ ⁀ TU m ⁀ RS = m ⁀ TU 3x = 2x + 27 x = 27 m ⁀ RS = 3 (27) = 81° ≅ chords have ≅ arcs. Def. of ≅ arcs Substitute the given measures. Subtract 2x from both sides. Substitute 27 for x. Simplify. B ⊙B ≅ ⊙E, and ⁀ AC ≅ ⁀ DF . Find m∠DEF. ∠ABC = ∠DEF m∠ABC = m∠DEF 5y + 5 = 7y - 43 5 = 2y - 43 ≅ arcs have ≅ central . Def. of ≅  Substitute the given measures. Subtract 5y from both sides. 48 = 2y 24 = y m∠DEF = 7 (24) - 43 = 125° Add 43 to both sides. Divide both sides by 2. Substitute 24 for y. Simplify. Find each measure. 3a.  PT bisects ∠RPS. Find RT. 3b. ⊙A ≅ ⊙B, and Find m ⁀ CD . ̶̶ CD ≅ ̶̶ EF . 758 758 Chapter 11 Circles �� Theorems THEOREM HYPOTHESIS CONCLUSION 11-2-3 In a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc. 11-2-4 In a circle, the perpendicular bisector of a chord is a radius (or diameter). ̶̶ CD bisects ̶̶ EF and ⁀ EF . ̶̶ CD ⊥ ̶̶ EF ̶̶ JK is a diameter of ⊙A. ̶̶ JK is ⊥ bisector of ̶̶̶ GH . You will prove Theorems 11-2-3 and 11-2-4 in Exercises 42 and 43. E X A M P L E 4 Using Radii and Chords Find BD. Step 1 Draw radius ̶̶ AD . AD = 5 Radii of a ⊙ are ≅. Step 2 Use the Pythagorean Theorem. CD 2 + AC 2 = AD 2 CD 2 + 3 2 = 5 2 CD 2 = 16 CD = 4 Step 3 Find BD. BD = 2(4 ) = 8 Substitute 3 for AC and 5 for AD. Subtract 3 2 from both sides. Take the square root of both sides. ̶̶ AE ⊥ ̶̶ BD , so ̶̶ AE bisects ̶̶ BD . 4. Find QR to the nearest tenth. THINK AND DISCUSS 1. What is true about the measure of an arc whose central angle is obtuse? 2. Under what conditions are two arcs the same measure but not congruent? 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write a definition and draw a sketch. 11-2 Arcs and Chords 759 759 �� 11-2 Exercises Exercises KEYWORD: MG7 11-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. An arc that joins the endpoints of a diameter is called a ? . (semicircle or ̶̶̶̶ major arc) 2. How do you recognize a central angle of a circle? ABC = 205°. Therefore ⁀ 3. In ⊙P m ⁀ ABC is a ? . (major arc or minor arc) ̶̶̶̶ 4. In a circle, an arc that is less than a semicircle is a ? . (major arc or minor arc) ̶̶̶̶ . 756 Consumer Application Use the following information for Exercises 5–10. The circle graph shows how a typical household spends money on energy. Find each of the following. 5. m∠PAQ 7. m∠SAQ 9. m ⁀ RQ 6. m∠VAU 8. m ⁀ UT 10. m ⁀ UPT . 757 Find each measure. 11. m ⁀ DF 12. m ⁀ DEB 13. m ⁀ JL 14. m ⁀ HLK 15. ∠QPR ≅ ∠RPS. Find QR. 16. ⊙A ≅ ⊙B, and ⁀ CD ≅ ⁀ EF . Find m∠EBF. p. 758 Multi-Step Find each length to the nearest tenth. p. 759 17. RS 18. EF 760 760 Chapter 11 Circles �� Independent Practice For See Exercises Example 19–24 25–28 29–30 31–32 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S24 Application Practice p. S38 PRACTICE AND PROBLEM SOLVING Sports Use the following information for Exercises 19–24. The key shows the number of medals won by U.S. athletes at the 2004 Olympics in Athens. Find each of the following to the nearest tenth. �� 19. m∠ADB 21. m ⁀ AB 21. 23. m ⁀ ACB 20. m∠ADC 22. m ⁀ BC 24. m ⁀ CAB Find each measure. 25. m ⁀ MP 26. m ⁀ QNL 29. ⊙A ≅ ⊙B, and ⁀ CD ≅ ⁀ EF . Find m∠CAD. �� 27. m ⁀ WT 28. m ⁀ WTV 30. ̶̶ JK ≅ ̶̶̶ LM . Find m ⁀ JK . Multi-Step Find each length to the nearest tenth. 31. CD 32. RS Determine whether each statement is true or false. If false, explain why. 33. The central angle of a minor arc is an acute angle. 34. Any two points on a circle determine a minor arc and a major arc. 35. In a circle, the perpendicular bisector of a chord must pass through the center of the circle. 36. Data Collection Use a graphing calculator, a pH probe, and a data-collection device to collect information about the pH levels of ten different liquids. Then create a circle graph with the following sectors: strong basic (9 < pH < 14) , weak basic (7 < pH < 9) , neutral (pH = 7) , weak acidic (5 < pH < 7) , and strong acidic (0 < pH < 5) . 37. In ⊙E, the measures of ∠AEB, ∠BEC, and ∠CED are in the ratio 3 : 4 : 5. Find m ⁀ AB , m ⁀ BC , and m ⁀ CD . 11-2 Arcs and Chords 761 761 �� Algebra Find the indicated measure. 38. m ⁀ JL 39. m∠SPT 40. Prove ≅ chords have ≅ arcs. ̶̶ Given: ⊙A, BC ≅ Prove: ⁀ BC ≅ ⁀ DE ̶̶ DE 41. Prove ≅ arcs have ≅ central . Given: ⊙A, ⁀ BC ≅ ⁀ DE Prove: ∠BAC ≅ ∠DAE 42. Prove Theorem 11-2-3. Given: ⊙C, Prove: ̶̶ EF ̶̶ EF ̶̶ CD ⊥ ̶̶ CD bisects and ⁀ EF . ̶̶ CE and ̶̶ CF (Hint: Draw and use the HL Theorem.) 43. Prove Theorem 11-2-4. ̶̶ JK ⊥ ̶̶̶ GH Given: ⊙A, bisector of Prove: (Hint: Use the Converse of the ⊥ Bisector Theorem.) ̶̶ JK is a diameter 44. Critical Thinking Roberto folds a circular piece of paper as shown. When he unfolds the paper, how many different-sized central angles will be formed? One fold Two folds Three folds 45. /////ERROR ANALYSIS///// Below are two solutions to find the value of x. Which solution is incorrect? Explain the error. 46. Write About It According to a school survey, 40% of the students take a bus to school, 35% are driven to school, 15% ride a bike, and the remainder walk. Explain how to use central angles to create a circle graph from this data. 47. This problem will prepare you for the Multi-Step TAKS This problem will prepare you for the Multi-Step TAKS Prep on page 770. Chantal’s bike has wheels with a 27 in. diameter. a. What are AC and AD if DB is 7 in.? b. What is CD to the nearest tenth of an inch? c. What is CE, the length of the top of the bike stand? � � � � � 762 762 Chapter 11 Circles �� 48. Which of these arcs of ⊙Q has the greatest measure? ⁀ WT ⁀ UW ⁀ VR ⁀ TV 49. In ⊙A, CD = 10. Which of these is closest ̶̶ AE ? to the length of 3.3 cm 4 cm 5 cm 7.8 cm 50. Gridded Response ⊙P has center P (2, 1) and radius 3. What is the measure, in degrees, of the minor arc with endpoints A (-1, 1) and B (2, -2)? CHALLENGE AND EXTEND ̶̶ 51. In the figure, AB ⊥ ̶̶ CD . Find m ⁀ BD to the nearest tenth of a degree. 52. Two points on a circle determine two distinct arcs. How many arcs are determined by n points on a circle? (Hint: Make a table and look for a pattern.) 53. An angle measure other than degrees is radian measure. 360° converts to 2π radians, or 180°converts to π radians. , π _ , π _ a. Convert the following radian angle measures to degrees: π _ . 4 3 2 b. Convert the following angle measures to radians: 135°, 270°. SPIRAL REVIEW Simplify each expression. (Previous course) 54. (3x) 3 ( 2y 2 ) ( 3 -2 y 2 ) 55. a 4 b 3 (-2a) -4 2 56. ( -2t 3 s 2 ) ( 3ts 2 ) Find the ne
xt term in each pattern. (Lesson 2-1) 57. 1, 3, 7, 13, 21, … 58. C, E, G, I, K, ... 59. 1, 6, 15, … In the figure, (Lesson 11-1) ̶̶ QP and ̶̶̶ QM are tangent to ⊙N. Find each measure. 60. m∠NMQ 61. MQ Construction Circle Through Three Noncollinear Points    Draw three noncollinear points. Construct m and n, the ⊥ bisectors of ̶̶ PQ and ̶̶ QR . Label the intersection O. Center the compass at O. Draw a circle through P. 1. Explain why ⊙O with radius ̶̶ OP also contains Q and R. 11-2 Arcs and Chords 763 763 �� 11-3 Sector Area and Arc Length TEKS G.8.B Congruence and the geometry of size: find areas of sectors and arc lengths of circles using proportional reasoning. Objectives Find the area of sectors. Find arc lengths. Vocabulary sector of a circle segment of a circle arc length Who uses this? Farmers use irrigation radii to calculate areas of sectors. (See Example 2.) The area of a sector is a fraction of the circle containing the sector. To find the area of a sector whose central angle measures m°, multiply the area of the circle by m°____ 360° . Sector of a Circle TERM NAME DIAGRAM AREA A sector of a circle is a region bounded by two radii of the circle and their intercepted arc. sector ACB Also G.1A, G.1.B, G.9.C A = πr 2 ( m° _ ) 360° E X A M P L E 1 Finding the Area of a Sector Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. A sector MPN 360° A = π r 2 ( m° _ ) = π (3 ) 2 ( 80° _ ) = 2π in 2 ≈ 6.28 in 2 360° B sector EFG 360° A = π r 2 ( m° _ ) = π (6) 2 ( 120° _ ) = 12π ≈ 37.70 cm 2 360° Use formula for area of a sector. Substitute 3 for r and 80 for m. Simplify. Use formula for area of a sector. Substitute 6 for r and 120 for m. Simplify. Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. 1a. sector ACB 1b. sector JKL Write the degree symbol after m in the formula to help you remember to use degree measure not arc length. 764 764 Chapter 11 Circles �� E X A M P L E 2 Agriculture Application A circular plot with a 720 ft diameter is watered by a spray irrigation system. To the nearest square foot, what is the area that is watered as the sprinkler rotates through an angle of 50°? 360° A = π r 2 ( m° _ ) = π (360) 2 ( 50° _ ) ≈ 56,549 ft 2 360° d = 720 ft, r = 360 ft. Simplify. 2. To the nearest square foot, what is the area watered in Example 2 as the sprinkler rotates through a semicircle? A segment of a circle is a region bounded by an arc and its chord. The shaded region in the figure is a segment. Area of a Segment area of segment = area of sector - area of triangle E X A M P L E 3 Finding the Area of a Segment Find the area of segment ACB to the nearest hundredth. Step 1 Find the area of sector ACB. In a 30°-60°-90° triangle, the length of the leg opposite the 60° angle is √  3 times the length of the shorter leg. 360° A = π r 2 ( m° _ ) = π (12) 2 ( 60° _ ) = 24π in 2 360° Use formula for area of a sector. Substitute 12 for r and 60 for m. Step 2 Find the area of △ACB. ̶̶ AD . Draw altitude bh = 1 _ A = 1 _ (12) (6 √  3 ) 2 2 = 36 √  3 in 2 CD = 6 in., and h = 6 √  3 in. Simplify. Step 3 area of segment = area of sector ACB - area of △ACB = 24π - 36 √  3 ≈ 13.04 in 2 3. Find the area of segment RST to the nearest hundredth. 11-3 Sector Area and Arc Length 765 765 �� In the same way that the area of a sector is a fraction of the area of the circle, the length of an arc is a fraction of the circumference of the circle. Arc Length TERM DIAGRAM LENGTH Arc length is the distance along an arc measured in linear units. L = 2πr ( m° _ ) 360° E X A M P L E 4 Finding Arc Length Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. A ⁀CD L = 2πr ( m°_ 360°) = 2π(10)( 90°_ 360°) Use formula for arc length. Substitute 10 for r and 90 for m. = 5π ft ≈ 15.71 ft Simplify. B an arc with measure 35° in a circle with radius 3 in. L = 2πr ( m°_ 360°) = 2π(3)( 35°_ 360°) = 7_ 12 in. ≈ 1.83 in. Use formula for arc length. Substitute 3 for r and 35 for m. Simplify. Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. 4a. ⁀GH 4b. an arc with measure 135° in a circle with radius 4 cm THINK AND DISCUSS 1. What is the difference between arc measure and arc length? 2. A slice of pizza is a sector of a circle. Explain what measurements you would need to make in order to calculate the area of the slice. 3. GET ORGANIZED Copy and complete the graphic organizer. 766 766 Chapter 11 Circles �� 11-3 Exercises Exercises KEYWORD: MG7 11-3 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In a circle, the region bounded by a chord and an arc is called a ? . (sector or segment) ̶̶̶̶ . 764 Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. 2. sector PQR 3. sector JKL 4. sector ABC . Navigation The beam from a lighthouse is visible for a distance of 3 mi. p. 765 To the nearest square mile, what is the area covered by the beam as it sweeps in an arc of 150°? Multi-Step Find the area of each segment to the nearest hundredth. p. 765 6. 7. 8. 766 Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. 9. ⁀ EF 10. ⁀ PQ Independent Practice For See Exercises Example 12–14 15 16–18 19–21 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S24 Application Practice p. S38 11. an arc with measure 20° in a circle with radius 6 in. PRACTICE AND PROBLEM SOLVING Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. 12. sector DEF 13. sector GHJ 14. sector RST 15. Architecture A lunette is a semicircular window that is sometimes placed above a doorway or above a rectangular window. To the nearest square inch, what is the area of the lunette? �� 11-3 Sector Area and Arc Length 767 767 �� Multi-Step Find the area of each segment to the nearest hundredth. 16. 17. 18. Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. 19. ⁀ UV 20. ⁀ AB Math History 21. an arc with measure 9° in a circle with diameter 4 ft 22. Math History Greek mathematicians studied the salinon, a figure bounded by four semicircles. What is the perimeter of this salinon to the nearest tenth of an inch? Hypatia lived 1600 years ago. She is considered one of history’s most important mathematicians. She is credited with contributions to both geometry and astronomy. Tell whether each statement is sometimes, always, or never true. 23. The length of an arc of a circle is greater than the circumference of the circle. 24. Two arcs with the same measure have the same arc length. 25. In a circle, two arcs with the same length have the same measure. Find the radius of each circle. 26. area of sector ABC = 9π 27. arc length of ⁀ EF = 8π 28. Estimation The fraction 22 __ 7 is an approximation for π. a. Use this value to estimate the arc length of ⁀ XY . b. Use the π key on your calculator to find the length of ⁀ XY to 8 decimal places. c. Was your estimate in part a an overestimate or an underestimate? 29. This problem will prepare you for the Multi-Step TAKS Prep on page 770. The pedals of a penny-farthing bicycle are directly connected to the front wheel. a. Suppose a penny-farthing bicycle has a front wheel with a diameter of 5 ft. To the nearest tenth of a foot, how far does the bike move when you turn the pedals through an angle of 90°? b. Through what angle should you turn the pedals in order to move forward by a distance of 4.5 ft? Round to the nearest degree. 768 768 Chapter 11 Circles �� 30. Critical Thinking What is the length of the radius that makes the area of ⊙A = 24 in 2 and the area of sector BAC = 3 in 2 ? Explain. 31. Write About It Given the length of an arc of a circle and the measure of the arc, explain how to find the radius of the circle. 32. What is the area of sector AOB? 4π 16π 32π 64π 33. What is the length of ⁀ AB ? 4π 2π 8π 16π 34. Gridded Response To the nearest hundredth, what is the area of the sector determined by an arc with measure 35° in a circle with radius 12? CHALLENGE AND EXTEND 35. In the diagram, the larger of the two concentric circles has radius 5, and the smaller circle has radius 2. What is the area of the shaded region in terms of π ? 36. A wedge of cheese is a sector of a cylinder. a. To the nearest tenth, what is the volume of the wedge with the dimensions shown? b. What is the surface area of the wedge of cheese to the nearest tenth? 37. Probability The central angles of a target measure 45°. The inner circle has a radius of 1 ft, and the outer circle has a radius of 2 ft. Assuming that all arrows hit the target at random, find the following probabilities. a. hitting a red region b. hitting a blue region c. hitting a red or blue region �� SPIRAL REVIEW Determine whether each line is parallel to y = 4x - 5, perpendicular to y = 4x - 5, or neither. (Previous course) 38. 8x - 2y = 6 39. line passing through the points ( 1 __ 2 , 0) and (1 1 __ 2 , 2) 40. line with x-intercept 4 and y-intercept 1 Find each measurement. Give your answer in terms of π. (Lesson 10-8) 41. volume of a sphere with radius 3 cm 42. circumference of a great circle of a sphere whose surface area is 4π cm 2 Find the indicated measure. (Lesson 11-2) 43. m∠KLJ 44. m ⁀ KJ 45. m ⁀ JFH 11-3 Sector Area and Arc Length 769 769 �� SECTION 11A Lines and Arcs in Circles As the Wheels Turn The bicycle was invented in the 1790s. The first models didn’t even have pedals—riders moved forward b
y pushing their feet along the ground! Today the bicycle is a high-tech machine that can include hydraulic brakes and electronic gear changers. 1. A road race bicycle wheel is 28 inches in diameter. A manufacturer makes metal bicycle stands that are 10 in. tall. How long should a stand be to the nearest tenth in order to support a 28 in. wheel? (Hint: Consider the triangle formed by the radii and the top of the stand.) 2. The chain of a bicycle loops around a large gear connected to the bike’s pedals and a small gear attached to the rear wheel. In the diagram, the distance AB between the centers of the gears the nearest tenth is 15 in. Find CD, the length of the chain between the two gears to the nearest tenth. (Hint: Draw a segment from B to ̶̶ AD that is parallel to ̶̶ CD .) �� 3. By pedaling, you turn the large gear through an angle of 60°. How far does the chain move around the circumference of the gear to the nearest tenth? 4. As the chain moves, it turns the small gear. If you use the distance you calculated in Problem 3, through what angle does the small gear turn to the nearest degree? �� 770 770 Chapter 11 Circles SECTION 11A Quiz for Lessons 11-1 Through 11-3 11-1 Lines That Intersect Circles Identify each line or segment that intersects each circle. 1. 2. 3. The tallest building in Africa is the Carlton Centre in Johannesburg, South Africa. What is the distance from the top of this 732 ft building to the horizon to the nearest mile? (Hint: 5280 ft = 1 mi; radius of Earth = 4000 mi) 11-2 Arcs and Chords Find each measure. 4. ⁀ BC 5. ⁀ BED 6. ⁀ SR 7. ⁀ SQU Find each length to the nearest tenth. 8. JK 9. XY 11-3 Sector Area and Arc Length 10. As part of an art project, Peter buys a circular piece of fabric and then cuts out the sector shown. What is the area of the sector to the nearest square centimeter? Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. 11. ⁀ AB 12. ⁀ EF �� 13. an arc with measure 44° in a circle with diameter 10 in. 14. a semicircle in a circle with diameter 92 m Ready to Go On? 771 771 �� 11-4 Inscribed Angles TEKS G.5.B Geometric paterns: use … patterns to make generalizations about geometric properties, including … angle relationships in … circles. Objectives Find the measure of an inscribed angle. Use inscribed angles and their properties to solve problems. Vocabulary inscribed angle intercepted arc subtend Also G.1.A, G.2.A, G.2.B, G.9.C Why learn this? You can use inscribed angles to find measures of angles in string art. (See Example 2.) String art often begins with pins or nails that are placed around the circumference of a circle. A long piece of string is then wound from one nail to another. The resulting pattern may include hundreds of inscribed angles. An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. An intercepted arc consists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between them. A chord or arc subtends an angle if its endpoints lie on the sides of the angle. ∠DEF is an inscribed angle. ⁀ DF is the intercepted arc. ⁀ DF subtends ∠DEF. Theorem 11-4-1 Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. m∠ABC = 1 _ 2 m ⁀ AC Case 1 Case 2 Case 3 You will prove Cases 2 and 3 of Theorem 11-4-1 in Exercises 30 and 31. PROOF PROOF Inscribed Angle Theorem Given: ∠ABC is inscribed in ⊙X. Prove: m∠ABC = 1 __ 2 m ⁀ AC Proof Case 1: ̶̶ BC . Draw ̶̶ XA . m ⁀ AC = m∠AXC. ̶̶ XA and ̶̶ XB are radii of the circle, ∠ABC is inscribed in ⊙X with X on By the Exterior Angle Theorem m∠AXC = m∠ABX + m∠BAX. Since △AXB is isosceles. Thus m∠ABX = m∠BAX. By the Substitution Property, m ⁀ AC = 2m∠ABX or 2m∠ABC. Thus 1 __ 2 m ⁀ AC = m∠ABC. ̶̶ XA ≅ ̶̶ XB . Then by definition 772 772 Chapter 11 Circles �� E X A M P L E 1 Finding Measures of Arcs and Inscribed Angles Find each measure. A m∠RST m∠RST = 1 _ m ⁀ RT 2 = 1 _ (120°) = 60° 2 Inscribed ∠ Thm. Substitute 120 for m ⁀ RT . B m ⁀ SU m∠SRU = 1 _ m ⁀ SU 2 40° = 1 _ m ⁀ SU 2 m ⁀ SU = 80° Inscribed ∠ Thm. Substitute 40 for m∠SRU. Mult. both sides by 2. Find each measure. 1a. m ⁀ ADC 1b. m∠DAE Corollary 11-4-2 COROLLARY HYPOTHESIS CONCLUSION If inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc, then the angles are congruent. ∠ACB ≅ ∠ADB ≅ ∠AEB (and ∠CAE ≅ ∠CBE) ∠ACB, ∠ADB, and ∠AEB intercept ⁀ AB . You will prove Corollary 11-4-2 in Exercise 32. E X A M P L E 2 Hobby Application Find m∠DEC, if m ⁀ AD = 86°. ∠BAC ≅ ∠BDC m∠BAC = m∠BDC m∠BDC = 60° m∠ACD = 1 _ m ⁀ AD 2 = 1 _ (86°) 2 = 43° ∠BAC and ∠BDC intercept ⁀ BC . Def. of ≅ Substitute 60 for m∠BDC. Inscribed ∠ Thm. Substitute 86 for m ⁀ AD . Simplify. m∠DEC + 60 + 43 = 180 △ Sum Theorem m∠DEC = 77° Simplify. 2. Find m∠ABD and m ⁀ BC in the string art. 11-4 Inscribed Angles 773 773 �� Theorem 11-4-3 An inscribed angle subtends a semicircle if and only if the angle is a right angle. You will prove Theorem 11-4-3 in Exercise 43. E X A M P L E 3 Finding Angle Measures in Inscribed Triangles Find each value. A x ∠RQT is a right angle ∠RQT is inscribed in a m∠RQT = 90° 4x + 6 = 90 4x = 84 x = 21 B m∠ADC semicircle. Def. of rt. ∠ Substitute 4x + 6 for m∠RQT. Subtract 6 from both sides. Divide both sides by 4. m∠ABC = m∠ADC ∠ABC and ∠ADC both 10y - 28 = 7y - 1 3y - 28 = -1 3y = 27 y = 9 intercept ⁀ AC . Substitute the given values. Subtract 7y from both sides. Add 28 to both sides. Divide both sides by 3. m∠ADC = 7 (9 ) -1 = 62° Substitute 9 for y. Find each value. 3a. z 3b. m∠EDF Construction Center of a Circle     Draw a circle and ̶̶ AB . chord Draw chord ̶̶ AC . Construct a line ̶̶ perpendicular to AB at B. Where the line and the circle intersect, label the point C. 774 774 Chapter 11 Circles ̶̶ DF . Repeat steps to draw ̶̶ chords DE and The intersection of ̶̶ and DF is the center of the circle. ̶̶ AC �� Theorem 11-4-4 THEOREM HYPOTHESIS CONCLUSION If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. ∠A and ∠C are supplementary. ∠B and ∠D are supplementary. ABCD is inscribed in ⊙E. You will prove Theorem 11-4-4 in Exercise 44. E X A M P L E 4 Finding Angle Measures in Inscribed Quadrilaterals Find the angle measures of PQRS. Step 1 Find the value of y. m∠P + m∠R = 180° PQRS is inscribed in a ⊙. 6y + 1 + 10y + 19 = 180 16y + 20 = 180 16y = 160 y = 10 Substitute the given values. Simplify. Subtract 20 from both sides. Divide both sides by 16. Step 2 Find the measure of each angle. m∠P = 6 (10) + 1 = 61° m∠R = 10 (10) + 19 = 119° m∠Q = 10 2 + 48 = 148° m∠Q + m∠S = 180° 148° + m∠S = 180° m∠S = 32° Substitute 10 for y in each expression. ∠Q and ∠S are supp. Substitute 148 for m∠Q. Subtract 148 from both sides. 4. Find the angle measures of JKLM. THINK AND DISCUSS 1. Can ABCD be inscribed in a circle? Why or why not? 2. An inscribed angle intercepts an arc that is 1 __ 4 of the circle. Explain how to find the measure of the inscribed angle. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box write a definition, properties, an example, and a nonexample. 11-4 Inscribed Angles 775 775 �� 11-4 Exercises Exercises KEYWORD: MG7 11-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary A, B, and C lie on ⊙P. ∠ABC is an example of an ? angle. ̶̶̶̶ (intercepted or inscribed Find each measure. p. 773 2. m∠DEF 3. m ⁀ EG 4. m ⁀ JKL 5. m∠LKM . Crafts A circular loom can be used � p. 773 for knitting. What is the m∠QTR in the knitting loom Find each value. �� p. 774 7. x 8. y 9. m∠XYZ Multi-Step Find the angle measures of each quadrilateral. p. 775 10. PQRS 11. ABCD Independent Practice For See Exercises Example 12–15 16 17–20 21–22 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S25 Application Practice p. S38 PRACTICE AND PROBLEM SOLVING Find each measure. 12. m ⁀ ML 14. m ⁀ EGH 13. m∠KMN 15. m∠GFH 16. Crafts An artist created a stained 16. glass window. If m∠BEC = 40° and m ⁀ AB = 44°, what is m∠ADC? 776 776 Chapter 11 Circles ABCDEge07sec11l04004a�� Algebra Find each value. 17. y 18. z 19. m ⁀ AB 20. m∠MPN Multi-Step Find the angle measures of each quadrilateral. 21. BCDE 22. TUVW Tell whether each statement is sometimes, always, or never true. 23. Two inscribed angles that intercept the same arc of a circle are congruent. 24. When a right triangle is inscribed in a circle, one of the legs of the triangle is a diameter of the circle. 25. A trapezoid can be inscribed in a circle. Multi-Step Find each angle measure. 26. m∠ABC if m∠ADC = 112° 27. m∠PQR if m ⁀ PQR = 130° 28. Prove that the measure of a central angle subtended by a chord is twice the measure of the inscribed angle subtended by the chord. Given: In ⊙H Prove: m∠JHK = 2m∠JLK ̶̶ JK subtends ∠JHK and ∠JLK. 29. This problem will prepare you for the Multi-Step TAKS Prep on page 806. A Native American sand painting could be used to indicate the direction of sunrise on the winter and summer solstices. You can make this design by placing six equally spaced points around the circumference of a circle and connecting them as shown. a. Find m∠BAC. b. Find m∠CDE. c. What type of triangle is △FBC? Why? 11-4 Inscribed Angles 777 777 ��ABEDCFge07sec11l0
4006a 30. Given: ∠ABC is inscribed in ⊙X with X in the interior of ∠ABC. Prove: m∠ABC = 1 _ m ⁀ AC 2 (Hint: Draw  BX and use Case 1 of the Inscribed Angle Theorem.) History The Winchester Round Table, probably built in the late thirteenth century, is 18 ft across and weighs 1.25 tons. King Arthur’s Round Table of English legend would have been much larger—it could seat 1600 men. 31. Given: ∠ABC is inscribed in ⊙X with X in the exterior of ∠ABC. Prove: m∠ABC = 1 _ m ⁀ AC 2 32. Prove Corollary 11-4-2. Given: ∠ACB and ∠ADB intercept ⁀ AB . Prove: ∠ACB ≅ ∠ADB 33. Multi-Step In the diagram, m ⁀ JKL = 198°, and m ⁀ of quadrilateral JKLM. KLM = 216°. Find the measures of the angles 34. Critical Thinking A rectangle PQRS is inscribed in a circle. What can you conclude about ̶̶ PR ? Explain. 35. History The diagram shows the Winchester Round Table with inscribed △ABC. The table may have been made at the request of King Edward III, who created the Order of Garter as a return to the Round Table and an order of chivalry. ̶̶ a. Explain why BC must be a diameter of the circle. b. Find m ⁀ AC . 36. To inscribe an equilateral triangle in a circle, draw a ̶̶ BC . Open the compass to the radius of the circle. diameter Place the point of the compass at C and make arcs on the circle at D and E, as shown. Draw ̶̶ DE . Explain why △BDE is an equilateral triangle. ̶̶ BD , ̶̶ BE , and 37. Write About It A student claimed that if a parallelogram contains a 30° angle, it cannot be inscribed in a circle. Do you agree or disagree? Explain. 38. Construction Circumscribe a circle about a triangle. (Hint: Follow the steps for the construction of a circle through three given noncollinear points.) 39. What is m∠BAC? 38° 43° 66° 81° 40. Equilateral △XCZ is inscribed in a circle. ̶̶ CY bisects ∠C, what is m ⁀ XY ? If 15° 30° 60° 120° 41. Quadrilateral ABCD is inscribed in a circle. The ratio of m∠A to m∠C is 4 : 5. What is m∠A? 20° 40° 80° 100° 42. Which of these angles has the greatest measure? ∠STR ∠QPR ∠QSR ∠PQS 778 778 Chapter 11 Circles �� CHALLENGE AND EXTEND 43. Prove that an inscribed angle subtends a semicircle if and only if the angle is a right angle. (Hint: There are two parts.) 44. Prove that if a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. (Hint: There are two parts.) 45. Find m ⁀ PQ to the nearest degree. 46. Find m∠ABD. 47. Construction To circumscribe an equilateral triangle ̶̶ AB parallel to the horizontal about a circle, construct diameter of the circle and tangent to the circle. Then use a 30°-60°-90° triangle to draw 60° angles with ̶̶ AB and are tangent to the circle. ̶̶ BC so that they form ̶̶ AC and SPIRAL REVIEW 48. Tickets for a play cost $15.00 for section C, $22.50 for section B, and $30.00 for section A. Amy spent a total of $255.00 for 12 tickets. If she spent the same amount on section C tickets as section A tickets, how many tickets for section B did she purchase? (Previous course) Write a ratio expressing the slope of the line through each pair of points. (Lesson 7-1) 49. (4 1 _ , -6) and (8, 1 _ ) 2 2 50. (-9, -8) and (0, -2) 51. (3, -14) and (11, 6) Find each of the following. (Lesson 11-2) 52. m ⁀ ST 53. area of △ABD Construction Tangent to a Circle From an Exterior Point     Draw ⊙C and locate P in the exterior of the circle. ̶̶ CP . Construct M, Draw the midpoint of ̶̶ CP . 1. Can you draw ̶̶ CR ⊥   RP ? Explain. Center the compass at M. Draw a circle through C and P. It will intersect ⊙C at R and S. R and S are the tangent points. Draw   PR and    PS tangent to ⊙C. 11-4 Inscribed Angles 779 779 �� 11-5 Use with Lesson 11-5 Activity 1 Explore Angle Relationships in Circles In Lesson 11-4, you learned that the measure of an angle inscribed in a circle is half the measure of its intercepted arc. Now you will explore other angles formed by pairs of lines that intersect circles. TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures …. Also G.1.A, G.2.B, G.3.B, G.4.A, G.5.A, G.5.B, G.9.C 1 Create a circle with center A. Label the point on the circle as B. Create a radius segment from A to a new point C on the circle. 2 Construct a line through C perpendicular to ̶̶ radius AC . Create a new point D on this line, which is tangent to circle A at C. Hide radius ̶̶ AC . 3 Create a new point E on the circle and then ̶̶ CE . construct secant 4 Measure ∠DCE and measure ⁀ CBE . (Hint: To measure an arc in degrees, select the three points and the circle and then choose Arc Angle from the Measure menu.) 5 Drag E around the circle and examine the changes in the measures. Fill in the angle and arc measures in a chart like the one below. Try to create acute, right, and obtuse angles. Can you make a conjecture about the relationship between the angle measure and the arc measure? m∠DCE m ⁀ CBE Angle Type Activity 2 1 Construct a new circle with two secants   CD and   EF that intersect inside the circle at G. 2 Create two new points H and I that are on the circle as shown. These will be used to measure the arcs. Hide B if desired. (It controls the circle’s size.) 3 Measure ∠DGF formed by the secant lines and measure ⁀ CHE and ⁀ DIF . 4 Drag F around the circle and examine the changes in measures. Be sure to keep H between C and E and I between D and F for accurate arc measurement. Move them if needed. 780 780 Chapter 11 Circles 5 Fill in the angle and arc measures in a chart like the one below. Try to create acute, right, and obtuse angles. Can you make a conjecture about the relationship between the angle measure and the two arc measures? m∠DGF m ⁀ CHE m ⁀ DIF Sum of Arcs Activity 3 1 Use the same figure from Activity 2. Drag points around the circle so that the intersection G is now outside the circle. Move H so it is between E and D and I is between C and F, as shown. 2 Measure ∠FGC formed by the secant lines and measure ⁀ CIF and ⁀ DHE . 3 Drag points around the circle and examine the changes in measures. Fill in the angle and arc measures in a chart like the one below. Can you make a conjecture about the relationship between the angle measure and the two arc measures? m∠FGC m ⁀ CIF m ⁀ DHE Number of Arcs Try This 1. How does the relationship you observed in Activity 1 compare to the relationship between an inscribed angle and its intercepted arc? 2. Why do you think the radius ̶̶ AC is needed in Activity 1 for the construction of the tangent line? What theorem explains this? 3. In Activity 3, try dragging points so that the secants become tangents. What conclusion can you make about the angle and arc measures? 4. Examine the conjectures and theorems about the relationships between angles and arcs in a circle. What is true of an angle with a vertex on the circle? What is true of an angle with a vertex inside the circle? What is true of an angle with a vertex outside the circle? Summarize your findings. 5. Does using geometry software to compare angle and arc measures constitute a formal proof of the relationship observed? 11- 5 Technology Lab 781 781 11-5 Angle Relationships in Circles TEKS G.5.B Geometric paterns: use … patterns to make generalizations about geometric properties, including … angle relationships in … circles. Also G.1.A, G.2.B, G.5.A, G.9.C Objectives Find the measures of angles formed by lines that intersect circles. Use angle measures to solve problems. Who uses this? Circles and angles help optometrists correct vision problems. (See Example 4.) Theorem 11-5-1 connects arc measures and the measures of tangent-secant angles with tangent-chord angles. Theorem 11-5-1 THEOREM HYPOTHESIS CONCLUSION If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. m∠ABC = 1 _ m ⁀ AB 2 Tangent   BC and secant   BA intersect at B. You will prove Theorem 11-5-1 in Exercise 45. E X A M P L E 1 Using Tangent-Secant and Tangent-Chord Angles Find each measure. A m∠BCD m∠BCD = 1 _ m ⁀ BC 2 m∠BCD = 1 _ (142°) 2 = 71° B m ⁀ ABC m∠ACD = 1 _ m ⁀ ABC 2 90° = 1 _ m ⁀ ABC 2 180° = m ⁀ ABC Find each measure. 1a. m∠STU 1b. m ⁀ SR 782 782 Chapter 11 Circles �� Theorem 11-5-2 THEOREM HYPOTHESIS CONCLUSION If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs. (m ⁀ AB + m ⁀ CD ) m∠1 = 1 _ 2 ̶ AD and ̶ BC Chords intersect at E. PROOF PROOF Theorem 11-5-2 ̶ ̶ BC intersect at E. AD and Given: (m ⁀ AB + m ⁀ CD ) Prove: m∠1 = 1 _ 2 Proof: Statements Reasons 1. ̶ AD and ̶ BD . 2. Draw ̶ BC intersect at E. 3. m∠1 = m∠EDB + m∠EBD 4. m∠EDB = 1 __ m ⁀ AB , 2 m∠EBD = 1 __ m ⁀ CD 2 m ⁀ AB + 1 __ 5. m∠1 = 1 __ m ⁀ CD 2 2 (m ⁀ AB + m ⁀ CD ) 6. m∠1 = 1 __ 2 1. Given 2. Two pts. determine a line. 3. Ext. ∠ Thm. 4. Inscribed ∠ Thm. 5. Subst. 6. Distrib. Prop. E X A M P L E 2 Finding Angle Measures Inside a Circle Find each angle measure. m∠SQR (m ⁀ PT + m ⁀ SR ) m∠SQR = 1 _ 2 = 1 _ (32° + 100° ) 2 = 1 _ (132°) 2 = 66° Find each angle measure. 2a. m∠ABD 2b. m∠RNM 11-5 Angle Relationships in Circles 783 783 �� Theorem 11-5-3 If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs. (m ⁀ AD - m ⁀ BD ) m∠2 = 1 _ m∠1 = 1 _ 2 2 (m ⁀ EHG - m ⁀ EG ) m∠3 = 1 _ 2 (m ⁀ JN - m ⁀ KM ) You will prove Theorem 11-5-3 in Exercises 34–36. E X A M P L E 3 Finding Measures Using Tangents and Secants Find the value of x. A B EHG and ⁀ EG joined ⁀ together make a whole circle. So m ⁀ EHG = 360° - 132° = 228° (m ⁀RS - m ⁀QS ) (174° - 98°) x = 1_ 2 = 1_ 2 = 38° 3. Find the value of x. E X A M P L E 4 Biology Application (m ⁀EHG - m ⁀EG ) (228° - 132°) x =
1_ 2 = 1_ 2 = 48° When a person is farsighted, light rays enter the eye and are focused behind the retina. In the eye shown, light rays converge at R. If m ⁀PS = 60° and m ⁀QT = 14°, what is m∠PRS? (m ⁀PS - m ⁀QT ) m∠PRS = 1_ 2 = 1_ 2 = 1_ 2 (60° - 14° ) (46°) = 23° 4. Two of the six muscles that control eye movement are attached to the eyeball and intersect behind the eye. If m ⁀AEB = 225°, what is m∠ACB? 784 784 Chapter 11 Circles �� Angle Relationships in Circles VERTEX OF THE ANGLE MEASURE OF ANGLE On a circle Half the measure of its intercepted arc DIAGRAMS m∠1 = 60° m∠2 = 100° (44° + 86° ) m∠1 = 1 _ 2 = 65° Inside a circle Half the sum of the measures of its intercepted arcs Outside a circle Half the difference of the measures of its intercepted arcs (202° - 78°) m∠1 = 1_ 2 = 62° (125° - 45°) m∠2 = 1_ 2 = 40° E X A M P L E 5 Finding Arc Measures Find m ⁀ AF . Step 1 Find m ⁀ ADB . m∠ABC = 1 _ m ⁀ ADB 2 110° = 1 _ m ⁀ ADB 2 m ⁀ ADB = 220° Step 2 Find m ⁀ AD . If a tangent and secant intersect on a ⊙ at the pt. of tangency, then the measure of the ∠ formed is half the measure of its intercepted arc. Substitute 110 for m∠ABC. Mult. both sides by 2. ADB = m ⁀ AD + m ⁀ DB Arc Add. Post. m ⁀ 220° = m ⁀ AD + 160° m ⁀ AD = 60° Substitute. Subtract 160 from both sides. Step 3 Find m ⁀ AF . m ⁀ AF = 360° - (m ⁀ AD + m ⁀ DB + m ⁀ BF ) = 360° - (60° + 160° + 48° ) = 92° Def. of a ⊙ Substitute. Simplify. 5. Find m ⁀ LP . 11-5 Angle Relationships in Circles 785 785 �� THINK AND DISCUSS 1. Explain how the measure of an angle formed by two chords of a circle is related to the measure of the angle formed by two secants. 2. GET ORGANIZED Copy and complete the graphic organizer. In each box write a theorem and draw a diagram according to where the angle’s vertex is in relationship to the circle. 11-5 Exercises Exercises GUIDED PRACTICE Find each measure. p. 782 1. m∠DAB 2. m ⁀ AC 3. m ⁀ PN 4. m∠MNP KEYWORD: MG7 11-5 KEYWORD: MG7 Parent . m∠STU 6. m∠HFG 7. m∠NPK p. 783 Find the value of x. p. 784 8. 9. 10. 784 11. Science A satellite orbits Mars. When it reaches S it is about 12,000 km above the planet. How many arc degrees of the planet are visible to a camera in the satellite? 786 786 Chapter 11 Circles ��ABCSx°38°ge07se_c11l05004aAB�� . 785 Multi-Step Find each measure. 12. m ⁀ DF 13. m ⁀ CD 14. m ⁀ PN 15. m ⁀ KN PRACTICE AND PROBLEM SOLVING Find each measure. 16. m∠BCD 17. m∠ABC 18. m∠XZW 19. m ⁀ XZV Independent Practice For See Exercises Example 16–19 20–22 23–25 26 27–30 1 2 3 4 5 TEKS TEKS TAKS TAKS 20. m∠QPR 21. m∠ABC 22. m∠MKJ Skills Practice p. S25 Application Practice p. S38 Find the value of x. 23. 24. 25. Archaeology Outside of Hunt, Texas, is a replica of Stonehenge. It is 60 percent as tall as the original and 90 percent as large in circumference. 26. Archaeology Stonehenge is a circular arrangement of massive stones near Salisbury, England. A viewer at V observes the monument from a point where two of the stones A and B are aligned with stones at the endpoints of a diameter of the circular shape. Given that m ⁀ AB = 48°, what is m∠AVB? Multi-Step Find each measure. 27. m ⁀ EG 28. m ⁀ DE 29. m ⁀ PR 30. m ⁀ LP 11-5 Angle Relationships in Circles 787 787 �� In the diagram, m∠ABC = x °. Write an expression in terms of x for each of the following. 31. m ⁀ AB 33. m ⁀ AEB 32. m∠ABD 34. Given: Tangent CD and secant CA (m ⁀ AD - m ⁀ BD ) Prove: m∠ACD = 1 _ 2 Plan: Draw auxiliary line segment ̶ BD . Use the Exterior Angle Theorem to show that m∠ACD = m∠ABD - m∠BDC. Then use the Inscribed Angle Theorem and Theorem 11-5-1. 35. Given: Tangents  FG (m ⁀ Prove: m∠EFG = 1 _ 2  FE and EHG - m ⁀ EG ) 36. Given: Secants ̶ LN (m ⁀ JN - m ⁀ KM ) Prove: m∠JLN = 1 _ 2 ̶ LJ and 37. Critical Thinking Suppose two secants intersect in the exterior of a circle as shown. What is greater, m∠1 or m∠2? Justify your answer. 38. Write About It The diagrams show the intersection of perpendicular lines on a circle, inside a circle, and outside a circle. Explain how you can use these to help you remember how to calculate the measures of the angles formed. Algebra Find the measures of the three angles of △ABC. 39. 40. 41. This problem will prepare you for the Multi-Step TAKS Prep on page 806. The design was made by placing six equally-spaced points on a circle and connecting them. a. Find m∠BHC. b. Find m∠EGD. c. Classify △EGD by its angle measures and by its side lengths. 788 788 Chapter 11 Circles ��ABEDCFge07sec11l05007aABHG 42. What is m∠DCE? 19° 21° 79° 101° 43. Which expression can be used to calculate m∠ABC? 1 _ (m ⁀ DE - m ⁀ AF ) 2 1 _ (m ⁀ AD - m ⁀ AF ) 2 1 _ (m ⁀ AD + m ⁀ AF ) 2 1 _ (m ⁀ DE + m ⁀ AF ) 2 44. Gridded Response In ⊙Q, m ⁀ MN = 146° and m∠JLK = 45°. Find the degree measure of ⁀ JK . CHALLENGE AND EXTEND 45. Prove Theorem 11-5-1.  BC and secant Given: Tangent Prove: m∠ABC = 1 _ m ⁀ AB 2 (Hint: Consider two cases, one where  BA ̶ AB is a diameter and one where ̶ AB is not a diameter.) ̶ WZ are tangent to ⊙X. m ⁀ WY = 90° 46. Given: ̶ YZ and Prove: WXYZ is a square. 47. Find x. 48. Find m ⁀ GH . SPIRAL REVIEW Determine whether the ordered pair (7, -8) is a solution of the following functions. (Previous course) 49. g (x) = 2 x 2 - 15x - 1 50. f (x) = 29 - 3x 51. y = - 7 _ x 8 Find the volume of each pyramid or cone. Round to the nearest tenth. (Lesson 10-7) 52. regular hexagonal pyramid with a base edge of 4 m and a height of 7 m 53. right cone with a diameter of 12 cm and lateral area of 60π cm 2 54. regular square pyramid with a base edge of 24 in. and a surface area of 1200 in 2 In ⊙P, find each angle measure. (Lesson 11-4) 55. m∠BCA 56. m∠DBC 57. m∠ADC 11-5 Angle Relationships in Circles 789 789 �� 11-6 Use with Lesson 11-6 Activity 1 Explore Segment Relationships in Circles When secants, chords, or tangents of circles intersect, they create several segments. You will measure these segments and investigate their relationships. TEKS G.9.C Congruence and the geometry of size: formulate … conjectures about … circles and the lines that intersect them …. Also G.2.A, G.2.B, G.3.B, G.5.A KEYWORD: MG7 Lab11 1 Construct a circle with center A. Label the point on the circle as B. Construct two secants CD and   EF that intersect outside the circle at G. Hide B if desired. (It controls the circle’s size.) 2 Measure ̶̶ GC , ̶̶̶ GD , ̶̶ GE , and around the circle and examine the changes in the measurements. ̶̶ GF . Drag points 3 Fill in the segment lengths in a chart like the one below. Find the products of the lengths of segments on the same secant. Can you make a conjecture about the relationship of the segments formed by intersecting secants of a circle? GC GD GC ⋅ GD GE GF GE ⋅ GF Try This 1. Make a sketch of the diagram from Activity 1, ̶̶ DE to create △CFG and ̶̶ CF and and create △EDG as shown. 2. Name pairs of congruent angles in the diagram. How are △CFG and △EDG related? Explain your reasoning. 3. Write a proportion involving sides of the triangles. Cross-multiply and state the result. What do you notice? Activity 2 1 Construct a new circle with center A. Label the point on the circle as B. Create a radius segment from A to a new point C on the circle. 2 Construct a line through C perpendicular to radius point D on this line, which is tangent to circle A at C. Hide radius ̶̶ AC . Create a new ̶̶ AC . 790 790 Chapter 11 Circles 3 Create a secant line through D that intersects the circle at two new points E and F, as shown. 4 Measure ̶̶ DC , ̶̶ DE , and ̶̶ DF . Drag points around the circle and examine the changes in the measurements. Fill in the measurements in a chart like the one below. Can you make a conjecture about the relationship between the segments of a tangent and a secant of a circle? DE DF DE ⋅ DF DC ? Try This 4. How are the products for a tangent and a secant similar to the products for secant segments? 5. Try dragging E and F so they overlap (to make the secant segment look like a tangent segment). What do you notice about the segment lengths you measured in Activity 2? Can you state a relationship about two tangent segments from the same exterior point? 6. Challenge Write a formal proof of the relationship you found in Problem 2. Activity 3 1 Construct a new circle with two chords that intersect inside the circle at G. ̶̶ CD and ̶̶ EF 2 Measure ̶̶ GC , ̶̶̶ GD , ̶̶ GE , and the circle and examine the changes in the measurements. ̶̶ GF . Drag points around 3 Fill in the segment lengths in a chart like the ones used in Activities 1 and 2. Find the products of the lengths of segments on the same chord. Can you make a conjecture about the relationship of the segments formed by intersecting chords of a circle? Try This 7. Connect the endpoints of the chords to form two triangles. Name pairs of congruent angles. How are the two triangles that are formed related? Explain your reasoning. 8. Examine the conclusions you made in all three activities about segments formed by secants, chords, and tangents in a circle. Summarize your findings. 11- 6 Technology Lab 791 791 11-6 Segment Relationships in Circles TEKS G.5.A Geometric patterns: use numeric and geometric patterns to develop algebraic expressions representing geometric properties. Also G.1.A, G.2.B Objectives Find the lengths of segments formed by lines that intersect circles. Use the lengths of segments in circles to solve problems. Vocabulary se
cant segment external secant segment tangent segment Who uses this? Archaeologists use facts about segments in circles to help them understand ancient objects. (See Example 2.) In 1901, divers near the Greek island of Antikythera discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks. The following theorem describes the relationship among the four segments that are formed when two chords intersect in the interior of a circle. Theorem 11-6-1 Chord-Chord Product Theorem THEOREM HYPOTHESIS CONCLUSION If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal. AE ⋅ EB = CE ⋅ ED ̶̶ AB and ̶̶ CD Chords intersect at E. You will prove Theorem 11-6-1 in Exercise 28. E X A M P L E 1 Applying the Chord-Chord Product Theorem Find the value of x and the length of each chord. PQ ⋅ QR = SQ ⋅ QT 6 (4) = x ( 8) 24 = 8x 3 = x PR = 6 + 4 = 10 ST = 3 + 8 = 11 1. Find the value of x and the length of each chord. 792 792 Chapter 11 Circles �� E X A M P L E 2 Archaeology Application Archaeologists discovered a fragment of an ancient disk. To calculate its original diameter, they drew a chord ̶̶ PQ . Find the disk’s diameter. bisector ̶̶̶ PQ is the perpendicular bisector of ̶̶ PR is a diameter of the disk. ̶̶ AB and its perpendicular Since a chord, AQ ⋅ QB = PQ ⋅ QR 5 (5) = 3 (QR) 25 = 3QR 8 1 _ 3 in. = QR = 11 1 _ PR = 3 + 8 1 _ 3 3 in. 2. What if…? Suppose the length of chord ̶̶ AB that the archaeologists drew was 12 in. In this case how much longer is the disk’s diameter compared to the disk in Example 2? A secant segment is a segment of a secant with at least one endpoint on the circle. An external secant segment is a secant segment that lies in the exterior of the circle with one endpoint on the circle. ̶̶̶ NM , ̶̶̶ KM , and ̶̶ ̶̶̶ PM , JM are secant segments of ⊙Q. ̶̶ ̶̶̶ NM and JM are external secant segments. Theorem 11-6-2 Secant-Secant Product Theorem THEOREM HYPOTHESIS CONCLUSION If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (whole ⋅ outside = whole ⋅ outside) ̶̶ AE and ̶̶ CE Secants intersect at E. AE ⋅ BE = CE ⋅ DE PROOF PROOF Secant-Secant Product Theorem Given: Secant segments Prove: AE ⋅ BE = CE ⋅ DE ̶̶ AE and ̶̶ CE Proof: Draw auxiliary line segments ̶̶ AD and ̶̶ CB . ∠EAD and ∠ECB both intercept ⁀ BD , so ∠EAD ≅ ∠ECB. ∠E ≅ ∠E by the Reflexive Property of ≅. Thus △EAD ∼ △ECB by AA Similarity. Therefore corresponding sides are proportional, and AE ___ = DE ___ BE CE AE ⋅ BE = CE ⋅ DE. . By the Cross Products Property, 11-6 Segment Relationships in Circles 793 793 3 in.5 in.PBARQge07sec11l06002a�� E X A M P L E 3 Applying the Secant-Secant Product Theorem Find the value of x and the length of each secant segment. RT ⋅ RS = RQ ⋅ RP 10 (4) = (x + 5) 5 40 = 5x + 25 15 = 5x 3 = x RT = 4 + 6 = 10 RQ = 5 + 3 = 8 3. Find the value of z and the length of each secant segment. A tangent segment is a segment of a tangent with one endpoint on the circle. ̶̶ ̶̶ AC are tangent segments. AB and Theorem 11-6-3 Secant-Tangent Product Theorem THEOREM HYPOTHESIS CONCLUSION If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. (whole ⋅ outside = tangent 2 ) AC ⋅ BC = DC 2 ̶̶ AC and tangent ̶̶ DC Secant intersect at C. You will prove Theorem 11-6-3 in Exercise 29. E X A M P L E 4 Applying the Secant-Tangent Product Theorem Find the value of x. SQ ⋅ RQ = PQ 2 9 (4) = x 2 36 = x 2 ±6 = x The value of x must be 6 since it represents a length. 4. Find the value of y. 794 794 Chapter 11 Circles �� THINK AND DISCUSS 1. Does the Chord-Chord Product Theorem apply when both chords are diameters? If so, what does the theorem tell you in this case? 2. Given A in the exterior of a circle, how many different tangent segments can you draw with A as an endpoint? 3. GET ORGANIZED Copy and complete the graphic organizer. 11-6 Exercises Exercises KEYWORD: MG7 11-6 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary   AB intersects ⊙P at exactly one point. Point A is in the exterior ̶̶ AB is a(n) ? . (tangent segment or external ̶̶̶̶ of ⊙P, and point B lies on ⊙P. secant segment Find the value of the variable and the length of each chord. p. 792 2. 3. 4. Engineering A section of an aqueduct p. 793 is based on an arc of a circle as shown. ̶̶̶ ̶̶ EF is the perpendicular bisector of GH . GH = 50 ft, and EF = 20 ft. What is the diameter of the circle Find the value of the variable and the length of each secant segment. p. 794 6. 7. 8. 11-6 Segment Relationships in Circles 795 795 �� Find the value of the variable. p. 794 9. 10. 11. Independent Practice For See Exercises Example 12–14 15 16–18 19–21 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S25 Application Practice p. S38 PRACTICE AND PROBLEM SOLVING Find the value of the variable and the length of each chord. 12. 13. 14. 15. Geology Molokini is a small, crescent- shaped island 2 1 __ 2 miles from the Maui coast. It is all that remains of an extinct volcano. To approximate the diameter of the mouth of the volcano, a geologist can use a diagram like the one shown. What is the approximate diameter of the volcano’s mouth to the nearest foot? Find the value of the variable and the length of each secant segment. 16. 17. 18. Find the value of the variable. 19. 20. 21. Use the diagram for Exercises 22 and 23. 22. M is the midpoint of ̶̶ PQ . RM = 10 cm, and PQ = 24 cm. a. Find MS. b. Find the diameter of ⊙O. ̶̶ PQ .The diameter of ⊙O is 13 in., 23. M is the midpoint of and RM = 4 in. a. Find PM. b. Find PQ. 796 796 Chapter 11 Circles �� Multi-Step Find the value of both variables in each figure. 24. 25. 26. Meteorology A weather satellite S orbits Earth at a distance SE of 6000 mi. Given that the diameter of the earth is approximately 8000 mi, what is the distance from the satellite to P? Round to the nearest mile. 27. /////ERROR ANALYSIS///// The two solutions show how to find the value of x. Which solution is incorrect? Explain the error. Meteorology Satellites are launched to an area above the atmosphere where there is no friction. The idea is to position them so that when they fall back toward Earth, they fall at the same rate as Earth’s surface falls away from them. 28. Prove Theorem 11-6-1. ̶̶ AB and Given: Chords Prove: AE ⋅ EB = CE ⋅ ED ̶̶ CD intersect at point E. Plan: Draw auxiliary line segments ̶̶ AC and ̶̶ BD . Show that △ECA ∼ △EBD. Then write a proportion comparing the lengths of corresponding sides. 29. Prove Theorem 11-6-3. Given: Secant segment Prove: AC ⋅ BC = DC 2 ̶̶ AC , tangent segment ̶̶ DC 30. Critical Thinking A student drew a circle and two secant segments. By measuring with a ruler, he found ̶̶ PQ ≅ the student’s conclusion? Why or why not? ̶̶ PS . He concluded that ̶̶ ST . Do you agree with ̶̶ QR ≅ 31. Write About It The radius of ⊙A is 4. CD = 4, ̶̶ CB is a tangent segment. Describe two different and methods you can use to find BC. 32. This problem will prepare you for the Multi-Step TAKS Prep on page 806. Some Native American designs are based on eight points that are placed around the circumference of a circle. In ⊙O, BE = 3 cm. AE = 5.2 cm, and EC = 4 cm. a. Find DE to the nearest tenth. b. What is the diameter of the circle to the nearest tenth? c. What is the length of ̶̶ OE to the nearest hundredth? 11-6 Segment Relationships in Circles 797 797 ��ge07se_c11l06005aABSEP�� 33. Which of these is closest to the length of tangent ̶̶ PQ ? 6.9 9.2 9.9 10.6 34. What is the length of ̶̶ UT ? 5 7 12 14 35. Short Response In ⊙A, ̶̶ AB is the perpendicular ̶̶ CD . CD = 12, and EB = 3. Find the radius bisector of of ⊙A. Explain your steps. CHALLENGE AND EXTEND 36. Algebra ̶̶ KL is a tangent segment of ⊙N. 37. a. Find the value of x. b. Classify △KLM by its angle measures. Explain. ̶̶ PQ is a tangent segment of a circle with radius 4 in. Q lies on the circle, and PQ = 6 in. Find the distance from P to the circle. Round to the nearest tenth of an inch. 38. The circle in the diagram has radius c. Use this diagram and the Chord-Chord Product Theorem to prove the Pythagorean Theorem. 39. Find the value of y to the nearest hundredth. SPIRAL REVIEW 40. An experiment was conducted to find the probability of rolling two threes in a row on a number cube. The probability was 3.5%. How many trials were performed in this experiment if 14 favorable outcomes occurred? (Previous course) 41. Two coins were flipped together 50 times. In 36 of the flips, at least one coin landed heads up. Based on this experiment, what is the experimental probability that at least one coin will land heads up when two coins are flipped? (Previous course) Name each of the following. (Lesson 1-1) 42. two rays that do not intersect 43. the intersection of  AC and  CD 44. the intersection of  CA and  BD Find each measure. Give your answer in terms of π and rounded to the nearest hundredth. (Lesson 11-3) 45. area of the sector XZW 46. arc length of ⁀ XW 47. m∠YZX if the area of the sector YZW is 40π ft 2 798 798 Chapter 11 Circles �� 11-7 Circles in the Coordinate Plane TEKS G.2.B Geomet
ric structure: make conjectures about … circles … choosing from a variety of approaches such as coordinate …. Also G.1.A, G.4.A, G.5.A Objectives Write equations and graph circles in the coordinate plane. Use the equation and graph of a circle to solve problems. Who uses this? Meteorologists use circles and coordinates to plan the location of weather stations. (See Example 3.) The equation of a circle is based on the Distance Formula and the fact that all points on a circle are equidistant from the center = √  ( = √  (x - h ) 2 + (y - k ) 2 r 2 = (x - h) 2 + (y - k) 2 Distance Formula Substitute the given values. Square both sides. Theorem 11-7-1 Equation of a Circle The equation of a circle with center (h, k) and radius r is (x - h) 2 + (y - k Writing the Equation of a Circle Write the equation of each circle. A ⊙A with center A (4, -2) and radius 3 (x - 4) 2 + (y - (-2) ) (x - h) 2 + (y - kx - 4) 2 + (y + 2) 2 = 9 Equation of a circle Substitute 4 for h, -2 for k, and 3 for r. Simplify. B ⊙B that passes through (-2, 6) and has center B (-6, 3) r =  2 √ ( -2 - (-6) ) + (6 - 3) 2 Distance Formula = √  25 = 5 2 (x - (-6) ) Simplify. + (y - 3) 2 = 5 2 (x + 6) 2 + (y - 3) 2 = 25 Simplify. Substitute -6 for h, 3 for k, and 5 for r. Write the equation of each circle. 1a. ⊙P with center P (0, -3) and radius 8 1b. ⊙Q that passes through (2, 3) and has center Q (2, -1) 11-7 Circles in the Coordinate Plane 799 799 �� If you are given the equation of a circle, you can graph the circle by making a table or by identifying its center and radius. E X A M P L E 2 Graphing a Circle Graph each equation. x 2 + y 2 = 25 Step 1 Make a table of values. A Since the radius is √  25 , or 5, use ±5 and the values between for x-values. x y -5 -4 -3 0 3 4 0 ±3 ±4 ±5 ±4 ±3 5 0 Step 2 Plot the points and connect them to form a circle. Always compare the equation to the form (x - h) 2 + (y - k) 2 = r 2 . B (x + 1) 2 + (y - 2) 2 = 9 The equation of the given circle can be written as (x - (-1) ) + (y - 2) 2 = 3 2 . So h = -1, k = 2, and r = 3. The center is (-1, 2) , and the radius is 3. Plot the point (-1, 2) . Then graph a circle having this center and radius 3. 2 Graph each equation. 2a. x 2 + y 2 = 9 2b. (x - 3) 2 + (y + 2) 2 = 4 Graphing Circles I found a way to use my calculator to graph circles. You first need to write the circle’s equation in y = form. For example, to graph x 2 + y 2 = 16, first solve for y. y 2 = 16 - x 2 y = ± √  16 - x 2 Christina Avila Crockett High School Now enter and graph the two equations y 1 = √  16 - x 2 and y 2 = - √  16 - x 2 . 800 800 Chapter 11 Circles �� E X A M P L E 3 Meteorology Application Meteorologists are planning the location of a new weather station to cover Osceola, Waco, and Ireland, Texas. To optimize radar coverage, the station must be equidistant from the three cities which are located on a coordinate plane at A (2, 5) , B (3, -2) , and C (-5, -2) . a. What are the coordinates where the station should be built? b. If each unit of the coordinate plane represents 8.5 miles, what is the diameter of the region covered by the radar? The perpendicular bisectors of a triangle are concurrent at a point equidistant from each vertex. Step 1 Plot the three given points. Step 2 Connect A, B, and C to form a triangle. Step 3 Find a point that is equidistant from the three points by constructing the perpendicular bisectors of two of the sides of △ABC. The perpendicular bisectors of the sides of △ABC intersect at a point that is equidistant from A, B, and C. The intersection of the perpendicular bisectors is P (-1, 1) . P is the center of the circle that passes through A, B, and C. The weather station should be built at P (-1, 1) , Clifton, Texas. There are approximately 10 units across the circle. So the diameter of the region covered by the radar is approximately 85 miles. 3. What if…? Suppose the coordinates of the three cities in Example 3 are D (6, 2) , E (5, -5) , and F (-2, -4) . What would be the location of the weather station? THINK AND DISCUSS 1. What is the equation of a circle with radius r whose center is at the origin? 2. A circle has a diameter with endpoints (1, 4) and (-3, 4) . Explain how you can find the equation of the circle. 3. Can a circle have a radius of -6? Justify your answer. 4. GET ORGANIZED Copy and complete the graphic organizer. First select values for a center and radius. Then use the center and radius you wrote to fill in the other circles. Write the corresponding equation and draw the corresponding graph. 11-7 Circles in the Coordinate Plane 801 801 1800200020001800xyBCAWacoIrelandOsceolaxyBCAWacoIrelandOsceolaHolt, Rinehart & WinstonGeometry © 2007ge07sec11107003a Texas Grid map3rd proof1800200020001800xyPBCAWacoIrelandOsceolaCliftonxyPBCAWacoIrelandOsceolaCliftonHolt, Rinehart & WinstonGeometry © 2007ge07sec11107004a Texas Grid map3rd proof�� 11-7 Exercises Exercises GUIDED PRACTICE Write the equation of each circle. p. 799 1. ⊙A with center A (3, -5) and radius 12 2. ⊙B with center B (-4, 0) and radius 7 KEYWORD: MG7 11-7 KEYWORD: MG7 Parent 3. ⊙M that passes through (2, 0) and that has center M (4, 0) 4. ⊙N that passes through (2, -2) and that has center N (-1, 2 Multi-Step Graph each equation. p. 800 5. (x - 3) 2 + (y - 3) 2 = 4 7. (x + 3) 2 + (y + 4) 2 = 1 6. (x - 1) 2 + (y + 2) 2 = 9 8. (x - 3) 2 + (y + 4) 2 = 16 . 801 9. Communications A radio antenna tower is kept perpendicular to the ground by three wires of equal length. The wires touch the ground at three points on a circle whose center is at the base of the tower. The wires touch the ground at A (2, 6) , B (-2, -2) , and C (-5, 7) . a. What are the coordinates of the base of the tower? b. Each unit of the coordinate plane represents 1 ft. What is the diameter of the circle? PRACTICE AND PROBLEM SOLVING Independent Practice Write the equation of each circle. For See Exercises Example 10. ⊙R with center R (-12, -10) and radius 8 10–13 14–17 18 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S25 Application Practice p. S38 11. ⊙S with center S (1.5, -2.5) and radius √  3 12. ⊙C that passes through (2, 2) and that has center C (1, 1) 13. ⊙D that passes through (-5, 1) and that has center D (1, -2) 13. 15. (x + 1) 2 - y 2 = 16 17. x 2 + (y + 2) 2 = 4 Multi-Step Graph each equation. 14. x 2 + (y - 2) 2 = 9 16. x 2 + y 2 = 100 18. Anthropology Hundreds of stone circles can be found along the Gambia River in western Africa. The stones are believed to be over 1000 years old. In one of the circles at Ker Batch, three stones have approximate coordinates of A (3, 1) , B (4, -2) , and C (-6, -2) . a. What are the coordinates of the center of the stone circle? b. Each unit of the coordinate plane represents 1 ft. What is the diameter of the stone circle? 802 802 Chapter 11 Circles Entertainment The wooden carousel at Fair Park in Dallas, Texas, was manufactured by Gustav Dentzel and his family around 1914, It has 50 elaborately carved jumping horses, 16 standing horses, and 2 chariots. Algebra Write the equation of each circle. 19. 20. 21. Entertainment In 2004, the world’s largest carousel was located at the House on the Rock, in Spring Green, Wisconsin. Suppose that the center of the carousel is at the origin and that one of the animals on the circumference of the carousel has coordinates (24, 32) . a. If one unit of the coordinate plane equals 1 ft, what is the diameter of the carousel? b. As the carousel turns, the animals follow a circular path. Write the equation of this circle. Determine whether each statement is true or false. If false, explain why. 22. The circle x 2 + y 2 = 7 has radius 7. 23. The circle (x - 2) 2 + (y + 3) 2 = 9 passes through the point (-1, -3) . 24. The center of the circle (x - 6)2 + (y + 4)2 = 1 lies in the second quadrant. 25. The circle (x + 1) 2 + (y - 4) 2 = 4 intersects the y-axis. 26. The equation of the circle centered at the origin with diameter 6 is x 2 + y 2 = 36. 27. Estimation You can use the graph of a circle to estimate its area. a. Estimate the area of the circle by counting the number of squares of the coordinate plane contained in its interior. Be sure to count partial squares. b. Find the radius of the circle. Then use the area formula to calculate the circle’s area to the nearest tenth. c. Was your estimate in part a an overestimate or an underestimate? 28. Consider the circle whose equation is (x - 4) 2 + (y + 6) 2 = 25. Write, in point-slope form, the equation of the line tangent to the circle at (1, -10) . 29. This problem will prepare you for the Multi-Step TAKS Prep on page 806. A hogan is a traditional Navajo home. An artist is using a coordinate plane to draw the symbol for a hogan. The symbol is based on eight equally spaced points placed around the circumference of a circle. a. She positions the symbol at A (-3, 5) and C (0, 2) . What are the coordinates of E and G? b. What is the length of a diameter of the symbol? c. Use your answer from part b to write an equation of the circle. 11-7 Circles in the Coordinate Plane 803 803 �� Geology Geology The New Madrid earthquake of 1811 was the largest earthquake known in American history. Large areas sank into the earth, new lakes were formed, forests were destroyed, and the course of the Mississippi River was changed. Find the center and radius of each circle. 30. (x - 2)2 + (y + 3)2 = 81 31. x 2 + (y + 15)2 = 25 32. (x + 1)2 + y 2 = 7 Find the area and circumference of each circle. Express your answer in terms of π. Find the area and circumference of each circle. Express your answer in terms of 33. circle with equation (x + 2) 2 + (y - 7) 2 = 9 33. 34. circle with equation (x - 8)2 + (y + 5)2 = 7 34. 34. 35. circle with center (-1, 3) that passes through (2, -1) 36. Critical Thinking Describe the graph of the equation x 2 + y 2 = r 2 when r = 0. 37. Geology A se
ismograph measures ground motion during an earthquake. To find the epicenter of an earthquake, scientists take readings in three different locations. Then they draw a circle centered at each location. The radius of each circle is the distance the earthquake is from the seismograph. The intersection of the circles is the epicenter. Use the data below to find the epicenter of the New Madrid earthquake. Seismograph Location Distance to Earthquake A B C (-200, 200) (400, -100) (100, -500) 300 mi 600 mi 500 mi 38. For what value(s) of the constant k is the circle x 2 + (y - k) 2 = 25 tangent to the x-axis? 39. ⊙A has a diameter with endpoints (-3, -2) and (5, -2) . Write the equation of ⊙A. 40. Recall that a locus is the set of points that satisfy a given condition. Draw and describe the locus of points that are 3 units from (2, 2) . 41. Write About It The equation of ⊙P is (x - 2) 2 + (y - 1) 2 = 9. Without graphing, explain how you can determine whether the point (3, -1) lies on ⊙P, in the interior of ⊙P, or in the exterior of ⊙P. 42. Which of these circles intersects the x-axis? (x - 3) 2 + (y + 3) 2 = 4 (x + 1) 2 + (y - 4) 2 = 9 (x + 2) 2 + (y + 1) 2 = 1 (x + 1) 2 + (y + 4) 2 = 9 43. What is the equation of a circle with center (-3, 5) that passes through the point (1, 5) ? (x + 3) 2 + (y - 5) 2 = 4 (x - 3) 2 + (y + 5) 2 = 4 (x + 3) 2 + (y - 5) 2 = 16 (x - 3) 2 + (y + 5) 2 = 16 44. On a map of a park, statues are located at (4, -2) , (-1, 3) , and (-5, -5) . A circular path connects the three statues, and the circle has a fountain at its center. Find the coordinates of the fountain. (-1, -2) (2, 1) (-2, 1) (1, -2) 804 804 Chapter 11 Circles CharlestonDetroitMinneapolisEpicenterHolt, Rinehart & WinstonGeometry © 2007ge07sec11107009a Epicenter map2nd proof CHALLENGE AND EXTEND 45. In three dimensions, the equation of a sphere is similar to that of a circle. The equation of a sphere with center (h, j, k) and radius r is (x - h) 2 + (y - j) 2 + (z - k) 2 = r 2 . a. Write the equation of a sphere with center (2, -4, 3) that contains the point (1, -2, -5) . b.   AC and   BC are tangents from the same exterior point. If AC = 15 m, what is BC? Explain. 46. Algebra Find the point(s) of intersection of the line x + y = 5 and the circle x 2 + y 2 = 25 by solving the system of equations. Check your result by graphing the line and the circle. 47. Find the equation of the circle with center (3, 4) that is tangent to the line whose equation is y = 2x + 3. (Hint: First find the point of tangency.) SPIRAL REVIEW Simplify each expression. (Previous course) 18a + 4 (9a + 3) __ 6 2 x 2 - 2 (4 x 2 + 1) __ 2 49. 48. 50. 3 (x + 3y) - 4 (3x + 2y) - (x - 2y) In isosceles △DEF, EF = 4y - 1. Find the value of each variable. (Lesson 4-8) ̶̶ DE ≅ ̶̶ EF . m∠E = 60°, and m∠D = (7x + 4) °. DE = 2y + 10, and 51. x 52. y Find each measure. (Lesson 11-5) 53. m ⁀ LNQ 54. m∠NMP KEYWORD: MG7 Career Q: What math classes did you take in high school? A: I took Algebra 1 and Geometry. I also took Drafting and Woodworking. Those classes aren’t considered math classes, but for me they were since math was used in them. Q: What type of furniture do you make? A: I mainly design and make household furniture, such as end tables, bedroom furniture, and entertainment centers. Q: How do you use math? A: Taking appropriate and precise measurements is very important. If wood is not measured correctly, the end result doesn’t turn out as expected. Understanding angle measures is also important. Some of the furniture I build has 30° or 40° angles at the edges. Q: What are your future plans? A: Someday I would love to design all the furniture in my own home. It would be incredibly satisfying to know that all my furniture was made with quality and attention to detail. 11-7 Circles in the Coordinate Plane 805 805 Bryan Moreno Furniture Maker �� SECTION 11B Angles and Segments in Circles Native American Design The members of a Native American cultural center are painting a circle of colors on their gallery floor. They start by laying out the circle and chords shown. Before they apply their paint to the design, they measure angles and lengths to check for accuracy. 1. The circle design is based on twelve equally spaced points placed around the circumference of the circle. As the group lays out the design, what should be m∠AGB? 2. What should be m∠KAE? Why? 3. What should be m∠KMJ ? Why? 4. The diameter of the circle is 22 ft. KM ≈ 4.8 ft, and JM ≈ 6.4 ft. What should be the length of ̶̶̶ MB ? 5. The group members use a coordinate plane to help them position the design. Each square of a grid represents one square foot, and the center of the circle is at (20, 14) . What is the equation of the circle? 6. What are the coordinates of points L, C, F, and I? 806 806 Chapter 11 Circles FEGDHCIBJAKMLge07sec11ac2003a SECTION 11B Quiz for Lessons 11-4 Through 11-7 11-4 Inscribed Angles Find each measure. 1. m∠BAC 2. m ⁀ CD 3. m∠FGH 4. m ⁀ JFG 11-5 Angle Relationships in Circles Find each measure. 5. m∠ RST 6. m∠AEC 7. A manufacturing company is creating a plastic stand for DVDs. They want to make the stand with m ⁀ MN = 102°. What should be the measure of ∠MPN? 11-6 Segment Relationships in Circles Find the value of the variable and the length of each chord or secant segment. 8. 9. 10. An archaeologist discovers a portion of a circular stone wall, shown by ⁀ ST in the figure. ST = 12.2 m, and UR = 3.9 m. What was the diameter of the original circular wall? Round to the nearest hundredth. 11-7 Circles in the Coordinate Plane Write the equation of each circle. 11. ⊙A with center A (-2, -3) and radius 3 12. ⊙B that passes through (1, 1) and that has center B (4, 5) 13. A television station serves residents of three cities located at J (5, 2) , K (-7, 2) , and L (-5, -8) . The station wants to build a new broadcast facility that is equidistant from the three cities. What are the coordinates of the location where the facility should be built? Ready to Go On? 807 807 �� EXTENSION EXTENSION Polar Coordinates TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. Objectives Convert between polar and rectangular coordinates. Plot points using polar coordinates. Vocabulary polar coordinate system pole polar axis In a Cartesian coordinate system, a point is represented by the two coordinates x and y. In a polar coordinate system , a point A is represented by its distance from the origin r, and an angle θ. θ is measured counterclockwise from  OA . The ordered pair (r, θ ) the horizontal axis to represents the polar coordinates of point A. In a polar coordinate system, the origin is called the pole . The horizontal axis is called the polar axis . y You can use the equation of a circle r 2 = x 2 + y 2 and the tangent ratio θ = convert rectangular coordinates to polar coordinates. __ x to E X A M P L E 1 Converting Rectangular Coordinates to Polar Coordinates Convert (3, 4) to polar coordinates = 25 r = 5 tan θ = 4 _ 3 θ = tan -1(4_ 3) ≈ 53° The polar coordinates are (5, 53°) . 1. Convert (4, 1) to polar coordinates. You can use the relationships x = r cos θ and y = r sin θ to convert polar coordinates to rectangular coordinates. E X A M P L E 2 Converting Polar Coordinates to Rectangular Coordinates Convert (2, 130°) to rectangular coordinates. x = r cos θ x = 2 cos 130° ≈ -1.29 y = r sinθ y = 2 sin 130° ≈ 1.53 The rectangular coordinates are (-1.29, 1.53) . 2. Convert (4, 60°) to rectangular coordinates. 808 808 Chapter 11 Circles �� E X A M P L E 3 Plotting Polar Coordinates Plot the point (4, 225°) . Step 1 Measure 225° counterclockwise from the polar axis. Step 2 Locate the point on the ray that is 4 units from the pole. 3. Plot the point (4, 300°) . E X A M P L E 4 Graphing Polar Equations Graph r = 4. Make a table of values and plot the points. θ r 0° 4 45° 135° 270° 300° 4 4 4 4 4. Graph r = 2. EXTENSION Exercises Exercises Convert to polar coordinates. 1. (2, 2) 2. (1, 0) Convert to rectangular coordinates. 6. (5, 214°) 5. (3, 150°) 3. (3, 7) 4. (0, 15) 7. (4, 303°) 8. (4.5, 90°) Plot each point. 9. (4, 45°) 10. (3, 165°) 11. (1, 240°) 12. (3.5, 315°) 13. Critical Thinking Graph the equation r = 5. What can you say about the graph of an equation of the form r = a, where a is a positive real number? Technology Graph each equation. 14. r = -5 sin θ 17. r = 5 cos 3θ 15. r = 3 sin 4θ 18. r = 3 cos 2θ 16. r = -4 cos θ 19. r = 2 + 4 sin θ Chapter 11 Extension 809 809 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary adjacent arcs . . . . . . . . . . . . . . . 757 exterior of a circle . . . . . . . . . . . 746 secant segment . . . . . . . . . . . . . 793 arc . . . . . . . . . . . . . . . . . . . . . . . . . 756 external secant segment . . . . . 793 sector of a circle . . . . . . . . . . . . 764 arc length . . . . . . . . . . . . . . . . . . 766 inscribed angle . . . . . . . . . . . . . 772 segment of a circle . . . . . . . . . . 765 central angle . . . . . . . . . . . . . . . 756 intercepted arc . . . . . . . . . . . . . 772 semicircle . . . . . . . . . . . . . . . . . . 756 chord . . . . . . . . . . . . . . . . . . . . . . 746 interior of a circle . . . . . . . . . . . 746 subtend . . . . . . . . . . . . . . . . . . . . 772 common tangent . . . . . . . . . . . 748 major arc . . . . . . . . . . . . . . . . . . 756 tangent of a circle . . . . . . . . . . . 746 concentric circles . . . . . . . . . . . 747 minor arc . . . . . . . . . . . . . . . . . . 756 tangent circles . . . . . . . . . . . . . . 747 congruent arcs . . . . . . . . . . . . . 757 point of tangency . . . . . . . . . . . 746 tangent segment . . . . . . . . . . . . 79
4 congruent circles . . . . . . . . . . . 747 secant . . . . . . . . . . . . . . . . . . . . . 746 Complete the sentences below with vocabulary words from the list above. 1. A(n) ? is a region bounded by an arc and a chord. ̶̶̶̶ 2. An angle whose vertex is at the center of a circle is called a(n) ? . ̶̶̶̶ 3. The measure of a(n) ? is 360° minus the measure of its central angle. ̶̶̶̶ ? are coplanar circles with the same center. ̶̶̶̶ 4. 11-1 Lines That Intersect Circles (pp. 746–754) TEKS G.1.A, G.2.A, G.2.B, G.9.C E X A M P L E S ■ Identify each line or segment that intersects ⊙A. chord: ̶̶ DE tangent:   BC ̶̶ ̶̶ AD , and AE , radii: ̶̶ AB secant:   DE diameter: ̶̶ DE ̶̶ RW are tangent to ⊙T. RS = x + 5 and ■ ̶̶ RS and RW = 3x - 7. Find RS. RS = RW 2 segs. tangent to ⊙ from same ext. pt. → segs. ≅. Substitute the given values. Subtract 3x from both sides. Subtract 5 from both sides. Divide both sides by -2. Substitute 6 for y. Simplify. x + 5 = 3x - 7 -2x + 5 = -7 -2x = -12 x = 6 RS = 6 + 5 = 11 810 810 Chapter 11 Circles EXERCISES Identify each line or segment that intersects each circle. 5. 6. Given the measures of the following segments that are tangent to a circle, find each length. 7. AB = 9x - 2 and BC = 7x + 4. Find AB. 8. EF = 5y + 32 and EG = 8 - y. Find EG. 9. JK = 8m - 5 and JL = 2m + 4. Find JK. 10. WX = 0.8x + 1.2 and WY = 2.4x. Find WY. �� 11-2 Arcs and Chords (pp. 756–763) TEKS G.1.A, G.2.A, G.2.B, G.8.C, G.9.C E X A M P L E S Find each measure. ■ m ⁀ BF ∠BAF and ∠FAE are supplementary, so m∠BAF = 180° - 62° = 118°. m ⁀ BF = m∠BAF = 118° ■ m ⁀ DF Since m∠DAE = 90°, m ⁀ DE = 90°. m∠EAF = 62°, so m ⁀ EF = 62°. By the Arc Addition Postulate, m ⁀ DF = m ⁀ DE + m ⁀ EF = 90° + 62° = 152°. EXERCISES Find each measure. 11. m ⁀ KM 12. m ⁀ HMK 13. m ⁀ JK 14. m ⁀ MJK Find each length to the nearest tenth. 15. ST 16. CD 11-3 Sector Area and Arc Length (pp. 764–769) TEKS G.1.A, G.1.B, G.8.B, G.9.C E X A M P L E S ■ Find the area of sector PQR. Give your answer in terms of π and rounded to the nearest hundredth. EXERCISES Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. 17. sector DEF 18. sector JKL 360° A = π r 2 ( m° _ ) = π (4) 2 ( 135° _ ) = 16π ( 3 _ ) = 6π m 2 360 8 ≈ 18.85 m 2 ■ Find the length of ⁀ AB . Give your answer in terms of π and rounded to the nearest hundredth. 360° L = 2πr ( m° _ ) = 2π (9) ( 80° _ ) = 18π ( 4 _ ) 360° 9 Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. 19. ⁀ GH 20. ⁀ MNP = 8π ft ≈ 25.13 ft Study Guide: Review 811 811 �� 11-4 Inscribed Angles (pp. 772–779) TEKS G.1.A, G.2.A, G.2.B, G.5.B, G.9.C E X A M P L E S Find each measure. ■ m∠ABD By the Inscribed Angle Theorem, m∠ABD = 1 __ 2 m ⁀ AD , so m∠ABD = 1 __ 2 (108°) = 54°. ■ m ⁀ BE By the Inscribed Angle Theorem, m∠BAE = 1 __ 2 m ⁀ BE . So 28° = 1 __ 2 m ⁀ BE , and m ⁀ BE = 2 (28°) = 56°. EXERCISES Find each measure. 21. m ⁀JL 22. m∠MKL Find each value. 23. x 24. m∠RSP 11-5 Angle Relationships in Circles (pp. 782–789) TEKS G.1.A, G.2.B, G.5.A, G.5.B, G.9.C E X A M P L E S Find each measure. ■ m∠UWX m∠UWX = 1 _ m ⁀ UW 2 = 1 _ (160°) 2 = 80° EXERCISES Find each measure. 25. m ⁀ MR 26. m∠QMR ■ m ⁀ VW Since m∠UWX = 80°, m∠UWY = 100° and m∠VWY = 50°. m∠VWY = 1 __ 2 m ⁀ VW . So 50° = 1 __ 2 m ⁀ VW , and m ⁀ VW = 2 (50°) = 100°. 27. m∠GKH ■ m∠AED (m ⁀ AD + m ⁀ BC ) m∠AED = 1 _ 2 = 1 _ (31° + 87°) 2 = 1 _ (118°) 2 = 59° 812 812 Chapter 11 Circles 28. A piece of string art is made by placing 16 evenly spaced nails around the circumference of a circle. A piece of string is wound from A to B to C to D. What is m∠BXC? �� 11-6 Segment Relationships in Circles (pp. 792–798) TEKS G.1.A, G.2.B, G.5.A EXERCISES Find the value of the variable and the length of each chord. 29. 30. Find the value of the variable and the length of each secant segment. 31. 32. E X A M P L E S ■ Find the value of x and the length of each chord. AE ⋅ EB = DE ⋅ EC 12x = 8 (6) 12x = 48 x = 4 AB = 12 + 4 = 16 DC = 8 + 6 = 14 ■ Find the value of x and the length of each secant segment. FJ ⋅ FG = FK ⋅ FH 16 (4) = (6 + x) 6 64 = 36 + 6x 28 = 6x x = 4 2 _ 3 FJ = 12 + 4 = 16 FK = 4 2 _ + 6 = 10 2 _ 3 3 11-7 Circles in the Coordinate Plane (pp. 799–805) TEKS G.1.A, G.2.B, G.4.A, G.5.A E X A M P L E S EXERCISES ■ Write the equation of ⊙A that passes through (-1, 1) and that has center A (2, 3) . The equation of a circle with center (h, k) and radius r is (x - h)2 + (y - k)2 = r 2 .  2 r = √ (2 - (-1) ) + (3 - 1) 2 = √  3 2 + 2 2 = √  13 The equation of ⊙A is (x - 2) 2 + (y - 3) 2 = 13. ■ Graph (x - 2) 2 + (y + 1) 2 = 4. The center of the circle is (2, -1) , and the radius is √  4 = 2. Write the equation of each circle. 33. ⊙A with center (-4, -3) and radius 3 34. ⊙B that passes through (-2, -2) and that has center B (-2, 0) 35. ⊙C 36. Graph (x + 2)2 + (y - 2)2 = 1. Study Guide: Review 813 813 �� 1. Identify each line or segment that intersects the circle. 2. A jet is at a cruising altitude of 6.25 mi. To the nearest mile, what is the distance from the jet to a point on Earth’s horizon? (Hint: The radius of Earth is 4000 mi.) Find each measure. 3. m ⁀ JK 4. UV 5. Find the area of the sector. Give your answer in terms of π and rounded to the nearest hundredth. 6. Find the length of ⁀ BC . Give your answer in terms of π and rounded to the nearest hundredth. 7. If m∠SPR = 47° in the diagram of a logo, find m ⁀ SR . 8. A printer is making a large version of the logo for a banner. According to the specifications, m ⁀ PQ = 58°. What should the measure of ∠QTR be? Find each measure. 9. m∠ABC 10. m∠NKL 11. A surveyor S is studying the positions of four columns A, B, C, and D that lie on a circle. He finds that m∠CSD = 42° and m ⁀ CD = 124°. What is m ⁀ AB ? Find the value of the variable and the length of each chord or secant segment. 12. 13. 14. The illustration shows a fragment of a circular plate. AB = 8 in., and CD = 2 in. What is the diameter of the plate? 15. Write the equation of the circle that passes through (-2, 4) and that has center (1, -2) . � � � 16. An artist uses a coordinate plane to plan a mural. The mural will include portraits of civic leaders at X (2, 4) , Y (-6, 0) , and Z (2, -8) and a circle that passes through all three portraits. What are the coordinates of the center of the circle? � � � 814 814 Chapter 11 Circles �� FOCUS ON SAT MATHEMATICS SUBJECT TESTS The topics covered on the SAT Mathematics Subject Tests vary only slightly each time the test is administered. You can find out the general distribution of questions across topics, then determine which areas need more of your attention when you are studying for the test. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. To prepare for the SAT Mathematics Subject Tests, start reviewing course material a couple of months before your test date. Take sample tests to find the areas you might need to focus on more. Remember that you are not expected to have studied all topics on the test. 1. ̶̶ ̶̶ BD intersect at the center of the circle AC and shown. If m∠BDC = 30°, what is the measure of minor ⁀ AB ? 4. Circle D has radius 6, and m∠ABC = 25°. What is the length of minor ⁀ AC ? (A) 15° (B) 30° (C) 60° (D) 105° (E) 120° Note: Figure not drawn to scale. 2. Which of these is the equation of a circle that is tangent to the lines x = 1 and y = 3 and has radius 2? (A) (x + 1) 2 + (y - 1) 2 = 4 (B) (x - 1) 2 + (y + 1) 2 = 4 (C) x 2 + (y - 1) 2 = 4 (D) (x - 1) 2 + y 2 = 4 (E) x 2 + y 2 = 4 3. If LK = 6, LN = 10, and PK = 3, what is PM? (A) 7 (B) 8 (C) 9 (D) 10 (E) 11 Note: Figure not drawn to scale. (A) 5π _ 6 (B) 5π _ 4 (C) 5π _ 3 (D) 3π (E) 5π 5. A square is inscribed in a circle as shown. If the radius of the circle is 9, what is the area of the shaded region, rounded to the nearest hundredth? (A) 11.56 (B) 23.12 (C) 57.84 (D) 104.12 (E) 156.23 College Entrance Exam Practice 815 815 �� Multiple Choice: Recognize Distracters In a multiple choice test item, incorrect answer choices that may seem right are called distracters. Test writers create distracters by using common errors that students make. Be sure you always check your answer. The answer you get when you solve the problem may be one of the answer choices, but it may not be the correct answer. ̶̶̶ CD is tangent to ⊙B at C, and m ⁀ AC = 65°. What is m∠ABC? 130° 65° 32.5° 25° Look at each answer choice carefully. A This is a distracter. The m ⁀ ACE , not m∠ABC, is 130°. Doubling the arc length is a common error. B This is the correct answer. C This is a distracter. Using the Inscribed Angle Theorem to solve this problem is an error a student might make. m∠ABC is not equal to half the m ⁀ AC . D This is a distracter. The sum of m∠ACD and m∠ABC is not 90°. In a circle, the length of an arc intercepted by a central angle is 4π, and the radius is 16 inches. What is the measure of the central angle? 5.625 22.5° 45° 90° Look at each answer choice carefully. F This is a distracter. Students who use π r 2 instead of 2πr will get this answer. G This is a distracter. Students often make errors when dividing. This distracter was created by dividing 4 by 32 and getting a quotient of 1 __ . 16 H This is the correct answer. J This is a distracter. You would get this answer if you simplified the formula for arc length incorrectly. 816 816 Chapter 11 Circles �� When you calculate an answer to a multiplechoice test item, solve the problem again with a different method to make sure your answer is correct. Read e
ach test item and answer the questions that follow. Item A Which point is the center of the circle defined by the equation x 2 + (y - 9) 2 = 81? (9, 9) (-9, -9) (0, 9) (9, 0) Item C A regular hexagon is inscribed in a circle with a radius of 8 inches. What is the length of one arc of the circle intercepted by one side of the hexagon? 4 _ π 3 8 _ π 3 32 _ π 3 16π 7. What is the formula for arc length? 8. How do you determine the measure of the central angle? 9. Describe the errors a student might make to 1. What common error do the coordinates in get each of the distracters. choice B represent? 2. The y-coordinate in choice C is correct, but the x-coordinate is not. What error was made in finding the x-coordinate? 3. Which choice is the correct answer? What alternative method can you use to see whether your answer is correct? Item B What is m∠FHG? 50° 70° 100° 200° 4. How would you describe ̶̶ EH and ̶̶ HD ? 5. If a student chose choice H, what common mistake might the student have made? 6. What common error would lead to answer choice J? Item D What is the equation of a circle centered at (4, -5) with a radius of 6? (x + 5) 2 + (y - 4) 2 = 6 (x - 4) 2 + (y + 5) 2 = 6 (x + 5) 2 + (y - 4) 2 = 36 (x - 4) 2 + (y + 5) 2 = 36 10. What common error does the equation in choice G represent? 11. What common error does the equation in choice H represent? 12. Why is choice J the correct answer? Item E What is m∠ABC? 45° 90° 180° 360° 13. How does knowing what ̶̶ AC is help you determine m∠ABC? 14. What mistake would lead to choice C? TAKS Tackler 817 817 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–11 Multiple Choice 1. The composite figure is a right prism that shares a base with the regular pentagonal pyramid on top. If the lateral area of this figure is 328 square feet, what is the slant height of the pyramid? 6. △JKL is a right triangle where m∠K = 90° and tan J = 3 __ . Which of the following could be 4 the side lengths of △JKL? KL = 16, KJ = 12, and JL = 20 KL = 15, KJ = 25, and JL = 20 KL = 20, KJ = 16, and JL = 12 KL = 18, KJ = 24, and JL = 30 Use the diagram for Items 7 and 8. 2.5 feet 5.0 feet 8.4 feet 9.0 feet 2. What is the area of the polygon with vertices A (2, 3) , B (12, 3) , C (6, 0) , and D (2, 0) ? 12 square units 30 square units 21 square units 42 square units 7. What is m ⁀ QU ? Use the diagram for Items 3–5. 25° 42° 58° 71° 8. Which expression can be used to calculate the length of ̶̶ PS ? PR ⋅ PQ _ PU PR ⋅ PR _ PU PQ ⋅ QR _ PU PQ ⋅ PR _ PS 3. What is m ⁀ BC ? 36° 45° 54° 72° 4. If the length of ⁀ ED is 6π, what is the area of sector EFD? 20π square centimeters 72π square centimeters 120π square centimeters 240π square centimeters 5. Which of these line segments is NOT a chord of ⊙F? ̶̶ EC ̶̶ CA ̶̶ AF ̶̶ AE 818 818 Chapter 11 Circles 9. △ABC has vertices A (0, 0) , B (-1, 3) , and C (2, 4) . If △ABC ∼ △DEF and △DEF has vertices D (5, -3) , E (4, -2) , and F (3, y) , what is the value of y? -7 -5 -3 -1 10. What is the equation of the circle with ̶̶̶ MN that has endpoints M (-1, 1) diameter and N (3, -5) ? (x + 1) 2 + (y - 2) 2 = 13 (x - 1) 2 + (y + 2) 2 = 13 (x + 1) 2 + (y - 2) 2 = 26 (x - 1) 2 + (y + 2) 2 = 52 �� Remember that an important part of writing a proof is giving a justification for each step in the proof. Justifications may include theorems, postulates, definitions, properties, or the information that is given to you. STANDARDIZED TEST PREP Short Response 21. Use the diagram to find the value of x. Show your work or explain in words how you determined your answer. 11. Kite PQRS has diagonals ̶̶ PR and ̶̶ QS that intersect at T. Which of the following is the shortest ̶̶ PR ? segment from Q to ̶̶ PT ̶̶ QP ̶̶ RQ ̶̶ TQ 12. If the perimeter of an equilateral triangle is reduced by a factor of 1 __ , what is the effect on 2 the area of the triangle? The area remains constant. The area is reduced by a factor of 1 __ . 2 The area is reduced by a factor of 1 __ . 4 The area is reduced by a factor of 1 __ . 6 13. The area of a right isosceles triangle is 36 m 2 . What is the length of the hypotenuse of the triangle? 6 meters 6 √  2 meters 12 meters 12 √  2 meters Gridded Response 14. The ratio of the side lengths of a triangle is 4 : 5 : 8. If the perimeter is 38.25 centimeters, what is the length in centimeters of the shortest side? 22. Paul needs to rent a storage unit. He finds one that has a length of 10 feet, a width of 5 feet, and a height of 9 feet. He finds a second storage unit that has a length of 11 feet, a width of 4 feet, and a height of 8 feet. Suppose that the first storage unit costs $85.00 per month and that the second storage unit costs $70.00 per month. a. Which storage unit has a lower price per cubic foot? Show your work or explain in words how you determined your answer. b. Paul finds a third storage unit that charges $0.25 per cubic foot per month. What are possible dimensions of the storage unit if the charge is $100.00 per month? 23. The equation of ⊙C is x 2 + (y + 1) 2 = 25. a. Graph ⊙C. b. Write the equation of the line that is tangent to ⊙C at (3, 3) . Show your work or explain in words how you determined your answer. 15. What is the geometric mean of 4 and 16? 24. A tangent and a secant intersect on a circle at 16. For △HGJ and △LMK suppose that ∠H ≅ ∠L, HG = 4x + 5, KL = 9, HJ = 5x -1, and LM = 13. What must be the value of x to prove that △HGJ and △LMK are congruent by SAS? 17. If the length of a side of a regular hexagon is 2, what is the area of the hexagon to the nearest tenth? 18. What is the arc length of a semicircle in a circle with radius 5 millimeters? Round to the nearest hundredth. 19. What is the surface area of a sphere whose volume is 288π cubic centimeters? Round to the nearest hundredth. 20. Convert (6, 60°) to rectangular coordinates. What is the value of the x-coordinate? the point of tangency and form an acute angle. Explain how you would find the range of possible measures for the intercepted arc. Extended Response 25. Let ABCD be a quadrilateral inscribed in a circle such that ̶̶ AB ǁ ̶̶ DC . a. Prove that m ⁀ AD = m ⁀ BC . b. Suppose ABCD is a trapezoid. Show that ABCD must be isosceles. Justify your answer. c. If ABCD is not a trapezoid, explain why ABCD must be a rectangle. Cumulative Assessment, Chapters 1–11 819 819 �� Extending Transformational Geometry 12A Congruence Transformations 12-1 Reflections 12-2 Translations 12-3 Rotations Lab Explore Transformations with Matrices 12-4 Compositions of Transformations 12B Patterns 12-5 Symmetry 12-6 Tessellations Lab Use Transformations to Extend Tessellations 12-7 Dilations Ext Using Patterns to Generate Fractals KEYWORD: MG7 ChProj Each year, hundreds of millions of monarch butterflies pass through Texas during their annual migration. 820 820 Chapter 12 Vocabulary Match each term on the left with a definition on the right. 1. image A. a mapping of a figure from its original position to a new 2. preimage 3. transformation 4. vector position B. a ray that divides an angle into two congruent angles C. a shape that undergoes a transformation D. a quantity that has both a size and a direction E. the shape that results from a transformation of a figure Ordered Pairs Graph each ordered pair. 5. (0, 4) 8. (3, -1) Congruent Figures 6. (-3, 2) 9. (-1, -3) 7. (4, 3) 10. (-2, 0) Can you conclude that the given triangles are congruent? If so, explain why. 11. △PQS and △PRS 12. △DEG and △FGE Identify Similar Figures Can you conclude that the given figures are similar? If so, explain why. 13. △JKL and △JMN 14. rectangle PQRS and rectangle UVWX Angles in Polygons 15. Find the measure of each interior angle of a regular octagon. 16. Find the sum of the interior angle measures of a convex pentagon. 17. Find the measure of each exterior angle of a regular hexagon. 18. Find the value of x in hexagon ABCDEF. Extending Transformational Geometry 821 821 �� Key Vocabulary/Vocabulario composition of transformations composición de transformaciones glide reflection deslizamiento con inversión isometry symmetry tessellation isometría simetría teselado Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. A composition is something that has been put together. How can you use this idea to understand what is meant by a composition of transformations ? 2. The prefix iso- means “equal.” The suffix -metry means “measure.” What do you think might be true about the preimage and image of a figure under a transformation that is an isometry ? Geometry TEKS G.2.A Geometric structure* use constructions to explore attributes of geometric figures ... Les. 12-1 Les. 12-2 Les. 12-3 ★ ★ ★ 12-3 Tech. Lab Les. 12-4 Les. 12-5 Les. 12-6 12-6 Geo. Lab Les. 12-7 Ext. ★ G.2.B Geometric structure* make conjectures … and ★ ★ ★ ★ ★ determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic G.5.B Geometric patterns* use numeric and geometric ★ patterns to make generalizations about geometric properties … G.5.C Geometric patterns* use properties of transformations ★ ★ ★ ★ ★ and their compositions to make connections between mathematics and the real world such as tessellations G.7.A Dimensionality and the geometry of location* use one- ★ ★ ★ ★ and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures G.10.A Congruence and the geometry of size* use congruence transformations to make conjectures and justify properties of geometric figures … G.11.A Similarity and the geometry of shape* use and extend similarity properties and transformations to explore and justify conjectures … G.11.B Similarity and the geometry of shape* uses ratios to solve problems involvi
ng similar figures G.11.D Similarity and the geometry of shape* describe the effect on perimeter, area, and volume when one or more dimensions of figure are changed and apply this idea in solving problems ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 822 822 Chapter 12 Study Strategy: Prepare for Your Final Exam Math is a cumulative subject, so your final exam will probably cover all of the material you have learned since the beginning of the course. Preparation is essential for you to be successful on your final exam. It may help you to make a study timeline like the one below. 2 weeks before the final: • Look at previous exams and homework to determine areas I need to focus on; rework problems that were incorrect or incomplete. • Make a list of all formulas, postulates, and theorems I need to know for the final. • Create a practice exam using problems from the book that are similar to problems from each exam. 1 week before the final: • Take the practice exam and check it. For each problem I miss, find two or three similar ones and work those. • Work with a friend in the class to quiz each other on formulas, postulates, and theorems from my list. 1 day before the final: • Make sure I have pencils, calculator (check batteries!), ruler, compass, and protractor. Try This 1. Create a timeline that you will use to study for your final exam. Extending Transformational Geometry 823 823 12-1 Reflections TEKS G.10.A Congruence and the geometry of size: use congruence transformations to make conjectures and justify properties of geometric figures .... Objective Identify and draw reflections. Vocabulary isometry Who uses this? Trail designers use reflections to find shortest paths. (See Example 3.) An isometry is a transformation that does not change the shape or size of a figure. Reflections, translations, and rotations are all isometries. Isometries are also called congruence transformations or rigid motions. The Houston skyline Also G.2.A, G.2.B, G.7.A Recall that a reflection is a transformation that moves a figure (the preimage) by flipping it across a line. The reflected figure is called the image. A reflection is an isometry, so the image is always congruent to the preimage. E X A M P L E 1 Identifying Reflections Tell whether each transformation appears to be a reflection. Explain. A B To review basic transformations, see Lesson 1-7, pages 50−55. Yes; the image appears to be flipped across a line. No; the figure does not appear to be flipped. Tell whether each transformation appears to be a reflection. 1a. 1b. Construction Reflect a Figure Using Patty Paper    Draw a triangle and a line of reflection on a piece of patty paper. Fold the patty paper back along the line of reflection. Trace the triangle. Then unfold the paper. Draw a segment from each vertex of the preimage to the corresponding vertex of the image. Your construction should show that the line of reflection is the perpendicular bisector of every segment connecting a point and its image. 824 824 Chapter 12 Extending Transformational Geometry Reflections A reflection is a transformation across a line, called the line of reflection, so that the line of reflection is the perpendicular bisector of each segment joining each point and its image. E X A M P L E 2 Drawing Reflections For more on reflections, see the Transformation Builder on page xxiv. Copy the quadrilateral and the line of reflection. Draw the reflection of the quadrilateral across the line. Step 1 Through each vertex draw a line perpendicular to the line of reflection. Step 2 Measure the distance from each vertex to the line of reflection. Locate the image of each vertex on the opposite side of the line of reflection and the same distance from it. Step 3 Connect the images of the vertices. 2. Copy the quadrilateral and the line of reflection. Draw the reflection of the quadrilateral across the line. E X A M P L E 3 Problem-Solving Application A trail designer is planning two trails that connect campsites A and B to a point on the river. He wants the total length of the trails to be as short as possible. Where should the trail meet the river? Understand the Problem The problem asks you to locate point X on the river so that AX + XB has the least value possible. Make a Plan Let B' be the reflection of point B across the river. For any point X on the river, are collinear. ̶̶ XB , so AX + XB = AX + XB'. AX + XB' is least when A, X, and B' ̶̶̶ XB' ≅ Solve Reflect B across the river to locate B'. Draw locate X at the intersection of ̶̶̶ AB' and the river. ̶̶̶ AB' and Look Back To verify your answer, choose several possible locations for X and measure the total length of the trails for each location. 3. What if…? If A and B were the same distance from the river, ̶̶ AX and what would be true about ̶̶ BX ? 12-1 Reflections 825 825 ��123��4�� Reflections in the Coordinate Plane ACROSS THE x-AXIS ACROSS THE y-AXIS ACROSS THE LINE Drawing Reflections in the Coordinate Plane Reflect the figure with the given vertices across the given line. A M (1, 2) , N (1, 4) , P (3, 3) ; y-axis The reflection of (x, y) is (-x, y) . M (1, 2) → M' (-1, 2) N (1, 4) → N' (-1, 4) P (3, 3) → P' (-3, 3) Graph the preimage and image. B D (2, 0) , E (2, 2) , F (5, 2) , G (5, 1) ; y = x The reflection of (x, y) is (y, x) . D (2, 0) → D' (0, 2) E (2, 2) → E' (2, 2) F (5, 2) → F' (2, 5) G (5, 1) → G' (1, 5) Graph the preimage and image. 4. Reflect the rectangle with vertices S (3, 4) , T (3, 1) , U (-2, 1) , and V (-2, 4) across the x-axis. THINK AND DISCUSS 1. Acute scalene △ABC is reflected across ̶̶ BC . Classify quadrilateral ABA'C. Explain your reasoning. 2. Point A' is a reflection of point A across line ℓ. What is the relationship of ℓ to ̶̶̶ AA' ? 3. GET ORGANIZED Copy and complete the graphic organizer. 826 826 Chapter 12 Extending Transformational Geometry �� 12-1 Exercises Exercises KEYWORD: MG7 12-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary If a transformation is an isometry, how would you describe the relationship between the preimage and the image Tell whether each transformation appears to be a reflection. p. 824 2. 4. 3. 5. 825 Multi-Step Copy each figure and the line of reflection. Draw the reflection of the figure across the line. 6. 7. 825 8. City Planning The towns of San Pablo and Tanner are located on the same side of Highway 105. Two access roads are planned that connect the towns to a point P on the highway. Draw a diagram that shows where point P should be located in order to make the total length of the access roads as short as possible Reflect the figure with the given vertices across the given line. p. 826 9. A (-2, 1) , B (2, 3) , C (5, 2) ; x-axis 10. R (0, -1) , S (2, 2) , T (3, 0) ; y-axis 11. M (2, 1) , N (3, 1) , P (2, -1) , Q (1, -1) ; y = x 12. A (-2, 2) , B (-1, 3) , C (1, 2) , D (-2, -2) ; y = x PRACTICE AND PROBLEM SOLVING Tell whether each transformation appears to be a reflection. 13. 15. 14. 16. Independent Practice For See Exercises Example 13–16 17–18 19 20–23 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S26 Application Practice p. S39 12-1 Reflections 827 827 �� Multi-Step Copy each figure and the line of reflection. Draw the reflection of the figure across the line. 17. 18. 19. Recreation Cara is playing pool. She wants to hit the ball at point A without hitting the ball at point B. She has to bounce the cue ball, located at point C, off the side rail and into her ball. Draw a diagram that shows the exact point along the rail that Cara should aim for. Reflect the figure with the given vertices across the given line. 20. A (-3, 2) , B (0, 2) , C (-2, 0) ; y-axis 21. M (-4, -1) , N (-1, -1) , P (-2, -2) ; y = x 22. J (1, 2) , K (-2, -1) , L (3, -1) ; x-axis 23. S (-1, 1) , T (1, 4) , U (3, 2) , V (1, -3) ; y = x Copy each figure. Then complete the figure by drawing the reflection image across the line. 24. 25. 26. Chemistry Louis Pasteur (1822– 1895) is best known for the pasteurization process, which kills germs in milk. He discovered chemical chirality when he observed that two salt crystals were mirror images of each other. 27. Chemistry In chemistry, chiral molecules are mirror images of each other. Although they have similar structures, chiral molecules can have very different properties. For example, the compound R- (+) -limonene smells like oranges, while its mirror image, S- (-) -limonene, smells like lemons. Use the figure and the given line of reflection to draw S- (-) -limonene. Each figure shows a preimage and image under a reflection. Copy the figure and draw the line of reflection. 28. 29. 30. Use arrow notation to describe the mapping of each point when it is reflected across the given line. 31. (5, 2) ; x-axis 32. (-3, -7) ; y-axis 33. (0, 12) ; x-axis 34. (-3, -6) ; y = x 35. (0, -5) ; y = x 36. (4, 4) ; y = x 828 828 Chapter 12 Extending Transformational Geometry ��ABCge07sec12l01006a 37. This problem will prepare you for the Multi-Step TAKS Prep on page 854. The figure shows one hole of a miniature golf course. a. Is it possible to hit the ball in a straight line from the tee T to the hole H? b. Find the coordinates of H', the reflection of H across ̶̶ BC . c. The point at which a player should aim in order to make a hole in one is the intersection ̶̶̶ TH' and of this point? ̶̶ BC . What are the coordinates of 38. Critical Thinking Sketch the next figure in the sequence below. 39. Critical Thinking Under a reflection in the coordinate plane, the point (3, 5) is mapped to the point (5, 3) . What is the line of reflection? Is this the only possible line of reflection? Explain. Draw the reflection of the graph of each function across the gi
ven line. 40. x-axis 41. y-axis 42. Write About It Imagine reflecting all the points in a plane across line ℓ. Which points remain fixed under this transformation? That is, for which points is the image the same as the preimage? Explain. Construction Use the construction of a line perpendicular to a given line through a given point (see page 179) and the construction of a segment congruent to a given segment (see page 14) to construct the reflection of each figure across a line. 43. a point 44. a segment 45. a triangle 46. Daryl is using a coordinate plane to plan a garden. He draws a flower bed with vertices (3, 1) , (3, 4) , (-2, 4) , and (-2, 1) . Then he creates a second flower bed by reflecting the first one across the x-axis. Which of these is a vertex of the second flower bed? (-2, -4) (-3, 1) (2, 1) (-3, -4) 12-1 Reflections 829 829 �� 47. In the reflection shown, the shaded figure is the preimage. Which of these represents the mapping? � � � MJNP → DSWG DGWS → MJNP JMPN → GWSD PMJN → SDGW 48. What is the image of the point (-3, 4) when it is reflected across the y-axis? (4, -3) (-3, -4) (3, 4) (-4, -3) � � � � � CHALLENGE AND EXTEND Find the coordinates of the image when each point is reflected across the given line. 49. (4, 2) ; y = 3 51. (3, 1) ; y = x + 2 50. (-3, 2) ; x = 1 52. Prove that the reflection image of a segment is congruent ̶̶̶ A'B' ̶̶ AB across line ℓ. to the preimage. ̶̶̶ A'B' is the reflection image of ̶̶ AB ≅ Given: Prove: Plan: Draw auxiliary lines that △ACD ≅ △A'CD. Then use CPCTC to conclude that ∠CDA ≅ ∠CDA'. Therefore ∠ADB ≅ ∠A'DB', which makes it possible to prove that △ADB ≅ △A'DB'. Finally use CPCTC to conclude that ̶̶̶ AA' and ̶̶̶ BB' as shown. First prove ̶̶ AB ≅ ̶̶̶ A'B' . � � �� Once you have proved that the reflection image of a segment is congruent to the preimage, how could you prove the following? Write a plan for each proof. 53. If ̶̶̶ A'B' is the reflection of ̶̶ AB , then AB = A'B'. 54. If ∠A'B'C' is the reflection of ∠ABC, then m∠ABC = m∠A'B'C'. 55. The reflection △A'B'C' is congruent to the preimage △ABC. 56. If point C is between points A and B, then the reflection C' is between A' and B'. 57. If points A, B, and C are collinear, then the reflections A', B', and C' are collinear. SPIRAL REVIEW A jar contains 2 red marbles, 6 yellow marbles, and 4 green marbles. One marble is drawn and replaced, and then a second marble is drawn. Find the probability of each outcome. (Previous course) 58. Both marbles are green. 59. Neither marble is red. 60. The first marble is yellow, and the second is green. The width of a rectangular field is 60 m, and the length is 105 m. Use each of the following scales to find the perimeter of a scale drawing of the field. (Lesson 7-5) 61. 1 cm : 30 m 62. 1.5 cm : 15 m 63. 1 cm : 25 m Find each unknown measure. Round side lengths to the nearest hundredth and angle measures to the nearest degree. (Lesson 8-3) 64. BC 65. m∠A 66. m∠C � � � �� 830 830 Chapter 12 Extending Transformational Geometry 12-2 Translations TEKS G.10.A Congruence and the geometry of size: use congruence transformations to make conjectures and justify properties ... Objective Identify and draw translations. Who uses this? Marching band directors use translations to plan their bands’ field shows. (See Example 4.) Also G.2.A, G.2.B, G.7.A A translation is a transformation where all the points of a figure are moved the same distance in the same direction. A translation is an isometry, so the image of a translated figure is congruent to the preimage. E X A M P L E 1 Identifying Translations Tell whether each transformation appears to be a translation. Explain. A B No; not all of the points have moved the same distance. Yes; all of the points have moved the same distance in the same direction. Tell whether each transformation appears to be a translation. 1a. 1b. Construction Translate a Figure Using Patty Paper    Draw a triangle and a translation vector on a sheet of paper. Place a sheet of patty paper on top of the diagram. Trace the triangle and vector. Slide the bottom paper in the direction of the vector until the head of the top vector aligns with the tail of the bottom vector. Trace the triangle. To review vectors, see Lesson 8-6, pages 559−567. Draw a segment from each vertex of the preimage to the corresponding vertex of the image. Your construction should show that every segment connecting a point and its image is the same length as the translation vector. These segments are also parallel to the translation vector. 12-2 Translations 831 831 Translations A translation is a transformation along a vector such that each segment joining a point and its image has the same length as the vector and is parallel to the vector. E X A M P L E 2 Drawing Translations Copy the triangle and the translation vector.  v . Draw the translation of the triangle along Step 1 Draw a line parallel to the vector through each vertex of the triangle. For more on translations, see the Transformation Builder on page xxiv. Step 2 Measure the length of the vector. Then, from each vertex mark off this distance in the same direction as the vector, on each of the parallel lines.w Step 3 Connect the images of the vertices. 2. Copy the quadrilateral and the translation vector. Draw the translation of the quadrilateral along  w . Recall that a vector in the coordinate plane can be written as 〈a, b〉, where a is the horizontal change and b is the vertical change from the initial point to the terminal point. Translations in the Coordinate Plane HORIZONTAL TRANSLATION ALONG VECTOR 〈a, 0〉 VERTICAL TRANSLATION ALONG VECTOR 〈0, b〉 GENERAL TRANSLATION ALONG VECTOR 〈a, b〉 832 832 Chapter 12 Extending Transformational Geometry �� E X A M P L E 3 Drawing Translations in the Coordinate Plane Translate the triangle with vertices A (-2, -4) , B (-1, -2) , and C (-3, 0) along the vector 〈2, 4〉. The image of (x, y) is (x + 2, y + 4) . A (-2, -4) → A' (-2 + 2, -4 + 4) = A' (0, 0) B (-1, -2) → B' (-1 + 2, -2 + 4) = B' (1, 2) C (-3, 0) → C' (-3 + 2, 0 + 4) = C' (-1, 4) Graph the preimage and image. 3. Translate the quadrilateral with vertices R (2, 5) , S (0, 2) , T (1, -1) , and U (3, 1) along the vector 〈-3, -3〉. E X A M P L E 4 Entertainment Application Entertainment In 1955, the University of Texas Longhorn Band, pictured above, became the proud owner of Big Bertha, the largest bass drum in the world. The drum is 54 inches wide and 8 feet in diameter. In a marching drill, it takes 8 steps to march 5 yards. A drummer starts 8 steps to the left and 8 steps up from the center of the field. She marches 16 steps to the right to her second position. Then she marches 24 steps down the field to her final position. What is the drummer’s final position? What single translation vector moves her from the starting position to her final position? The drummer’s starting coordinates are (-8, 8) . Her second position is (-8 + 16, 8) = (8, 8) . Her final position is (8, 8 - 24) = (8, -16) . The vector that moves her directly from her starting position to her final position is 〈16, 0〉 + 〈0, -24〉 = 〈16, -24〉. 4. What if…? Suppose another drummer started at the center of the field and marched along the same vectors as above. What would this drummer’s final position be? THINK AND DISCUSS 1. Point A' is a translation of point A along  v to ̶̶ AA '?  v . What is the 2. relationship of ̶̶̶ ̶̶ AB is translated to form A'B' . Classify quadrilateral AA'B'B. Explain your reasoning. 3. GET ORGANIZED Copy and complete the graphic organizer. 12-2 Translations 833 833 �� 12-2 Exercises Exercises GUIDED PRACTICE Tell whether each transformation appears to be a translation. p. 831 1. 3. 2. 4. KEYWORD: MG7 12-2 KEYWORD: MG7 Parent . 832 Multi-Step Copy each figure and the translation vector. Draw the translation of the figure along the given vector. 5. 6 Translate the figure with the given vertices along the given vector. p. 833 7. A (-4, -4) , B (-2, -3) , C (-1, 3) ; 〈5, 0〉 8. R (-3, 1) , S (-2, 3) , T (2, 3) , U (3, 1) ; 〈0, -4〉 9. J (-2, 2) , K (-1, 2) , L (-1, -2) , M (-3, -1) ; 〈3, 2〉 10. Art The Zulu people of southern Africa are known for their beadwork. To create a typical Zulu pattern, translate the polygon with vertices (1, 5) , (2, 3) , (1, 1) , and (0, 3) along the vector 〈0, -4〉. Translate the image along the same vector. Repeat to generate a pattern. What are the vertices of the fourth polygon in the pattern? PRACTICE AND PROBLEM SOLVING Tell whether each transformation appears to be a translation. 11. 13. 13. 12. 14. 833 Independent Practice For See Exercises Example 11–14 15–16 17–19 20 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S26 Application Practice p. S39 834 834 Chapter 12 Extending Transformational Geometry �� Multi-Step Copy each figure and the translation vector. Draw the translation of the figure along the given vector. 15. 16. Animation Each frame of a computer-animated feature represents 1 __ 24 of a second of film. Source: www.pixar.com Translate the figure with the given vertices along the given vector. 17. P (-1, 2) , Q (1, -1) , R (3, 1) , S (2, 3) ; 〈-3, 0〉 18. A (1, 3) , B (-1, 2) , C (2, 1) , D (4, 2) ; 〈-3, -3〉 19. D (0, 15) , E (-10, 5) , F (10, -5) ; 〈5, -20〉 20. Animation An animator draws the ladybug shown and then translates it along the vector 〈1, 1〉, followed by a translation of the new image along the vector 〈2, 2〉, followed by a translation of the second image along the vector 〈3, 3〉. a. Sketch the ladybug’s final position. b. What single vector moves the ladybug from its starting position to its final position? Draw the tran
slation of the graph of each function along the given vector. 21. 〈3, 0〉 22. 〈-1, -1〉 23. Probability The point P (3, 2) is translated along one of the following four vectors chosen at random: 〈-3, 0〉, 〈-1, -4〉, 〈3, -2〉, and 〈2, 3〉. Find the probability of each of the following. a. The image of P is in the fourth quadrant. b. The image of P is on an axis. c. The image of P is at the origin. 24. This problem will prepare you for the Multi-Step TAKS Prep on page 854. The figure shows one hole of a miniature golf course and the path of a ball from the tee T to the hole H. a. What translation vector represents the path of the ball from T to ̶̶ DC ? b. What translation vector represents the path of the ball from ̶̶ DC to H? c. Show that the sum of these vectors is equal to the vector that represents the straight path from T to H. 12-2 Translations 835 835 ��yx0–5–5ge07sec12l02008a4th pass5/11/5cmurphy�� Each figure shows a preimage (blue) and its image (red) under a translation. Copy the figure and draw the vector along which the polygon is translated. 25. 26. 27. Critical Thinking The points of a plane are translated along the given vector remain fixed under this transformation? That is, are there any points for which the image coincides with the preimage? Explain.  AB . Do any points 28. Carpentry Carpenters use a tool called adjustable parallels to set up level work areas and to draw parallel lines. Describe how a carpenter could use this tool to translate a given point along a given vector. What additional tools, if any, would be needed? Find the vector associated with each translation. Then use arrow notation to describe the mapping of the preimage to the image. 29. the translation that maps point A to point B 30. the translation that maps point B to point A 31. the translation that maps point C to point D 32. the translation that maps point E to point B 33. the translation that maps point C to the origin 34. Multi-Step The rectangle shown is translated two-thirds of the way along one of its diagonals. Find the area of the region where the rectangle and its image overlap. 35. Write About It Point P is translated along the vector 〈a, b〉. Explain how to find the distance between point P and its image. Construction Use the construction of a line parallel to a given line through a given point (see page 163) and the construction of a segment congruent to a given segment (see page 13) to construct the translation of each figure along a vector. 36. a point 37. a segment 38. a triangle 39. What is the image of P (1, 3) when it is translated along the vector 〈-3, 5〉? (0, 4) (-2, 8) (0, 6) (1, 3) 40. After a translation, the image of A (-6, -2) is B (-4, -4) . What is the image of the point (3, -1) after this translation? (-5, 1) (5, -3) (5, 1) (-5, -3) 836 836 Chapter 12 Extending Transformational Geometry �� 41. Which vector translates point Q to point P? 〈-2, -4〉〈4, -2〉〈-2, 4〉〈2, -4〉 � � � � � � � � � CHALLENGE AND EXTEND 42. The point M (1, 2) is translated along a vector that is parallel to the line y = 2x + 4. The translation vector has magnitude √  5 . What are the possible images of point M? 43. A cube has edges of length 2 cm. Point P is translated along  u ,  v , and  w as shown. a. Describe a single translation vector that maps point P to point Q. b. Find the magnitude of this vector to the nearest hundredth. 44. Prove that the translation image of a segment is congruent to the preimage. ̶̶̶ A'B' is the translation image of Given: ̶̶ AB ≅ Prove: (Hint: Draw auxiliary lines What can you conclude about ̶̶̶ BB' . ̶̶̶ AA' and ̶̶̶ AA' and ̶̶̶ A'B' ̶̶ AB . ̶̶̶ BB' ?) � � �� Once you have proved that the translation image of a segment is congruent to the preimage, how could you prove the following? Write a plan for each proof. 45. If ̶̶̶ A'B' is a translation of ̶̶ AB , then AB = A'B'. 46. If ∠A'B'C' is a translation of ∠ABC, then m∠ABC = m∠A'B'C'. 47. The translation △A'B'C' is congruent to the preimage △ABC. 48. If point C is between points A and B, then the translation C' is between A' and B'. 49. If points A, B, and C are collinear, then the translations A', B', and C' are collinear. SPIRAL REVIEW Solve each system of equations and check your solution. (Previous course) ⎧ -5x - 2y = 17 50. ⎨ 6x - 2y = -5 ⎩ ⎧ 2x - 3y = -7 51. ⎨ 6x + 5y = 49 ⎩ ⎧ 4x + 4y = -1 52. ⎨ 12x - 8y = -8 ⎩ Solve to find x and y in each diagram. (Lesson 3-4) 53. 54. �� △MNP has vertices M (-2, 0) , N (-3, 2) , and P (0, 4) . Find the coordinates of the vertices of △M'N'P' after a reflection across the given line. (Lesson 12-1) 55. x-axis 56. y-axis 57. y = x 12-2 Translations 837 837 Transformations of Functions Algebra Transformations can be used to graph complicated functions by using the graphs of simpler functions called parent functions. The following are examples of parent functions and their graphs. See Skills Bank page S63 y = ⎜x Transformation of Parent Function y = f (x) Reflection Vertical Translation Horizontal Translation Across x-axis: y = -f (x) y = f (x) + k y = f (x - h) Across y-axis: y = f (-x) Up k units if k > 0 Right h units if h > 0 Down k units if k < 0 Left h units if h < 0 Example For the parent function y = x 2 , write a function rule for the given transformation and graph the preimage and image. A a reflection across the x-axis B a translation up 2 units and right 3 units function rule: y = - x 2 graph: function rule: y = (x - 3) 2 + 2 graph: Try This TAKS Grades 9–11 Obj. 2, 5 For each parent function, write a function rule for the given transformation and graph the preimage and image. 1. parent function: y = x 2 transformation: a translation down 1 unit and right 4 units 2. parent function: y = √x transformation: a reflection across the x-axis 3. parent function: y = ⎜x⎟ transformation: a translation up 2 units and left 1 unit 838 838 Chapter 12 Extending Transformational Geometry �� 12-3 Rotations TEKS G.10.A Congruence and the geometry of size: use congruence transformations to make conjectures and justify properties of geometric figures .... Objective Identify and draw rotations. Who uses this? Astronomers can use properties of rotations to analyze photos of star trails. (See Exercise 35.) Also G.2.A, G.2.B, G.7.A Remember that a rotation is a transformation that turns a figure around a fixed point, called the center of rotation. A rotation is an isometry, so the image of a rotated figure is congruent to the preimage. E X A M P L E 1 Identifying Rotations Tell whether each transformation appears to be a rotation. Explain. A B Yes; the figure appears to be turned around a point. No; the figure appears to be flipped, not turned. Tell whether each transformation appears to be a rotation. 1a. 1b. Construction Rotate a Figure Using Patty Paper    On a sheet of paper, draw a triangle and a point. The point will be the center of rotation. Place a sheet of patty paper on top of the diagram. Trace the triangle and the point. Hold your pencil down on the point and rotate the bottom paper counterclockwise. Trace the triangle. Draw a segment from each vertex to the center of rotation. Your construction should show that a point’s distance to the center of rotation is equal to its image’s distance to the center of rotation. The angle formed by a point, the center of rotation, and the point’s image is the angle by which the figure was rotated. 12-3 Rotations 839 839 Rotations A rotation is a transformation about a point P, called the center of rotation, such that each point and its image are the same distance from P, and such that all angles with vertex P formed by a point and its image are congruent. In the figure, ∠APA' is the angle of rotation. E X A M P L E 2 Drawing Rotations Copy the figure and the angle of rotation. Draw the rotation of the triangle about point P by m∠A. Step 1 Draw a segment from each vertex to point P. Unless otherwise stated, all rotations in this book are counterclockwise. Step 2 Construct an angle congruent to ∠A onto each segment. Measure the distance from each vertex to point P and mark off this distance on the corresponding ray to locate the image of each vertex. Step 3 Connect the images of the vertices. 2. Copy the figure and the angle of rotation. Draw the rotation of the segment about point Q by m∠X. Rotations in the Coordinate Plane BY 90° ABOUT THE ORIGIN BY 180° ABOUT THE ORIGIN For more on rotations, see the Transformation Builder on page xxiv. If the angle of a rotation in the coordinate plane is not a multiple of 90°, you can use sine and cosine ratios to find the coordinates of the image. 840 840 Chapter 12 Extending Transformational Geometry �� E X A M P L E 3 Drawing Rotations in the Coordinate Plane Rotate △ABC with vertices A (2, -1) , B (4, 1) , and C (3, 3) by 90° about the origin. The rotation of (x, y) is (-y, x) . A (2, -1) → A' (1, 2) B (4, 1) → B' (-1, 4) C (3, 3) → C' (-3, 3) Graph the preimage and image. 3. Rotate △ABC by 180° about the origin. E X A M P L E 4 Engineering Application The Texas Star Ferris wheel has a radius of 106 ft and takes 40 seconds to make a complete rotation. A car starts at position (106, 0) . What are the coordinates of the car’s location after 5 seconds? Step 1 Find the angle of rotation. Five seconds is 5 __ 40 = 1 __ 8 of a complete revolution, or 1 __ 8 (360°) = 45°. Step 2 Draw a right triangle to represent the car’s location (x, y) after a rotation of 45° about the origin. To review the sine and cosine ratios, see Lesson 8-2, pages 525–532. Step 3 Use the cosine ratio to find the x-coordinate. cos 45° = x _ 106 cos = adj. _ hyp. Solve for x. x = 106 cos 45° ≈ 75.0 Step 4 Use the sine ratio to find the
y-coordinate. sin 45° = y _ 106 y = 106 sin 45° ≈ 75.0 sin = opp. _ hyp. Solve for y. The car’s location after 5 seconds is approximately (75.0, 75.0) . 4. The London Eye observation wheel has a radius of 67.5 m and takes 30 minutes to make a complete rotation. A car starts at position (67.5, 0) . Find the coordinates of the car after 6 minutes. Round to the nearest tenth. THINK AND DISCUSS 1. Describe the image of a rotation of a figure by an angle of 360°. 2. Point A' is a rotation of point A about point P. What is the relationship ̶̶ AP to ̶̶ A'P ? of 3. GET ORGANIZED Copy and complete the graphic organizer. 12-3 Rotations 841 841 �� 12-3 Exercises Exercises GUIDED PRACTICE Tell whether each transformation appears to be a rotation. p. 839 1. 3. 2. 4. KEYWORD: MG7 12-3 KEYWORD: MG7 Parent . 840 Copy each figure and the angle of rotation. Draw the rotation of the figure about point P by m∠A. 5. 6. 841 Rotate the figure with the given vertices about the origin using the given angle of rotation. 7. A (1, 0) , B (3, 2) , C (5, 0) ; 90° 8. J (2, 1) , K (4, 3) , L (2, 4) , M (-1, 2) ; 90° 9. D (2, 3) , E (-1, 2) , F (2, 1) ; 180° 10. P (-1, -1) , Q (-4, -2) , R (0, -2) ; 180 11. Animation An artist uses a coordinate plane to plan the motion of p. 841 an animated car. To simulate the car driving around a curve, the artist places the car at the point (10, 0) and then rotates it about the origin by 30°. Give the car’s final position, rounding the coordinates to the nearest tenth. PRACTICE AND PROBLEM SOLVING Tell whether each transformation appears to be a rotation. 12. 14. 14. 13. 15. Independent Practice For See Exercises Example 12–15 16–17 18–21 22 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S26 Application Practice p. S39 842 842 Chapter 12 Extending Transformational Geometry �� Copy each figure and the angle of rotation. Draw the rotation of the figure about point P by m∠A. 16. 17. Rotate the figure with the given vertices about the origin using the given angle of rotation. 18. E (-1, 2) , F (3, 1) , G (2, 3) ; 90° 19. A (-1, 0) , B (-1, -3) , C (1, -3) , D (1, 0) ; 90° 20. P (0, 2) , Q (2, 0) , R (3, -3) ; 180° 21. L (2, 0) , M (-1, -2) , N (2, -2) ; 180° 22. Architecture The CN Tower in Toronto, Canada, features a revolving restaurant that takes 72 minutes to complete a full rotation. A table that is 50 feet from the center of the restaurant starts at position (50, 0) . What are the coordinates of the table after 6 minutes? Round coordinates to the nearest tenth. Copy each figure. Then draw the rotation of the figure about the red point using the given angle measure. 23. 90° 24. 180° 25. 180° 26. Point Q has coordinates (2, 3) . After a rotation about the origin, the image of point Q lies on the y-axis. a. Find the angle of rotation to the nearest degree. b. Find the coordinates of the image of point Q. Round to the nearest tenth. Rectangle RSTU is the image of rectangle LMNP under a 180° rotation about point A. Name each of the following. 27. the image of point N 28. the preimage of point S 29. the image of ̶̶̶ MN 30. the preimage of ̶̶ TU 31. This problem will prepare you for the Multi-Step TAKS Prep on page 854. A miniature golf course includes a hole with a windmill. Players must hit the ball through the opening at the base of the windmill while the blades rotate. a. The blades take 20 seconds to make a complete rotation. Through what angle do the blades rotate in 4 seconds? b. Find the coordinates of point A after 4 seconds. (Hint: (4, 3) is the center of rotation.) � � � � � � 12-3 Rotations 843 843 �� Each figure shows a preimage and its image under a rotation. Copy the figure and locate the center of rotation. 32. 33. 34. 35. Astronomy The photograph was made by placing a camera on a tripod and keeping the camera’s shutter open for a long time. Because of Earth’s rotation, the stars appear to rotate around Polaris, also known as the North Star. a. Estimation Estimate the angle of rotation of the stars in the photo. b. Estimation Use your result from part a to estimate the length of time that the camera’s shutter was open. (Hint: If the shutter was open for 24 hours, the stars would appear to make one complete rotation around Polaris.) �� 36. Estimation In the diagram, △ABC → △A'B'C' under �� a rotation about point P. a. Estimate the angle of rotation. b. Explain how you can draw two segments and can then use a protractor to measure the angle of rotation. c. Copy the figure. Use the method from part b to find the angle of rotation. How does your result compare to your estimate? �� 37. Critical Thinking A student wrote the following in his math journal. “Under a rotation, every point moves around the center of rotation by the same angle measure. This means that every point moves the same distance.” Do you agree? Explain. �� Use the figure for Exercises 38–40. 38. Sketch the image of pentagon ABCDE under a rotation of 90° about the origin. Give the vertices of the image. 39. Sketch the image of pentagon ABCDE under a rotation of 180° about the origin. Give the vertices of the image. 40. Write About It Is the image of ABCDE under a rotation of 180° about the origin the same as its image under a reflection across the x-axis? Explain your reasoning. 41. Construction Copy the figure. Use the construction of an angle congruent to a given angle (see page 22) to construct the image of point X under a rotation about point P by m∠A. � � � � � �� 844 844 Chapter 12 Extending Transformational Geometry 42. What is the image of the point (-2, 5) when it is rotated about the origin by 90°? (-5, 2) (5, -2) (-5, -2) (2, -5) 43. The six cars of a Ferris wheel are located at the vertices of a regular hexagon. Which rotation about point P maps car A to car C? 60° 90° 120° 135° 44. Gridded Response Under a rotation about the origin, the point (-3, 4) is mapped to the point (3, -4) . What is the measure of the angle of rotation? CHALLENGE AND EXTEND 45. Engineering Gears are used to change the speed and direction of rotating parts in pieces of machinery. In the diagram, suppose gear B makes one complete rotation in the counterclockwise direction. Give the angle of rotation and direction for the rotation of gear A. Explain how you got your answer. 46. Given: Prove: ̶̶̶ A'B' is the rotation image of ̶̶ AB ≅ ̶̶̶ A'B' ̶̶ AB about point P. (Hint: Draw auxiliary lines show that △APB ≅ △A'PB'.) ̶̶ AP , ̶̶ BP , ̶̶ A'P , and ̶̶̶ B'P and Once you have proved that the rotation image of a segment is congruent to the preimage, how could you prove the following? Write a plan for each proof. ̶̶ AB , then AB = A'B'. ̶̶̶ A'B' is a rotation of 47. If 48. If ∠A'B'C' is a rotation of ∠ABC, then m∠ABC = m∠A'B'C'. 49. The rotation △A'B'C' is congruent to the preimage △ABC. 50. If point C is between points A and B, then the rotation C' is between A' and B'. 51. If points A, B, and C are collinear, then the rotations A', B', and C' are collinear. SPIRAL REVIEW Find the value(s) of x when y is 3. (Previous course) 53. y = 2x 2 - 5x - 9 52. y = x 2 - 4x + 7 54. y = x 2 - 2 Find each measure. (Lesson 6-6) 55. m∠XYR 56. QR Given the points A (1, 3) , B (5, 0) , C (-3, -2) , and D (5, -6) , find the vector associated with each translation. (Lesson 12-2) 57. the translation that maps point A to point D 58. the translation that maps point D to point B 59. the translation that maps point C to the origin 12-3 Rotations 845 845 ��ge07sec12l03010aABAB�� 12-3 Explore Transformations with Matrices Use with Lesson 12-3 TEKS G.10.A Congruence and the geometry of size: use congruence transformations to make conjectures and justify properties of geometric figures .... Also G.2.B, G.5.B, G.7.A, G.11.A KEYWORD: MG7 Lab12 The vertices of a polygon in the coordinate plane can be represented by a point matrix in which row 1 contains the x-values and row 2 contains the y-values. For example, the triangle with vertices (1, 2) , (-2, 0) , ⎡ 1 and (3, -4) can be represented by ⎢ ⎣ 2 -2 0 ⎤ 3 . ⎥ ⎦ -4 On the graphing calculator, enter a matrix using the Matrix Edit menu. Enter the number of rows and columns and then enter the values. Matrix operations can be used to perform transformations. Activity 1 1 Graph the triangle with vertices (1, 0) , (2, 4) , and (5, 3) on graph paper. Enter the point matrix that represents the vertices into matrix [B] on your calculator. ⎡ 1 2 Enter the matrix ⎢ ⎣ 0 ⎤ 0 into matrix [A] on your calculator. Multiply ⎥ ⎦ -1 [A] * [B] and use the resulting matrix to graph the image of the triangle. Describe the transformation. Try This ⎡ -1 1. Enter the matrix ⎢ ⎣ 0 ⎤ 0 into matrix [A]. Multiply [A] * [B] and use the resulting ⎥ ⎦ 1 matrix to graph the image of the triangle. Describe the transformation. ⎡ 0 2. Enter the matrix ⎢ ⎣ 1 ⎤ 1 into matrix [A]. Multiply [A] * [B] and use the resulting ⎥ ⎦ 0 matrix to graph the image of the triangle. Describe the transformation. 846 846 Chapter 12 Extending Transformational Geometry �� Activity 2 1 Graph the triangle with vertices (0, 0) , (3, 1) , and (2, 4) on graph paper. Enter the point matrix that represents the vertices into matrix [B] on your calculator. ⎡ 0 2 Enter the matrix ⎢ ⎣ 2 0 2 ⎤ 0 into matrix [A]. Add [A] + [B] and use the resulting ⎥ ⎦ 2 matrix to graph the image of the triangle. Describe the transformation. Try This ⎡ -1 3. Enter the matrix ⎢ ⎣ 4 -1 4 ⎤ -1 into matrix [A]. Add [A] + [B] and use the resulting ⎥ ⎦ 4 matrix to graph the image of the triangle. Describe the transformation. 4. Make a Conjecture How do you think you could use matrices to translate a triangle by the vector 〈a, b〉? Choose several values for a and b and test your conjecture. Activity 3 1 Graph the triangle with vertices (1, 1) , (4, 1) , and (1, 2) on graph paper. Enter the
point matrix that represents the vertices into matrix [B] on your calculator. 2 Enter the matrix ⎡ ⎣ 0 -1 1 0 ⎤ ⎦ into matrix [A]. Multiply [A] * [B] and use the resulting matrix to graph the image of the triangle. Describe the transformation. Try This ⎡ -1 5. Enter the values ⎢ ⎣ 0 ⎤ 0 into matrix [A]. Multiply [A] * [B] and use the resulting ⎥ ⎦ -1 matrix to graph the image of the triangle. Describe the transformation. ⎡ 0 6. Enter the values ⎢ ⎣ -1 ⎤ 1 into matrix [A]. Multiply [A] * [B] and use the resulting ⎥ ⎦ 0 matrix to graph the image of the triangle. Describe the transformation. 12-3 Technology Lab 847 847 12-4 Compositions of Transformations TEKS G.10.A Congruence and the geometry of size: use congruence transformations to make conjectures and justify properties of geometric figures .... Also G.5.C Objectives Apply theorems about isometries. Identify and draw compositions of transformations, such as glide reflections. Vocabulary composition of transformations glide reflection Why learn this? Compositions of transformations can be used to describe chess moves. (See Exercise 11.) A composition of transformations is one transformation followed by another. For example, a glide reflection is the composition of a translation and a reflection across a line parallel to the translation vector. The glide reflection that maps △JKL to △J'K'L' is the composition of a translation along followed by a reflection across line ℓ. v A life-sized chessboard in Galveston The image after each transformation is congruent to the previous image. By the Transitive Property of Congruence, the final image is congruent to the preimage. This leads to the following theorem. Theorem 12-4-1 A composition of two isometries is an isometry. E X A M P L E 1 Drawing Compositions of Isometries Draw the result of the composition of isometries. A Reflect △ABC across line ℓ and then translate it along  v . Step 1 Draw △A'B'C', the reflection image of △ABC. Step 2 Translate △A'B'C' along find the final image, △A''B''C''.  v to 848 848 Chapter 12 Extending Transformational Geometry �� B △RST has vertices R (1, 2) , S (1, 4) , and T (-3, 4) . Rotate △RST 90° about the origin and then reflect it across the y-axis. Step 1 The rotation image of (x, y) is (-y, x) . R (1, 2) → R' (-2, 1) , S (1, 4) → S' (-4, 1) , and T (-3, 4) → T ' (-4, -3) . Step 2 The reflection image of (x, y) is (-x, y) . R' (-2, 1) → R'' (2, 1) , S' (-4, 1) → S'' (4, 1) , and T ' (-4, -3) → T '' (4, -3) . Step 3 Graph the preimage and images. 1. △JKL has vertices J (1, -2) , K (4, -2) , and L (3, 0) . Reflect △JKL across the x-axis and then rotate it 180° about the origin. Theorem 12-4-2 The composition of two reflections across two parallel lines is equivalent to a translation. • The translation vector is perpendicular to the lines. • The length of the translation vector is twice the distance between the lines. The composition of two reflections across two intersecting lines is equivalent to a rotation. • The center of rotation is the intersection of the lines. • The angle of rotation is twice the measure of the angle formed by the lines. E X A M P L E 2 Art Application For more on the composition of two reflections, see the Transformation Builder on page xxiv. Tabitha is creating a design for an art project. She reflects a figure across line ℓ and then reflects the image across line m. Describe a single transformation that moves the figure from its starting position to its final position. By Theorem 12-4-2, the composition of two reflections across intersecting lines is equivalent to a rotation about the point of intersection. Since the lines are perpendicular, they form a 90° angle. By Theorem 12-4-2, the angle of rotation is 2 · 90° = 180°. 2. What if…? Suppose Tabitha reflects the figure across line n and then the image across line p. Describe a single transformation that is equivalent to the two reflections. 12-4 Compositions of Transformations 849 849 �� Theorem 12-4-3 Any translation or rotation is equivalent to a composition of two reflections. E X A M P L E 3 Describing Transformations in Terms of Reflections Copy each figure and draw two lines of reflection that produce an equivalent transformation. A translation: △ABC → △A'B'C' Step 1 Draw ̶̶̶ AA' and locate the midpoint M of ̶̶̶ AA' . Step 2 Draw the perpendicular ̶̶̶ AM and bisectors of ̶̶̶ A'M . B rotation with center P : △DEF → △D'E'F' To draw the perpendicular bisector of a segment, use a ruler to locate the midpoint, and then use a right angle to draw a perpendicular line. Step 1 Draw ∠DPD'. Draw the angle bisector  PX . Step 2 Draw the bisectors of ∠DPX and ∠D'PX. 3. Copy the figure showing the translation that maps LMNP → L'M'N'P'. Draw the lines of reflection that produce an equivalent transformation. THINK AND DISCUSS 1. Which theorem explains why the image of a rectangle that is translated and then rotated is congruent to the preimage? 2. Point A' is a glide reflection of point A along What is the relationship between would use to draw a glide reflection.  v and across line ℓ.  v and ℓ? Explain the steps you 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe an equivalent transformation and sketch an example. 850 850 Chapter 12 Extending Transformational Geometry �� 12-4 Exercises Exercises KEYWORD: MG7 12-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Explain the steps you would use to draw a glide reflection Draw the result of each composition of isometries. p. 848 2. Translate △DEF along  u and then reflect it across line ℓ. 3. Reflect rectangle PQRS across line m and then translate it along  v . 4. △ABC has vertices A (1, -1) , B (4, -1) , and C (3, 2) . Reflect △ABC across the y-axis and then translate it along the vector 〈0, -2〉. . Sports To create the opening graphics p. 849 for a televised football game, an animator reflects a picture of a football helmet across line ℓ. She then reflects its image across line m, which intersects line ℓ at a 50° angle. Describe a single transformation that moves the helmet from its starting position to its final position. 850 Copy each figure and draw two lines of reflection that produce an equivalent transformation. 6. translation: △EFG → △E'F'G' 7. rotation with center P: △ABC → △A'B'C' Independent Practice Draw the result of each composition of isometries. PRACTICE AND PROBLEM SOLVING For See Exercises Example 8–10 11 12–13 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S26 Application Practice p. S39 8. Translate △RST along translate it along  v .  u and then 9. Rotate △ABC 90° about point P and then reflect it across line ℓ. 10. △GHJ has vertices G (1, -1) , H (3, 1) , and J (3, -2) . Reflect △GHJ across the line y = x and then reflect it across the x-axis. 12-4 Compositions of Transformations 851 851 �� 11. Games In chess, a knight moves in the shape of the letter L. The piece moves two spaces horizontally or vertically. Then it turns 90° in either direction and moves one more space. a. Describe a knight’s move as a composition of transformations. b. Copy the chessboard with the knight. Label all the positions the knight can reach in one move. c. Label all the positions the knight can reach in two moves. Copy each figure and draw two lines of reflection that produce an equivalent transformation. 12. translation: ABCD → A'B'C'D' 13. rotation with center Q: �� △JKL → △J'K'L' �� 14. /////ERROR ANALYSIS///// The segment with endpoints A (4, 2) and B (2, 1) is reflected across the y-axis. The image is reflected across the x-axis. What transformation is equivalent to the composition of these two reflections? Which solution is incorrect? Explain the error. � � � �� 15. Equilateral △ABC is reflected across ̶̶ AB . Then its image is translated along  BC . Copy △ABC and draw its final image. � Tell whether each statement is sometimes, always, or never true. 16. The composition of two reflections is equivalent to a rotation. � � 17. An isometry changes the size of a figure. 18. The composition of two isometries is an isometry. 19. A rotation is equivalent to a composition of two reflections. 20. Critical Thinking Given a composition of reflections across two parallel lines, does the order of the reflections matter? For example, does reflecting △ABC across m and then its image across n give the same result as reflecting △ABC across n and then its image across m? Explain. � � � � � 21. Write About It Under a glide reflection, △RST → △R'S'T '. The vertices of △RST are R (-3, -2) , S (-1, -2) , and T (-1, 0) . The vertices of △R'S'T ' are R' (2, 2) , S' (4, 2) , and T ' (4, 0) . Describe the reflection and translation that make up the glide reflection. 852 852 Chapter 12 Extending Transformational Geometry 22. This problem will prepare you for the Multi-Step TAKS Prep on page 854. The figure shows one hole of a miniature golf course where T is the tee and H is the hole. a. Yuriko makes a hole in one as shown by the red arrows. Write the ball’s path as a composition of translations. b. Find a different way to make a hole in one, and write the ball’s path as a composition of translations. 23. △ABC is reflected across th
e y-axis. Then its image is rotated 90° about the origin. What are the coordinates of the final image of point A under this composition of transformations? (-1, -2) (-2, 1) (1, 2) (-2, -1) 24. Which composition of transformations maps △ABC into the fourth quadrant? Reflect across the x-axis and then reflect across the y-axis. Rotate about the origin by 180° and then reflect across the y-axis. Translate along the vector 〈-5, 0〉 and then rotate about the origin by 90°. Rotate about the origin by 90° and then translate along the vector 〈1, -2〉. 25. Which is equivalent to the composition of two translations? Reflection Rotation Translation Glide reflection CHALLENGE AND EXTEND 26. The point A (3, 1) is rotated 90° about the point P (-1, 2) and then reflected across the line y = 5. Find the coordinates of the image A'. 27. For any two congruent figures in a plane, one can be transformed to the other by a composition of no more than three reflections. Copy the figure. Show how to find a composition of three reflections that maps △MNP to △M'N'P'. 28. A figure in the coordinate plane is reflected across the line y = x + 1 and then across the line y = x + 3. Find a translation vector that is equivalent to the composition of the reflections. Write the vector in component form. SPIRAL REVIEW Determine whether the set of ordered pairs represents a function. (Previous course)   (-3, -1) , (1, 2) , (-3, 1) , (5, 10) 30. ⎬ ⎨     (-6, -5) , (-1, 0) , (0, -5) , (1, 0) 29. ⎬ ⎨   Find the length of each segment. (Lesson 11-6) 31. ̶̶ EJ 32. ̶̶ CD 33. ̶̶ FH Determine the coordinates of each point after a rotation about the origin by the given angle of rotation. (Lesson 12-3) 35. N (-1, -3) ; 180° 34. F (2, 3) ; 90° 36. Q (-2, 0) ; 90° 12-4 Compositions of Transformations 853 853 �� SECTION 12A Congruence Transformations A Hole in One The figure shows a plan for one hole of a miniature golf course. The tee is at point T and the hole is at point H. Each unit of the coordinate plane represents one meter. 1. When a player hits the ball in a straight line from T to H, the path of the ball can be represented by a translation. What is the translation vector? How far does the ball travel? Round to the nearest tenth. 2. The designer of the golf course decides to make the hole more difficult by placing a barrier between the tee and the hole, as shown. To make a hole in one, a player must hit the ball so that ̶̶ DC . What point it bounces off wall along the wall should a player aim for? Explain. 3. Write the path of the ball in Problem 2 as a composition of two translations. What is the total distance that the ball travels in this case? Round to the nearest tenth. 4. The designer decides to remove the barrier and put a revolving obstacle between the tee and the hole. The obstacle consists of a turntable with four equally spaced pillars, as shown. The designer wants the turntable to make one complete rotation in 16 seconds. What should be the coordinates of the pillar at (4, 2) after 2 seconds? 854 854 Chapter 12 Extending Transformational Geometry �� SECTION 12A Quiz for Lessons 12-1 Through 12-4 12-1 Reflections Tell whether each transformation appears to be a reflection. 1. 2. Copy each figure and the line of reflection. Draw the reflection of the figure across the line. 3. 4. 12-2 Translations Tell whether each transformation appears to be a translation. 5. 6. 7. A landscape architect represents a flower bed by a polygon with vertices (1, 0) , (4, 0) , (4, 2) , and (1, 2) . She decides to move the flower bed to a new location by translating it along the vector 〈-4, -3〉. Draw the flower bed in its final position. 12-3 Rotations Tell whether each transformation appears to be a rotation. 8. 9. Rotate the figure with the given vertices about the origin using the given angle of rotation. 10. A (1, 0) , B (4, 1) , C (3, 2) ; 180° 11. R (-2, 0) , S (-2, 4) , T (-3, 4) , U (-3, 0) ; 90° 12-4 Compositions of Transformations 12. Draw the result of the following composition of transformations. Translate GHJK along and then reflect it across line m.  v 13. △ABC with vertices A (1, 0) , B (1, 3) , and C (2, 3) is reflected across the y-axis, and then its image is reflected across the x-axis. Describe a single transformation that moves the triangle from its starting position to its final position. Ready to Go On? 855 855 �� 12-5 Symmetry TEKS G.10.A Congruence and the geometry of size: use congruence transformations to make conjectures and justify properties .... Objective Identify and describe symmetry in geometric figures. Who uses this? Marine biologists use symmetry to classify diatoms. Vocabulary symmetry line symmetry line of symmetry rotational symmetry Also G.2.B, G.5.C Diatoms are microscopic algae that are found in aquatic environments. Scientists use a system that was developed in the 1970s to classify diatoms based on their symmetry. A figure has symmetry if there is a transformation of the figure such that the image coincides with the preimage. Line Symmetry A figure has line symmetry (or reflection symmetry) if it can be reflected across a line so that the image coincides with the preimage. The line of symmetry (also called the axis of symmetry) divides the figure into two congruent halves. E X A M P L E 1 Identifying Line Symmetry Tell whether each figure has line symmetry. If so, copy the shape and draw all lines of symmetry. A B C yes; one line of symmetry no line symmetry yes; five lines of symmetry Tell whether each figure has line symmetry. If so, copy the shape and draw all lines of symmetry. 1a. 1b. 1c. 856 856 Chapter 12 Extending Transformational Geometry ge07se_c12l05005aABeckmann Rotational Symmetry A figure has rotational symmetry (or radial symmetry) if it can be rotated about a point by an angle greater than 0° and less than 360° so that the image coincides with the preimage. The angle of rotational symmetry is the smallest angle through which a figure can be rotated to coincide with itself. The number of times the figure coincides with itself as it rotates through 360° is called the order of the rotational symmetry. Angle of rotational symmetry: 90° Order Identifying Rotational Symmetry Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. A B C yes; 180°; order: 2 no rotational symmetry yes; 60°; order: 6 Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. 2a. 2b. 2c. E X A M P L E 3 Biology Application Describe the symmetry of each diatom. Copy the shape and draw any lines of symmetry. If there is rotational symmetry, give the angle and order. A B line symmetry and rotational symmetry; angle of rotational symmetry: 180°; order: 2 line symmetry and rotational symmetry; angle of rotational symmetry: 120°; order: 3 Describe the symmetry of each diatom. Copy the shape and draw any lines of symmetry. If there is rotational symmetry, give the angle and order. 3a. 3b. 12-5 Symmetry 857 857 �� A three-dimensional figure has plane symmetry if a plane can divide the figure into two congruent reflected halves. A three-dimensional figure has symmetry about an axis if there is a line about which the figure can be rotated (by an angle greater than 0° and less than 360°) so that the image coincides with the preimage. E X A M P L E 4 Identifying Symmetry in Three Dimensions Tell whether each figure has plane symmetry, symmetry about an axis, or neither. A trapezoidal prism B equilateral triangular prism plane symmetry plane symmetry and symmetry about an axis Tell whether each figure has plane symmetry, symmetry about an axis, or no symmetry. 4a. cone 4b. pyramid THINK AND DISCUSS 1. Explain how you could use scissors and paper to cut out a shape that has line symmetry. 2. Describe how you can find the angle of rotational symmetry for a regular polygon with n sides. 3. GET ORGANIZED Copy and complete the graphic organizer. In each region, draw a figure with the given type of symmetry. 858 858 Chapter 12 Extending Transformational Geometry �� 12-5 Exercises Exercises KEYWORD: MG7 12-5 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Describe the line of symmetry of an isosceles triangle. 2. The capital letter T has ? . (line symmetry or rotational symmetry) ̶̶̶̶ . 856 Tell whether each figure has line symmetry. If so, copy the shape and draw all lines of symmetry. 3. 4. 5. 857 Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. 6. 7. 8. 857 9. Architecture The Pentagon in Alexandria, Virginia, is the world’s largest office building. Copy the shape of the building and draw all lines of symmetry. Give the angle and order of rotational symmetry Tell whether each figure has plane symmetry, symmetry about an axis, or neither. p. 858 10. prism 11. cylinder 12. rectangular prism PRACTICE AND PROBLEM SOLVING Tell whether each figure has line symmetry. If so, copy the shape and draw all lines of symmetry. 13. 14. 15. Independent Practice For See Exercises Example 13–15 16–18 19 20–22 1 2 3 4 TEKS TEKS TAKS TAKS Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. Skills Practice p. S27 Application Practice p. S39 16. 17. 18. 12-5 Symmetry 859 859 �� 19. Art Op art is a style of art that uses optical effects to create an impression of movement in a painting or sculpture. The painting at right, Vega-Tek, by Victor Vasarely, is an example of op art. Sketch the shape in the painting and draw any lines of symmetry. If there is rotational symmetry, give the angle and order. Tell whether each figure has pl
ane symmetry, symmetry about an axis, or neither. 20. sphere 21. triangular pyramid 22. torus Draw a triangle with the following number of lines of symmetry. Then classify the triangle. 23. exactly one line of symmetry 24. three lines of symmetry 25. no lines of symmetry Data Analysis The graph shown, called the standard normal curve, is used in statistical analysis. The area under the curve is 1 square unit. There is a vertical line of symmetry at x = 0. The areas of the shaded regions are indicated on the graph. 26. Find the area under the curve for x > 0. � �� 27. Find the area under the curve for x > 3. �� 28. If a point under the curve is selected at random, what is the probability that the x-value of the point will be between -1 and 1? Tell whether the figure with the given vertices has line symmetry and/or rotational symmetry. Give the angle and order if there is rotational symmetry. Draw the figure and any lines of symmetry. 29. A (-2, 2) , B (2, 2) , C (1, -2) , D (-1, -2) 30. R (-3, 3) , S (3, 3) , T (3, -3) , U (-3, -3) 31. J (4, 4) , K (-2, 2) , L (2, -2) 32. A (3, 1) , B (0, 2) , C (-3, 1) , D (-3, -1) , E (0, -2) , F (3, -1) 33. Art The Chokwe people of Angola are known for their traditional sand designs. These complex drawings are traced out to illustrate stories that are told at evening gatherings. Classify the symmetry of the Chokwe design shown. Algebra Graph each function. Tell whether the graph has line symmetry and/or rotational symmetry. If there is rotational symmetry, give the angle and order. Write the equations of any lines of symmetry. 35. y = (x - 2) 2 34. y = x 2 36. y = x 3 860 860 Chapter 12 Extending Transformational Geometry 37. This problem will prepare you for the Multi-Step TAKS Prep on page 880. This woodcut, entitled Circle Limit III, was made by Dutch artist M. C. Escher. a. Does the woodcut have line symmetry? If so, describe the lines of symmetry. If not, explain why not. b. Does the woodcut have rotational symmetry? If so, give the angle and order of the symmetry. If not, explain why not. c. Does your answer to part b change if color is not taken into account? Explain. Classify the quadrilateral that meets the given conditions. First make a conjecture and then verify your conjecture by drawing a figure. 38. two lines of symmetry perpendicular to the sides and order-2 rotational symmetry 39. no line symmetry and order-2 rotational symmetry 40. two lines of symmetry through opposite vertices and order-2 rotational symmetry 41. four lines of symmetry and order-4 rotational symmetry 42. one line of symmetry through a pair of opposite vertices and no rotational symmetry 43. Physics High-speed photography makes it possible to analyze the physics behind a water splash. When a drop lands in a bowl of liquid, the splash forms a crown of evenly spaced points. What is the angle of rotational symmetry for a crown with 24 points? 44. Critical Thinking What can you conclude about a rectangle that has four lines of symmetry? Explain. 45. Geography The Isle of Man is an island in the Irish Sea. The island’s symbol is a triskelion that consists of three running legs radiating from the center. Describe the symmetry of the triskelion. 46. Critical Thinking Draw several examples of figures that have two perpendicular lines of symmetry. What other type of symmetry do these figures have? Make a conjecture based on your observation. Each figure shows part of a shape with a center of rotation and a given rotational symmetry. Copy and complete each figure. 47. order 4 48. order 6 49. order 2 50. Write About It Explain the connection between the angle of rotational symmetry and the order of the rotational symmetry. That is, if you know one of these, explain how you can find the other. 12-5 Symmetry 861 861 51. What is the order of rotational symmetry for the hexagon shown? 2 3 4 6 52. Which of these figures has exactly four lines of symmetry? Regular octagon Equilateral triangle Isosceles triangle Square 53. Consider the graphs of the following equations. Which graph has the y-axis as a line of symmetry? y = (x - 3x + 3⎟ 54. Donnell designed a garden plot that has rotational symmetry, but not line symmetry. Which of these could be the shape of the plot? CHALLENGE AND EXTEND 55. A regular polygon has an angle of rotational symmetry of 5°. How many sides does the polygon have? 56. How many lines of symmetry does a regular n-gon have if n is even? if n is odd? Explain your reasoning. Find the equation of the line of symmetry for the graph of each function. 57. y = (x + 4) 2 58. y = ⎜x - 2⎟ 59. y = 3x 2 + 5 Give the number of axes of symmetry for each regular polyhedron. Describe all axes of symmetry. 60. cube 61. tetrahedron 62. octahedron SPIRAL REVIEW 63. Shari worked 16 hours last week and earned $197.12. The amount she earns in one week is directly proportional to the number of hours she works in that week. If Shari works 20 hours one week, how much does she earn? (Previous course) Find the slant height of each figure. (Lesson 10-5) 64. a right cone with radius 5 in. and surface area 61π in 2 65. a square pyramid with lateral area 45 cm 2 and surface area 65.25 cm 2 66. a regular triangular pyramid with base perimeter 24 √  3 m and surface area 120 √  3 m 2 Determine the coordinates of the final image of the point P (-1, 4) under each composition of isometries. (Lesson 12-4) 67. Reflect point P across the line y = x and then translate it along the vector 〈2, -4〉. 68. Rotate point P by 90° about the origin and then reflect it across the y-axis. 69. Translate point P along the vector 〈1, 0〉 and then rotate it 180° about the origin. 862 862 Chapter 12 Extending Transformational Geometry 12-6 Tessellations TEKS G.5.C Geometric patterns: use ... compositions [of transformations] to make connections between mathematics and the real world .... Objectives Use transformations to draw tessellations. Identify regular and semiregular tessellations and figures that will tessellate. Vocabulary translation symmetry frieze pattern glide reflection symmetry tessellation regular tessellation semiregular tessellation Who uses this? Repeating patterns play an important role in traditional Native American art. A pattern has translation symmetry if it can be translated along a vector so that the image coincides with the preimage. A frieze pattern is a pattern that has translation symmetry along a line. Both of the frieze patterns shown below have translation symmetry. The pattern on the right also has glide reflection symmetry. A pattern with glide reflection symmetry coincides with its image after a glide reflection. E X A M P L E 1 Art Application Identify the symmetry in each frieze pattern. A B When you are given a frieze pattern, you may assume that the pattern continues forever in both directions. translation symmetry and glide reflection symmetry translation symmetry Identify the symmetry in each frieze pattern. 1a. 1b. A tessellation , or tiling, is a repeating pattern that completely covers a plane with no gaps or overlaps. The measures of the angles that meet at each vertex must add up to 360°. In the tessellation shown, each angle of the quadrilateral occurs once at each vertex. Because the angle measures of any quadrilateral add to 360°, any quadrilateral can be used to tessellate the plane. Four copies of the quadrilateral meet at each vertex. 12-6 Tessellations 863 863 The angle measures of any triangle add up to 180°. This means that any triangle can be used to tessellate a plane. Six copies of the triangle meet at each vertex, as shown. m∠1 + m∠2 + m∠3 = 180° m∠1 + m∠2 + m∠3 + m∠1 + m∠2 + m∠3 = 360° E X A M P L E 2 Using Transformations to Create Tessellations Copy the given figure and use it to create a tessellation. Step 1 Rotate the triangle 180° about the midpoint of one side. Step 2 Translate the resulting pair of triangles to make a row of triangles. Step 3 Translate the row of triangles to make a tessellation. A B Step 1 Rotate the quadrilateral 180° about the midpoint of one side. Step 2 Translate the resulting pair of quadrilaterals to make a row of quadrilaterals. Step 3 Translate the row of quadrilaterals to make a tessellation. 2. Copy the given figure and use it to create a tessellation. A regular tessellation is formed by congruent regular polygons. A semiregular tessellation is formed by two or more different regular polygons, with the same number of each polygon occurring in the same order at every vertex. Every vertex has two squares and three triangles in this order: square, triangle, square, triangle, triangle. 864 864 Chapter 12 Extending Transformational Geometry �� Tessellations When I need to decide if given figures can be used to tessellate a plane, I look at angle measures. To form a regular tessellation, the angle measures of a regular polygon must be a divisor of 360°. To form a semiregular tessellation, the angle measures around a vertex must add up to 360°. For example, regular octagons and equilateral triangles cannot be used to make a semiregular tessellation because no combination of 135° and 60° adds up to exactly 360°. Ryan Gray Sunset High School E X A M P L E 3 Classifying Tessellations Classify each tessellation as regular, semiregular, or neither. A B C Two regular octagons and one square meet at each vertex. The tessellation is semiregular. Only squares are used. The tessellation is regular. Irregular hexagons are used in the tessellation. It is neither regular nor semiregular. Classify each tessellation as regular, semiregular, or neither. 3a. 3b. 3c. E X A M P L E 4 Determining Whether Polygons Will Tessellate Determine whether the given regular polygon(s) can be used to form a tessellation. If so, draw the tessellation. A B No; each angle of the pentagon measures 108°, and 108 is not a divisor of 360. Yes; two octagons and one square meet at each vertex.
135° + 135° + 90° = 360° Determine whether the given regular polygon(s) can be used to form a tessellation. If so, draw the tessellation. 4a. 4b. 12-6 Tessellations 865 865 �� THINK AND DISCUSS 1. Explain how you can identify a frieze pattern that has glide reflection symmetry. 2. Is it possible to tessellate a plane using circles? Why or why not? 3. GET ORGANIZED Copy and complete the graphic organizer. 12-6 Exercises Exercises KEYWORD: MG7 12-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Sketch a pattern that has glide reflection symmetry. 2. Describe a real-world example of a regular tessellation. 863 Transportation The tread of a tire is the part that makes contact with the ground. Various tread patterns help improve traction and increase durability. Identify the symmetry in each tread pattern. 3. 4 Copy the given figure and use it to create a tessellation. p. 864 6. 7. 5. 8 Classify each tessellation as regular, semiregular, or neither. p. 865 9. 10. 11. 865 Determine whether the given regular polygon(s) can be used to form a tessellation. If so, draw the tessellation. 12. 13. 14. 866 866 Chapter 12 Extending Transformational Geometry �� Independent Practice For See Exercises Example 15–17 18–20 21–23 24–26 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S27 Application Practice p. S39 PRACTICE AND PROBLEM SOLVING Interior Decorating Identify the symmetry in each wallpaper border. 15. 16. Copy the given figure and use it to create a tessellation. 18. 18. 19. 17. 20. Classify each tessellation as regular, semiregular, or neither. 21. 22. 23. Determine whether the given regular polygon(s) can be used to form a tessellation. If so, draw the tessellation. 24. 25. 26. 27. Physics A truck moving down a road creates whirling pockets of air called a vortex train. Use the figure to classify the symmetry of a vortex train. Identify all of the types of symmetry (translation, reflection, and/or rotation) in each tessellation. 28. 29. 30. Tell whether each statement is sometimes, always, or never true. 31. A triangle can be used to tessellate a plane. 32. A frieze pattern has glide reflection symmetry. 33. The angles at a vertex of a tessellation add up to 360°. 34. It is possible to use a regular pentagon to make a regular tessellation. 35. A semiregular tessellation includes scalene triangles. 12-6 Tessellations 867 867 36. This problem will prepare you for the Multi-Step TAKS Prep on page 880. Many of the patterns in M. C. Escher’s works are based on simple tessellations. For example, the pattern at right is based on a tessellation of equilateral triangles. Identify the figure upon which each pattern is based. a. a. b. Use the given figure to draw a frieze pattern with the given symmetry. 37. translation symmetry 38. glide reflection symmetry 39. translation symmetry 40. glide reflection symmetry 41. Optics A kaleidoscope is formed by three mirrors joined to form the lateral surface of a triangular prism. Small objects are reflected in the mirrors to form a tessellation. Copy the triangle and reflect the triangle over each side. Repeat to form a tessellation. Describe the symmetry of the tessellation. 42. Critical Thinking The pattern on a soccer ball is a tessellation of a sphere using regular hexagons and regular pentagons. Can these two shapes be used to tessellate a plane? Explain your reasoning. 43. Chemistry A polymer is a substance made of repeating chemical units or molecules. The repeat unit is the smallest structure that can be repeated to create the chain. Draw the repeat unit for polypropylene, the polymer shown below. 44. The dual of a tessellation is formed by connecting the centers of adjacent polygons with segments. Copy or trace the semiregular tessellation shown and draw its dual. What type of polygon makes up the dual tessellation? 45. Write About It You can make a regular tessellation from an equilateral triangle, a square, or a regular hexagon. Explain why these are the only three regular tessellations that are possible. 868 868 Chapter 12 Extending Transformational Geometry �� 46. Which frieze pattern has glide reflection symmetry? 47. Which shape CANNOT be used to make a regular tessellation? Equilateral triangle Square Regular pentagon Regular hexagon 48. Which pair of regular polygons can be used to make a semiregular tessellation? CHALLENGE AND EXTEND 49. Some shapes can be used to tessellate a plane in more than one way. Three tessellations that use the same rectangle are shown. Draw a parallelogram and draw at least three tessellations using that parallelogram. Determine whether each figure can be used to tessellate three-dimensional space. 50. 51. 52. SPIRAL REVIEW 53. A book is on sale for 15% off the regular price of $8.00. If Harold pays with a $10 bill and receives $2.69 in change, what is the sales tax rate on the book? (Previous course) 54. Louis lives 5 miles from school and jogs at a rate of 6 mph. Andrea lives 3.9 miles from school and jogs at a rate of 6.5 mph. Andrea leaves her house at 7:00 A.M. When should Louis leave his house to arrive at school at the same time as Andrea? (Previous course) Write the equation of each circle. (Lesson 11-7) 55. ⊙P with center (-2, 3) and radius √  5 56. ⊙Q that passes through (3, 4) and has center (0, 0) 57. ⊙T that passes through (1, -1) and has center (5, -3) Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. (Lesson 12-5) 58. 59. 60. 12-6 Tessellations 869 869 12-6 Use Transformations to Extend Tessellations In Lesson 12-6, you saw that you can use any triangle or quadrilateral to tessellate a plane. In this lab, you will learn how to use transformations to turn these basic patterns into more-complex tessellations. Use with Lesson 12-6 TEKS G.5.C Geometric patterns: use properties of transformations and their compositions to make connections between mathematics and the real world .... Activity 1 1 Cut a rectangle out of heavy paper. 2 Cut a piece from one side of the rectangle and translate it to the opposite side. Tape it into place. 3 Repeat the process with the other pair of sides. 4 The resulting shape will tessellate the plane. Trace around the shape to create a tessellation. Try This 1. Repeat Activity 1, starting with a parallelogram. 2. Repeat Activity 1, starting with a hexagon whose opposite sides are congruent and parallel. 3. Add details to one of your tessellations to make it look like a pattern of people, animals, flowers, or other objects. 870 870 Chapter 12 Extending Transformational Geometry Activity 2 1 Cut a triangle out of heavy paper. 2 Find the midpoint of one side. Cut a piece from one half of this side of the triangle and rotate the piece 180°. Tape it to the other half of this side. 3 Repeat the process with the other two sides. 4 The resulting shape will tessellate the plane. Trace around the shape to create a tessellation. Try This 4. Repeat Activity 2, starting with a quadrilateral. 5. How is this tessellation different from the ones you created in Activity 1? 6. Add details to one of your tessellations to make it look like a pattern of people, animals, flowers, or other objects. 12-6 Geometry Lab 871 871 12-7 Dilations TEKS G.11.A Similarity and the geometry of shape: use and extend similarity properties and transformations .... Objective Identify and draw dilations. Vocabulary center of dilation enlargement reduction Also G.2.A, G.11.B, G.11.D Who uses this? Artists use dilations to turn sketches into large-scale paintings. (See Example 2.) Recall that a dilation is a transformation that changes the size of a figure but not the shape. The image and the preimage of a figure under a dilation are similar. E X A M P L E 1 Identifying Dilations For a dilation with scale factor k, if k > 0, the figure is not turned or flipped. If k < 0, the figure is rotated by 180°. Tell whether each transformation appears to be a dilation. Explain. A B Yes; the figures are similar,and the image is not turned or flipped. No; the figures are not similar. Tell whether each transformation appears to be a dilation. 1a. 1b. Construction Dilate a Figure by a Scale Factor of 2     Draw a triangle and a point outside the triangle. The point is the center of dilation. Use a straightedge to draw a line through the center of dilation and each vertex of the triangle. Set the compass to the distance from the center of dilation to a vertex. Mark this distance along the line for each vertex as shown. Connect the vertices of the image. In the construction, the lines connecting points of the image with the corresponding points of the preimage all intersect at the center of dilation. Also, the distance from the center to each point of the image is twice the distance to the corresponding point of the preimage. 872 872 Chapter 12 Extending Transformational Geometry Dilations is the same for every point P. A dilation, or similarity transformation, is a transformation in which the lines connecting every point P with its image P' all intersect at a point C, called the center of dilation . CP' ___ CP The scale factor k of a dilation is the ratio of a linear measurement of the image to a corresponding measurement of the preimage. In the figure, k = P'Q' ___ . PQ A dilation enlarges or reduces all dimensions proportionally. A dilation with a scale factor greater than 1 is an enlargement , or expansion. A dilation with a scale factor greater than 0 but less than 1 is a reduction , or contraction. E X A M P L E 2 Drawing Dilations Copy the triangle and the center of dilation P. Draw the image of △ABC under a dilation with a scale factor of 1 __ . 2 Step 1 Draw a line through P and each vertex. Step 2 On each line, mark half the distance from P to the vertex. Step 3 Connect the vertices of the ima
ge. 2. Copy the figure and the center of dilation. Draw the dilation of RSTU using center Q and a scale factor of 3. E X A M P L E 3 Art Application An artist is creating a large painting from a photograph by dividing the photograph into squares and dilating each square by a scale factor of 4. If the photograph is 20 cm by 25 cm, what is the perimeter of the painting? The scale factor of the dilation is 4, so a 1 cm by 1 cm square on the photograph represents a 4 cm by 4 cm square on the painting. Find the dimensions of the painting. b = 4 (25) = 100 cm h = 4 (20) = 80 cm Multiply each dimension by the scale factor, 4. Find the perimeter of the painting. P = 2 (100 + 80) = 360 cm P = 2 (b + h) 3. What if…? In Example 3, suppose the photograph is a square with sides of length 10 in. Find the area of the painting. 12-7 Dilations 873 873 �� Dilations in the Coordinate Plane If P (x, y) is the preimage of a point under a dilation centered at the origin with scale factor k, then the image of the point is P' (kx, ky) . If the scale factor of a dilation is negative, the preimage is rotated by 180°. For k > 0, a dilation with a scale factor of -k is equivalent to the composition of a dilation with a scale factor of k that is rotated 180° about the center of dilation. E X A M P L E 4 Drawing Dilations in the Coordinate Plane Draw the image of a triangle with vertices A (-1, 1) , B (-2, -1) , and C (-1, -2) under a dilation with a scale factor of -2 centered at the origin. The dilation of (x, y) is (-2x, -2y) . A (-1, 1) → A' (-2 (-1) , -2 (1) ) = A' (2, -2) B (-2, -1) → B' (-2 (-2) , -2 (-1) ) = B' (4, 2) C (-1, -2) → C' (-2 (-1) , -2 (-2) ) = C' (2, 4) Graph the preimage and image. 4. Draw the image of a parallelogram with vertices R (0, 0) , S (4, 0) , T (2, -2) , and U (-2, -2) under a dilation centered at the origin with a scale factor of - 1 __ 2 . THINK AND DISCUSS 1. Given a triangle and its image under a dilation, explain how you could use a ruler to find the scale factor of the dilation. 2. A figure is dilated by a scale factor of k, and then the image is rotated 180° about the center of dilation. What single transformation would produce the same image? 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe the dilation with the given scale factor. 874 874 Chapter 12 Extending Transformational Geometry �� 12-7 Exercises Exercises KEYWORD: MG7 12-7 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary What are the center of dilation and scale factor for the transformation (x, y) → (3x, 3y Tell whether each transformation appears to be a dilation. p. 872 2. 4. 3. 5. 873 Copy each triangle and center of dilation P. Draw the image of the triangle under a dilation with the given scale factor. 6. Scale factor: 2 7. Scale factor: 1 __ . 873 8. Architecture A blueprint shows a reduction of a room using a scale factor of 1 __ 50 . In the blueprint, the room’s length is 8 in., and its width is 6 in. Find the perimeter of the room. 874 Draw the image of the figure with the given vertices under a dilation with the given scale factor centered at the origin. 9. A (1, 0) , B (2, 2) , C (4, 0) ; scale factor: 2 10. J (-2, 2) , K (4, 2) , L (4, -2) , M (-2, -2) ; scale factor: 1 _ 2 11. D (-3, 3) , E (3, 6) , F (3, 0) ; scale factor: - 1 _ 3 12. P (-2, 0) , Q (-1, 0) , R (0, -1) , S (-3, -1) ; scale factor: -2 PRACTICE AND PROBLEM SOLVING Tell whether each transformation appears to be a dilation. 13. Independent Practice For See Exercises Example 13–16 17–18 19 20–23 1 2 3 4 TEKS TEKS TAKS TAKS 15. Skills Practice p. S27 Application Practice p. S39 14. 16. 12-7 Dilations 875 875 �� Copy each rectangle and the center of dilation P. Draw the image of the rectangle Copy each rectangle and the center of dilation under a dilation with the given scale factor. under a dilation with the given scale factor. 17. 17. scale factor: 3 18. scale factor: 1 __ 2 Mosaics 19. Art Jeff is making a mosaic by gluing 1 cm square tiles onto a photograph. He starts with a 6 cm by 8 cm rectangular photo and enlarges it by a scale factor of 1.5. How many tiles will Jeff need in order to cover the enlarged photo? This mosaic of the seal of the Republic of Texas is one of six tile mosaics that were installed on the front façade of the Sam Houston Regional Library and Research Center in Liberty, Texas, in fall 2001. Draw the image of the figure with the given vertices under a dilation with the given scale factor centered at the origin. 20. M (0, 3) , N (6, 0) , P (0, -3) ; scale factor: - 1_ 3 21. A (-1, 3) , B (1, 1) , C (-4, 1) ; scale factor: -1 22. R (1, 0) , S (2, 0) , T (2, -2) , U (-1, -2) ; scale factor: -2 23. D (4, 0) , E (2, -4) , F (-2, -4) , G (-4, 0) , H (-2, 4) , J (2, 4) ; scale factor: - 1 _ 2 Each figure shows the preimage (blue) and image (red) under a dilation. Write a similarity statement based on the figure. 24. 25. 26. The rectangular prism shown is enlarged by a dilation with scale factor 4. Find the surface area and volume of the image. Copy each figure and locate the center of dilation. 27. 28. 29. 30. This problem will prepare you for the Multi-Step TAKS Prep on page 880. This lithograph, Drawing Hands, was made by M. C. Escher in 1948. a. In the original drawing, the rectangular piece of paper from which the hands emerge measures 27.6 cm by 19.9 cm. On a poster of the drawing, the paper is 82.8 cm long. What is the scale factor of the dilation that was used to make the poster? b. What is the area of the paper on the poster? 876 876 Chapter 12 Extending Transformational Geometry �� 31. /////ERROR ANALYSIS///// Rectangle A'B'C'D' is the image of rectangle ABCD under a dilation. Which calculation of the area of rectangle A'B'C'D' is incorrect? Explain the error. 32. Optometry The pupil is the circular opening that allows light into the eye. a. An optometrist dilates a patient’s pupil from 6 mm to 8 mm. What is the scale factor for this dilation? b. To the nearest tenth, find the area of the pupil before and after the dilation. c. As a percentage, how much more light is admitted to the eye after the dilation? 33. Estimation In the diagram, △ABC → △A'B'C' under a dilation with center P. a. Estimate the scale factor of the dilation. b. Explain how you can use a ruler to make measurements and to calculate the scale factor. c. Use the method from part b to calculate the scale factor. How does your result compare to your estimate? 34. △ABC has vertices A (-1, 1) , B (2, 1) , and C (2, 2) . a. Draw the image of △ABC under a dilation centered at the origin with scale factor 2 followed by a reflection across the x-axis. b. Draw the image of △ABC under a reflection across the x-axis followed by a dilation centered at the origin with scale factor 2. c. Compare the results of parts a and b. Does the order of the transformations matter? 35. Astronomy The image of the sun projected through the hole of a pinhole camera (the center of dilation) has a diameter of 1 __ 4 in. The diameter of the sun is 870,000 mi. What is the scale factor of the dilation? 12-7 Dilations 877 877 �� Multi-Step △ABC with vertices A (-2, 2) , B (1, 3) , and C (1, -1) is transformed by a dilation centered at the origin. For each given image point, find the scale factor of the dilation and the coordinates of the remaining image points. Graph the preimage and image on a coordinate plane. 36. A' (-4, 4) 38. B' (-1, -3) 37. C' (-2, 2) 39. Critical Thinking For what values of the scale factor is the image of a dilation congruent to the preimage? Explain. 40. Write About It When is a dilation equivalent to a rotation by 180°? Why? 41. Write About It Is the composition of a dilation with scale factor m followed by a dilation with scale factor n equivalent to a single dilation with scale factor mn? Explain your reasoning. Construction Copy each figure. Then use a compass and straightedge to construct the dilation of the figure with the given scale factor and point P as the center of dilation. 42. scale factor: 1 _ 2 43. scale factor: 2 44. scale factor: -1 45. scale factor: -2 46. Rectangle ABCD is transformed by a dilation centered at the origin. Which scale factor produces an image that has a vertex at (0, -2) ? - 1 _ 2 -1 -2 -4 47. Rectangle ABCD is enlarged under a dilation centered at the origin with scale factor 2.5. What is the perimeter of the image? 15 24 30 50 48. Gridded Response What is the scale factor of a dilation centered at the origin that maps the point (-2, 3) to the point (-8.4, 12.6) ? 49. Short Response The rules for a photo contest state that entries must have an area no greater than 100 cm 2 . Amber has a 6 cm by 8 cm digital photo, and she uses software to enlarge it by a scale factor of 1.5. Does the enlargement meet the requirements of the contest? Show the steps you used to decide your answer. 878 878 Chapter 12 Extending Transformational Geometry �� CHALLENGE AND EXTEND 50. Rectangle ABCD has vertices A (0, 2) , B (1, 2) , C (1, 0) , and D (0, 0) . a. Draw the image of ABCD under a dilation centered at point P with scale factor 2. b. Describe the dilation in part a as a composition of a dilation centered at the origin followed by a translation. c. Explain how a dilation with scale factor k and center of dilation (a, b) can be written as a composition of a dilation centered at the origin and a translation. 51. The equation of line ℓ is y = -x + 2.
Find the equation of the image of line ℓ after a dilation centered at the origin with scale factor 3. SPIRAL REVIEW 52. Jerry has a part-time job waiting tables. He kept records comparing the number of customers served to his total amount of tips for the day. If this trend continues, how many customers would he need to serve in order to make $68.00 in tips for the day? (Previous course) Customers per Day Tips per Day ($) 15 20 20 28 25 36 30 44 Find the perimeter and area of each polygon with the given vertices. (Lesson 9-4) 53. J (-3, -2) , K (0, 2) , L (7, 2) , and M (4, -2) 54. D (-3, 0) , E (1, 2) , and F (-1, -4) Determine whether the polygons can be used to tessellate a plane. (Lesson 12-6) 55. a right triangle and a square 56. a regular nonagon and an equilateral triangle Using Technology Use a graphing calculator to complete the following. 1. △ABC with vertices A (3, 4) , B (5, 2) , and C (1, 1) can be represented ⎡ 3 ⎢ by the point matrix ⎣ 4 5 2 ⎤ 1 . Enter these values into matrix [B] on ⎥ ⎦ 1 your calculator. (See page 846.) ⎤ 0 can be used to perform a dilation with scale factor ⎥ ⎦ 2 ⎡ 2 2. The matrix ⎢ ⎣ 0 2. Enter these values into matrix [A] on your calculator and find [A] * [B]. Graph the triangle represented by the resulting point matrix. 3. Make a conjecture about the matrix that could be used to perform a dilation with scale factor − 1 __ 2 . Enter the values into matrix [A] on your calculator. 4. Test your conjecture by finding [A] * [B] and graphing the triangle represented by the resulting point matrix. 12-7 Dilations 879 879 �� SECTION 12B Patterns Tessellation Fascination A museum is planning an exhibition of works by the Dutch artist M. C. Escher (1898– 1972). The exhibit will include the five drawings shown here. 1. Tell whether each drawing has parallel lines of symmetry, intersecting lines of symmetry, or no lines of symmetry. 2. Tell whether each drawing has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. 3. Tell whether each drawing is a tessellation. If so, identify the basic figure upon which the tessellation is based. Drawing A Drawing B Drawing C Drawing D 4. The entrance to the exhibit will include a large mural based on drawing E. In the original drawing, the cover of the book measures 13.2 cm by 11.1 cm. In the mural, the book cover will have an area of 21,098.88 cm 2 . What is the scale factor of the dilation that will be used to make the mural? Drawing E 880 880 Chapter 12 Extending Transformational Geometry SECTION 12B Quiz for Lessons 12-5 Through 12-7 12-5 Symmetry Explain whether each figure has line symmetry. If so, copy the figure and draw all lines of symmetry. 1. 2. 3. Explain whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. 4. 5. 6. 12-6 Tessellations Copy the given figure and use it to create a tessellation. 7. 8. Classify each tessellation as regular, semiregular, or neither. 10. 11. 9. 12. 13. Determine whether it is possible to tessellate a plane with regular octagons. If so, draw the tessellation. If not, explain why. 12-7 Dilations Tell whether each transformation appears to be a dilation. 14. 15. 16. Draw the image of the figure with the given vertices under a dilation with the given scale factor centered at the origin. 17. A (0, 2) , B (-1, 0) , C (0, -1) , D (1, 0) ; scale factor: 2 18. P (-4, -2) , Q (0, -2) , R (0, 0) , S (-4, 0) ; scale factor: - 1 _ 2 Ready to Go On? 881 881 EXTENSION EXTENSION Using Patterns to Generate Fractals TEKS G.5.C Geometric patterns: use properties of transformations and their compositions to make connections between mathematics and the real world .... Objective Describe iterative patterns that generate fractals. Vocabulary self-similar iteration fractal Look closely at one of the large spirals in the Romanesco broccoli. You will notice that it is composed of many smaller spirals, each of which has the same shape as the large one. This is an example of self-similarity. A figure is self-similar if it can be divided into parts that are similar to the entire figure. You can draw self-similar figures by iteration , the repeated application of a rule. To create a self-similar tree, start with the figure shown in stage 0. Replace each of its branches with the original figure to get the figure in stage 1. Again replace the branches with the original figure to get the figure in stage 2. Continue the pattern to generate the tree. Stage 0 Stage 1 Stage 2 Stage 3 Stage 8 A figure that is generated by iteration is called a fractal . E X A M P L E 1 Creating Fractals Continue the pattern to draw stages 3 and 4 of this fractal, which is called the Sierpinski triangle. To go from one stage to the next, remove an equilateral triangle from each remaining black triangle. Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 1. Explain how to go from one stage to the next to create the Koch snowflake fractal. 882 882 Chapter 12 Extending Transformational Geometry Stage 0 Stage 1 Stage 2 Stage 3 EXTENSION Exercises Exercises Explain how to go from one stage to the next to generate each fractal. 1. Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 Stage 10 2. Stage 0 Stage 1 Stage 2 Stage 3 3. The three-dimensional figure in the photo is called a Sierpinski tetrahedron. a. Describe stage 0 for this fractal. b. Explain how to go from one stage to the next to generate the Sierpinski tetrahedron. 4. A fractal is generated according to the following rules. Stage 0 is a segment. To go from one stage to the next, replace each segment with the figure at right. Draw Stage 2 of this fractal. 5. The first four rows of Pascal’s triangle are shown in the hexagonal tessellation at right. The beginning and end of each row is a 1. To find each remaining number, add the two numbers to the left and right from the row above. a. Continue the pattern to write the first eight rows of Pascal’s triangle. b. Shade all the hexagons that contain an odd number. c. What fractal does the resulting pattern resemble? 6. Write About It Explain why the fern leaf at right is an example of self-similarity. Chapter 12 Extension 883 883 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary center of dilation . . . . . . . . . . . . . . . . . . . . . . . . . 873 line of symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 856 composition of transformations . . . . . . . . . . . 848 reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873 enlargement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873 regular tessellation . . . . . . . . . . . . . . . . . . . . . . . 864 frieze pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 rotational symmetry . . . . . . . . . . . . . . . . . . . . . . 857 glide reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . 848 semiregular tessellation . . . . . . . . . . . . . . . . . . . 864 glide reflection symmetry . . . . . . . . . . . . . . . . . 863 symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856 isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824 tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 line symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856 translation symmetry . . . . . . . . . . . . . . . . . . . . . 863 Complete the sentences below with vocabulary words from the list above. 1. A(n) ? is a pattern formed by congruent regular polygons. ̶̶̶̶ 2. A pattern that has translation symmetry along a line is called a(n) ? . ̶̶̶̶ 3. A transformation that does not change the size or shape of a figure is a(n) ? . ̶̶̶̶ 4. One transformation followed by another is called a(n) ? . ̶̶̶̶ 12-1 Reflections (pp. 824–830) TEKS G.2.A, G.2.B, G.7.A, G.10.A E X A M P L E EXERCISES ■ Reflect the figure with the given vertices across the given line. A (1, -2) , B (4, -3) , C (3, 0) ; y = x To reflect across the line y = x, interchange the x- and y-coordinates of each point. The images of the vertices are A' (-2, 1) , B' (-3, 4) , and C' (0, 3) . Tell whether each transformation appears to be a reflection. 5. 6. 7. 8. Reflect the figure with the given vertices across the given line. 9. E (-3, 2) , F (0, 2) , G (-2, 5) ; x-axis 10. J (2, -1) , K (4, -2) , L (4, -3) , M (2, -3) ; y-axis 11. P (2, -2) , Q (4, -2) , R (3, -4) ; y = x 12. A (2, 2) , B (-2, 2) , C (-1, 4) ; y = x 884 884 Chapter 12 Extending Transformational Geometry �� 12-2 Translations (pp. 831–837) TEKS G.2.A, G.2.B, G.7.A, G.10.A E X A M P L E EXERCISES ■ Translate the figure with the given vertices along the given vector. D (-4, 4) , E (-4, 2) , F (-1, 1) , G (-2, 3) ; 〈5, -5〉 To translate along 〈5, -5〉, add 5 to the x-coordinate of each point and add -5 to the y-coordinate of each point. The vertices of the image are D' (1, -1) , E' (1, -3) , F' (4, -4) , and G' (3, -2) . Tell whether each transformation appears to be a translation. 13. 14. 15. 16. Translate the figure with the given vertices along the given vector. 17. R (1, -1) , S (1, -3) , T (4, -3) , U (4, -1) ; 〈-5, 2〉 18. A (-4, -1) , B (-3, 2) , C (-1, -2) ; 〈6, 0〉 19. M (1, 4) , N (4, 4) , P (3, 1) ; 〈-3, -3〉 20. D (3, 1) , E (2, -2) , F (3, -4) , G (4, -2) ; 〈-6, 2〉 12-3 Rotations (pp. 839–845) TEKS G.2.A, G.2.B, G.7.A, G.10.A E X A M P L E EXERCISES ■ Rotate the figure with the given vertices about the origin using the given angle of rotation. A (-2, 0) , B (-1, 3) , C (-4, 3) ; 180° To rotate by 180°, find the opposite of the x- and y-coordinate of each point. The vertices of the image are A' (2, 0) , B' (1, -3) , and C' (4, -3) . Tell whether each transformation appears to be a rotation. 21. 22. 23. 24. Rotate the figure with the given vertices about the origin using the given angle of rotation. 25. A (1, 3) , B (4, 1) , C (4, 4) ; 90° 26. A (1, 3) , B (4, 1) , C (4, 4) ; 180° 27. M (2, 2
) , N (5, 2) , P (3, -2) , Q (0, -2) ; 90° 28. G (-2, 1) , H (-3, -2) , J (-1, -4) ; 180° Study Guide: Review 885 885 �� 12-4 Compositions of Transformations (pp. 848–853) TEKS G.5.C, G.10.A E X A M P L E EXERCISES ■ Draw the result of the composition of isometries. Translate △MNP along across line ℓ. v and then reflect it Draw the result of the composition of isometries. 29. Translate ABCD along v and then reflect it across line m. First draw △M'N'P', the translation image of △MNP. Then reflect △M'N'P' across line ℓ to find the final image, △M''N''P''. 30. Reflect △JKL across line m and then rotate it 90° about point P. 12-5 Symmetry (pp. 856–862) TEKS G.2.B, G.5.C, G.10.A E X A M P L E S EXERCISES Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. Tell whether each figure has line symmetry. If so, copy the figure and draw all lines of symmetry. 31. 32. ■ no rotational symmetry ■ Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of symmetry. 33. 34. The figure coincides with itself when it is rotated by 90°. Therefore the angle of rotational symmetry is 90°. The order of symmetry is 4. 35. 36. 886 886 Chapter 12 Extending Transformational Geometry �� 12-6 Tessellations (pp. 863–869) TEKS G.5.C E X A M P L E S EXERCISES ■ Copy the given figure and use it to create a tessellation. Rotate the quadrilateral 180° about the midpoint of one side. Copy the given figure and use it to create a tessellation. 37. 38. Translate the resulting pair of quadrilaterals to make a row. 39. 40. Translate the row to make a tessellation. Classify each tessellation as regular, semiregular, or neither. 41. ■ Classify the tessellation as regular, semiregular, or neither. The tessellation is made of two different regular polygons, and each vertex has the same polygons in the same order. Therefore the tessellation is semiregular. 42. 12-7 Dilations (pp. 872–879) TEKS G.2.A, G.11.A, G.11.B, G.11.D E X A M P L E EXERCISES ■ Draw the image of the figure with the given vertices under a dilation centered at the origin using the given scale factor. A (0, -2) , B (2, -2) , C (2, 0) ; scale factor: 2 Tell whether each transformation appears to be a dilation. 43. 44. Multiply the x- and y-coordinates of each point by 2. The vertices of the image are A' (0, -4) , B' (4, -4) , and C' (4, 0) . Draw the image of the figure with the given vertices under a dilation centered at the origin using the given scale factor. 45. R (0, 0) , S (4, 4) , T (4, -4) ; scale factor: - 1_ 2 46. D (0, 2) , E (-2, 2) , F (-2, 0) ; scale factor: -2 Study Guide: Review 887 887 �� Tell whether each transformation appears to be a reflection. 1. 2. Tell whether each transformation appears to be a translation. 3. 4. 5. An interior designer is using a coordinate grid to place furniture in a room. The position of a sofa is represented by a rectangle with vertices (1, 3) , (2, 2) , (5, 5) , and (4, 6) . He decides to move the sofa by translating it along the vector 〈-1, -1〉. Draw the sofa in its final position. Tell whether each transformation appears to be a rotation. 6. 7. 8. Rotate rectangle DEFG with vertices D (1, -1) , E (4, -1) , F (4, -3) , and G (1, -3) about the origin by 180°. 9. Rectangle ABCD with vertices A (3, -1) , B (3, -2) , C (1, -2) , and D (1, -1) is reflected across the x-axis, and then its image is reflected across the line y = 4. Describe a single transformation that moves the rectangle from its starting position to its final position. 10. Tell whether the “no entry” sign has line symmetry. If so, copy the sign and draw all lines of symmetry. 11. Tell whether the “no entry” sign has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. Copy the given figure and use it to create a tessellation. 12. 13. 14. 15. Classify the tessellation shown as regular, semiregular, or neither. Tell whether each transformation appears to be a dilation. 16. 17. 18. Draw the image of △ABC with vertices A (2, -1) , B (1, -4) , and C (4, -4) under a dilation centered at the origin with scale factor - 1 __ 2 . 888 888 Chapter 12 Extending Transformational Geometry FOCUS ON ACT No question on the ACT Mathematics Test requires the use of a calculator, but you may bring certain types of calculators to the test. Check www.actstudent.org for a descriptive list of calculators that are prohibited or allowed with slight modifications. You may want to time yourself as you take this practice test. It should take you about 5 minutes to complete. If you are not sure how to solve a problem, looking through the answer choices may provide you with a clue to the solution method. It may take longer to work backward from the answers provided, so make sure you are monitoring your time. 1. Which of the following functions has a graph that is symmetric with respect to the y-axis? (A) f (x) = x 4 - 2 (B) f (x) = (x + 2) 4 (C) f (x) = 2x - 4 (D) f (x) = x 2 + 4x (E) f (x) = (x - 4) 2 2. What is the image of the point (-4, 5) after the translation that maps the point (1, -3) to the point (-1, -7) ? (F) (4, 1) (G) (-6, 1) (H) (-8, 3) (-2, 9) (J) (K) (0, 7) 3. When the point (-2, -5) is reflected across the x-axis, what is the resulting image? (A) (-5, -2) (B) (2, 5) (C) (2, -5) (D) (-2, 5) (E) (5, 2) 4. After a composition of transformations, the line segment from A (1, 4) to B (4, 2) maps to the line segment from C (-1, -2) to D (-4, -4) . Which of the following describes the composition that is applied to ̶̶ AB to obtain ̶̶ CD ? (F) Translate 5 units to the left and then reflect across the y-axis. (G) Reflect across the y-axis and then reflect across the x-axis. (H) Reflect across the y-axis and then translate 6 units down. (J) Reflect across the x-axis and then reflect across the y-axis. (K) Translate 6 units down and then reflect across the x-axis. 5. What is the image of the following figure after rotating it counterclockwise by 270°? (A) (B) (C) (D) (E) College Entrance Exam Practice 889 889 Any Question Type: Highlight Main Ideas Before answering a test item, identify the important information given in the problem and make sure you clearly identify the question being asked. Outlining the question or breaking a problem into parts can help you to understand the main idea. A common error in answering multi-step questions is to complete only the first step. In multiple-choice questions, partial answers are often used as the incorrect answer choices. If you start by outlining all steps needed to solve the problem, you are less likely to choose one of these incorrect answers. Gridded Response A blueprint shows a rectangular building’s layout reduced using a scale factor of 1 __ 30 . On the blueprint, the building’s width is 15 in. and its length is 7 in. Find the area of the actual building in square feet. What are you asked to find? the area of the actual building in square feet List the given information you need to solve the problem. The scale factor is 1 _ . 30 On the blueprint, the width is 15 in. and the length is 7 in. The area of the building is 656.25 square feet. Multiple Choice An animator uses a coordinate plane to show the motion of a flying bird. The bird begins at the point (12, 0) and is then rotated about the origin by 15° every 0.005 second. Give the bird’s position after 0.015 second. Round the coordinates to the nearest tenth. (8.49, 8.49) (-12, 0) (0, 12) (-8.49, 8.49) What are you asked to find? the coordinates of the bird’s position after 0.015 seconds, to the nearest tenth What information are you given? the initial position of the bird and the angle of rotation for every 0.005 second The correct answer is A. 890 890 Chapter 12 Extending Transformational Geometry �� Sometimes important information is given in a diagram. Read each test item and answer the questions that follow. Item A Multiple Choice Jonas is using a coordinate plane to plan an archaeological dig. He outlines a rectangle with vertices at (5, 2) , (5, 9) , (10, 9) , and (10, 2) . Then he outlines a second rectangle by reflecting the first area across the x-axis and then across the y-axis. Which is a vertex of the second outlined rectangle? (-5, 2) (-5, -9) (-2, -10) (10, -9) 1. Identify the sentence that gives the information regarding the coordinates of the initial rectangle. 2. What are you being asked to do? 3. How many transformations does Jonas perform before he sketches the second rectangle? Which sentence leads you to this answer? 4. A student incorrectly marked choice A as her response. What part of the test item did she fail to complete? Item B Gridded Response Gabby has a digital photo with dimensions 3.5 in. by 5 in., and she uses software to enlarge it by a scale factor of 5. How large must a frame be, in square inches, in order for the enlarged photo to fit? 5. Make a list stating the information given and what you are being asked to do. 6. Are there any intermediate steps you have to take to obtain a solution for the problem? If so, describe the steps. Item C Gridded Response Rectangle A'B'C'D' is the image of rectangle ABCD under a dilation. Once you have identified the scale factor, determine the area of rectangle A'B'C'D'. 7. What do you need to find before you can find the area of the rectangle? 8. Where in the test item can you find the important information (data) needed to solve the problem? Make a list of this information. Item D Multiple Choice △ABC is reflected across the x-axis. Then its image is rotated 180° about the origin. What are the coordinates of the image of point B after the reflection? (-4, -1) (-1, 4) (1, -4) (4, -1) 9. Identify the transformations described in the problem statement. 10. What are you being asked to do? 11. Identify any part of the problem statement that you will not
use to answer the question. 12. There are only two pieces of information given in this test item that are important to answering this question. What are they? TAKS Tackler 891 891 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–12 Multiple Choice 1. Which of the following best represents the area of the shaded figure if each square in the grid has a side length of 1 centimeter? 5. Marty conjectures that the sum of any two prime numbers is even. Which of the following is a counterexample that shows Marty’s conjecture is false = 11 3 + 5 = 8 Use the graph for Items 6–8. 17 square centimeters 21 square centimeters 25 square centimeters 29 square centimeters 2. Which of the following expressions represents the number of edges of a polyhedron with n vertices and n faces? n - 2 2n - 1 2 (n - 1) 2 (n + 1) 3. The image of point A under a 90° rotation about the origin is A' (10, -4) . What are the coordinates of point A? 6. What are the coordinates of the image of point C under the same translation that maps point D to point B? (4, 4) (0, 4) (0, 8) (4, -8) 7. △PQR is the image of a triangle under a dilation centered at the origin with scale factor - 1 __ . Which 2 point is a vertex of the preimage of △PQR under this dilation? (-10, -4) (-10, 4) (-4, -10) (4, 10) A B C D 4. A cylinder has a volume of 24 cubic centimeters. The height of a cone with the same radius is two times the height of the cylinder. What is the volume of the cone? 8 cubic centimeters 12 cubic centimeters 16 cubic centimeters 48 cubic centimeters 8. What is the measure of ∠PRQ? Round to the nearest degree. 63° 127° 117° 45° 9. Which mapping represents a rotation of 270° about the origin? (x, y) → (-x, -y) (x, y) → (x, -y) (x, y) → (-y, -x) (x, y) → (y, -x) 892 892 Chapter 12 Extending Transformational Geometry �� When problems involve geometric figures in the coordinate plane, it may be useful to describe properties of the figures algebraically. For example, you can use slope to verify that sides of a figure are parallel or perpendicular, or you can use the Distance Formula to find side lengths of the figure. 10. What are the coordinates of the center of the circle (x + 1) 2 + (y + 4) 2 = 4? (-1, -4) (-1, -2) (1, 2) (1, 4) STANDARDIZED TEST PREP Short Response 18. A (-4, -2) , B (-2, -3) , and C (-3, -5) are three of the vertices of rhombus ABCD. Show that ABCD is a square. Justify your answer. 19. ABCD is a square with vertices A (3, -1) , B (3, -3) , C (1, -3) , and D (1, -1) . ⊙P is a circle with equation (x - 2) 2 + (y - 2) 2 = 4. a. What is the center and radius of ⊙P? b. Describe a reflection and dilation of ABCD so that ⊙P is inscribed in the image of ABCD. Justify your answer. 11. Which regular polygon can be used with an equilateral triangle to tessellate a plane? 20. Determine the value of x if △ABC ≅ △BDC. Justify your answer. Heptagon Octagon Nonagon Dodecagon 12. What is the measure of ∠TSV in ⊙P? 24° 42° 45° 48° 13. Given the points B (-1, 2) , C (-7, y) , D (1, -3) , and E (-3, -2) , what is the value of y if ̶̶ BD ǁ ̶̶ CE ? -12 -8 3.5 8 Gridded Response 14. △ABC is a right triangle such that m∠B = 90°. If AC = 12 and BC = 9, what is the perimeter of △ABC? Round to the nearest tenth. 15. A blueprint for an office space uses a scale of 3 inches: 20 feet. What is the area in square inches of the office space on the blueprint if the actual office space has area 1300 square feet? 16. How many lines of symmetry does a regular hexagon have? 17. What is the x-coordinate of the image of the point A (12, -7) if A is reflected across the x-axis? 21. △ABC is reflected across line m. a. What observations can be made about △ABC and its reflected image △A'B'C' regarding the following properties: collinearity, betweenness, angle measure, triangle congruence, and orientation? b. Explain. 22. Given the coordinates of points A, B, and C, explain how you could demonstrate that the three points are collinear. 23. Proving that the diagonals of rectangle KLMN are equal using a coordinate proof involves placement of the rectangle and selection of coordinates. a. Is it possible to always position rectangle KLMN so that one vertex coincides with the origin? b. Why is it convenient to place rectangle KLMN so that one vertex is at the origin? Extended Response 24. ̶̶ AB has endpoints A (0, 3) and B (2, 5) . ̶̶ a. Draw AB and its image, ̶̶̶ A'B' , under the translation 〈0, -8〉. b. Find the equations of two lines such that the composition of the two reflections across the lines will also map work or explain in words how you found your answer. ̶̶̶ A'B' . Show your ̶̶ AB to c. Show that any glide reflection is equivalent to a composition of three reflections. Cumulative Assessment, Chapters 1–12 893 893 �� T E X A S TAKS Grades 9–11 Obj. 10 Point Isabel Lighthouse The Point Isabel Lighthouse was built in 1853 on a prominent bluff on the mainland. Today, the fully restored lighthouse is the only one in Texas that is open for climbing and viewing. Port Isabel Port Isabel ge07ts_c12psl001a Choose one or more strategies to solve each problem. 1st pass 9/06/05 dtrevino 15 miles at sea. To the nearest square mile, what is the area of water covered by the beam as it rotates by an angle of 60°? 1. Suppose the beam from the lighthouse is visible for up to 2. Given that Earth’s radius is approximately 4000 miles and that the top of the tower of a lighthouse is 65 ft above sea level, find the distance from the top of the tower to the horizon. Round your answer to the nearest mile. (Hint: 65 feet = 0.01 miles) For 3, use the table. 3. Most lighthouses use Fresnel lenses, named after their inventor, Augustine Fresnel. The table shows the sizes, or orders, of the circular lenses. The diagram shows some measurements of the Fresnel lens used in the Point Isabel Lighthouse. What is the order of the lens? Fresnel Lenses Order Lens Diameter First Second Third Fourth Fifth Sixth 6 ft 1 in. 4 ft 7 in. 3 ft 3 in. 1 ft 8 in. 1 ft 3 in. 1 ft 0 in. 894 894 Chapter 12 Extending Transformational Geometry �� Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Tule Lake Lift Bridge Moveable Bridges Texas is home to many moveable bridges. A moveable bridge has a section that can be lifted, tilted, or swung out of the way so that tall boats can pass. 1. The Quintana Swing Bridge is a swing bridge. Part of the roadbed can pivot horizontally to let boats pass. What transformation describes the motion of the bridge? The pivoting section moves through an angle of 90°. How far does a point 10 ft from the pivot travel as the bridge opens? Round to the nearest tenth of a foot. A lift bridge contains a section that can be translated vertically. For 2–4, use the table. Lift Bridges Name Vertical Clearance (Lowered Position) Vertical Clearance (Raised Position) Tule Lake Lift Bridge Rio Hondo Lift Bridge 10 ft 27 ft 138 ft 73 ft 2. It takes about 7 min to completely lift the roadbed of the Tule Lake Lift Bridge. At what speed, in feet per minute, does the lifting mechanism translate the roadbed? Round your answer to the nearest foot per minute. 3. To the nearest second, how long does it take the Tule Lake Lift Bridge’s lifting mechanism to translate the roadbed 10 ft? 4. The Rio Hondo Lift Bridge can be raised at 10.2 feet per minute. To the nearest second, how long does it take to completely lift its roadbed? 5. Corpus Christi once had a bascule bridge. Weights were used to raise the bridge at an angle so that boats could pass through the channel. The moveable section of the Bascule Bridge in Corpus Christi was 121 ft long. Find the height of the bridge above the roadway after it had been rotated by an angle of 20°. Round to the nearest inch. Problem Solving on Location 895 895 �� Student Handbook TEKS TAKS Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S4 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S4 Application Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S28 Problem-Solving Handbook . . . . . . . . . . . . . . . . . . . . . . . S40 Draw a Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S40 Make a Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S41 Guess and Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S42 Work Backward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S43 Find a Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S44 Make a Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S45 Solve a Simpler Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S46 Use Logical Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S47 Use a Venn Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S48 Make an Organized List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S49 Skills Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S50 Number and Operations Operations with Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S50 Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S50 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S51 Estimat
ion, Rounding, and Reasonableness . . . . . . . . . . . . . . . . . . . . . . S52 Classify Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S53 Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S53 Properties of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S54 Powers of 10 and Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S54 Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S55 Simplifying Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S55 Algebra The Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S56 Connecting Words with Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S57 Variables and Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S57 Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S58 Solving Equations for a Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S59 Writing and Graphing Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S59 S2S2 S2 Student Handbook Solving Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S60 Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S61 Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S61 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S62 Direct Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S62 Functional Relationships in Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . S63 Transformations of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S63 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S64 Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S65 Factoring to Solve Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . S66 The Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S66 Solving Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S67 Solving Systems of Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . S68 Solving Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S68 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S69 Measurement Structure of Measurement Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S70 Rates and Derived Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S70 Unit Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S71 Accuracy, Precision, and Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S72 Relative and Absolute Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S73 Significant Digits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S73 Choose Appropriate Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S74 Nonstandard Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S74 Use Tools for Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S75 Choose Appropriate Measuring Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . S75 Data Analysis and Probability Measures of Central Tendency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S76 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S77 Organizing and Describing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S78 Displaying Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S78 Scatter Plots and Trend Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S79 Quartiles and Box-and-Whisker Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S80 Circle Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S80 Misleading Graphs and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S81 Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S81 Postulates, Theorems, and Corollaries . . . . . S82 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S87 Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S88 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S115 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S161 Symbols and Formulas . . . . . . . . . . inside back cover S3S3 TEKS TAKS Practice Chapter 1 Skills Practice Lesson 1-1 Name each of the following. 1. two points 2. two lines 3. two planes 4. a point on   IH 5. a line that contains L and J 6. a plane that contains L, K, and H Draw and label each of the following. 7. a ray with endpoint A that passes through B 8. a line   PQ that intersects plane D Lesson 1-2 Find each length. 9. MN 10. MO 11. Segments that have the same length are ? . ̶̶̶̶ 12. Construct a segment congruent to AB. Then construct the midpoint M. 13. M is the midpoint of ̶̶ PR , PM = 2x + 5, and MR = 4x - 7. Solve for x and find PR. Lesson 1-3 Z is in the interior of ∠WXY. Find each of the following. 14. m∠WXY if ∠WXZ = 23° and m∠ZXY = 51° 15. m∠WXZ if m∠WXY = 44° and m∠ZXY = 20°  EH bisects ∠DEF. Find each of the following. 16. m∠DEH if m∠DEH = (10z - 2) ° and m∠HEF = (6z + 10) ° 17. m∠DEF if m∠DEH = (9x + 3) ° and m∠HEF = (5x + 11) ° 18. A ? is formed by two opposite rays and measures ̶̶̶̶ ? °. ̶̶̶̶ 19. There are ? ° in a circle. ̶̶̶̶ Lesson 1-4 Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 20. ∠AOB and ∠DOE 21. ∠AOE and ∠DOE 22. ∠COE and ∠EOA 23. ∠AOB and ∠BOD 24. Name a pair of vertical angles. Given m∠A = 41.7° and m∠B = (24.2 - x) °, find the measure of each of the following. 25. complement of ∠A 26. supplement of ∠A 27. supplement of ∠B S4 S4 TEKS TAKS Practice �� Lesson 1-5 Find the perimeter and area of each figure. 28. 29. 30. Find the circumference and area of each circle. Give your answer to the nearest hundredth. 31. 33. 32. Lesson 1-6 34. The formula to find the midpoint M of ̶̶ AB with endpoints A ( x 1 , y 1 ) and B ( x 2 , y 2 ) is ? . ̶̶̶̶ Find the coordinates of the midpoint of each segment. 35. ̶̶̶ WX with endpoints W (-4, 1) and X (2, 9) ̶̶ YZ with midpoints Y (4, 8) and Z (-1, -4) 36. 37. M is the midpoint of ̶̶ RS . R has coordinates (-7, -3) , and M has coordinates (1, 1) . Find the coordinates of S. Find the length of the given segments and determine if they are congruent. 38. ̶̶̶ VW and ̶̶ RS and ̶̶ PQ ̶̶ TU 39. Lesson 1-7 Identify each transformation. Then use arrow notation to describe the transformation. 40. 41. 42. A figure has vertices at (1, 1) , (2, 4) , and (5, 3) . After a transformation, the image of the figure has vertices at (-3, -2) , (-2, 1) , and (1, 0) . Draw the preimage and image. Then describe the transformation. 43. A figure has vertices at (5, 5) , (2, 6) , (1, 5) , and (2, 4) . After a transformation, the image of the figure has vertices at (5, 5) , (6, 8) , (5, 9) , and (4, 8) . Draw the preimage and image. Then describe the transformation. 44. The coordinates of the vertices of quadrilateral DEFG are (3, 0) , (2, 3) , (-3, 2) , and (-2, -1) . Find the coordinates for the image of rectangle DEFG after the translation (x, y) → (x, -y) . Draw the preimage and image. Then describe the transformation. TEKS TAKS Practice S5S5 �� Chapter 2 Skills Practice Lesson Lesson Lesson 2-1 2-5 2-5 Lesson 2-2 Find the next item in each pattern. 1. 3, 7, 11, 15, … 2. -3, 6, -12, 24, … 3. Complete the conjecture “The product of two negative numbers is ? .” ̶̶̶̶ 4. Show that the conjecture “The quotient of two integers is an integer” is false by finding a counterexample. Identify the hypothesis and conclusion of each conditional. 5. A number is divisible by 10 if it ends in zero. 6. If the temperature reaches 100° F, it will rain. Write a conditional statement from each of the following. 7. Perpendicular lines intersect to form 90° angles. 8. 9. The sum of two supplementary angles is 180°. Lesson 2-3 Determine if each conditional is true. If false, give a counterexample. 10. If a figure has four sides, then it is a square. 11. If x = 3 , then 5x = 15 . 12. Does the conclusion use inductive or deductive reasoning? To rent a boat, you must take a boating safety course. Jason rented a boat, so Jessica concludes that he has taken a boating safety course. 13. Determine if the conjecture is valid by the Law of Detachment. Given: If a student is in tenth grade, then the student may participate in student council. Eric is a tenth-grader. Conjecture: Eric may participate in student council. 14. Determine if the conjecture is valid by the Law of Syllogism. Gi
ven: If a triangle is isosceles, then it has two congruent sides. If a triangle has two congruent angles, then it has two congruent sides. Conjecture: If a triangle is isosceles, then it has two congruent angles. 15. Draw a conclusion from the given information. Given: If the sum of the angles of a polygon is 360°, then it is a quadrilateral. If a polygon is a quadrilateral, then it has four sides. The sum of the angles of polygon R is 360°. Lesson 2-4 16. Write the conditional statement and converse within the biconditional “A triangle is equilateral if and only if it has three congruent sides.” 17. For the conditional “If a triangle is scalene, then its sides have different lengths,” write the converse and a biconditional statement. 18. Determine if the biconditional “n + 3 = -1 ↔ n = -4” is true. If false, give a counterexample. Write each definition as a biconditional. 19. A parallelogram is a quadrilateral with two pairs of parallel sides. 20. Congruent angles have equal measures. S6 S6 TEKS TAKS Practice �� Lesson 2-5 Lesson 2-6 Lesson 2-7 Solve each equation. Write a justification for each step. x + 2 _ 5 21. 2x + 3 = 9 22. = 3 Write a justification for each step. 23. AC = AB + BC 9x - 5 = (3x + 6) + (5x + 2) 9x - 5 = 8x + 8 x - 5 = 8 x = 13 24. Fill in the blanks to complete the two-column proof. Given: ∠HMK and ∠JML are right angles. Prove: ∠1 ≅ ∠3 Proof: Statements Reasons 1. a. 2. b. c. ? ̶̶̶̶̶ ? ̶̶̶̶̶ ? ̶̶̶̶̶ 1. Given 2. Adjacent angles that form a right angle are complementary. 3. ∠1 ≅ ∠3 3. d. ? ̶̶̶̶̶ 25. Use the given plan to write a two-column proof of the Transitive Property of Congruence. ̶̶ AB ≅ ̶̶ AB ≅ ̶̶ CD , ̶̶ EF ̶̶ CD ≅ ̶̶ EF Given: Prove: Plan: Use the definition of congruent segments to write the given congruence statements as statements of equality. Then use the Transitive Property of Equality to show that AB = EF. So definition of congruent segments. ̶̶ EF by the ̶̶ AB ≅ 26. Use the given two-column proof to write a flowchart proof. Given: ∠2 ≅ ∠3 Prove: m∠1 = m∠4 Proof: Statements Reasons 1. ∠2 ≅ ∠3 1. Given 2. ∠1 and ∠2 are supplementary. 2. Lin. Pair Thm. ∠3 and ∠4 are supplementary. 3. ∠1 ≅ ∠4 4. m∠1 = m∠4 3. ≅ Supps. Thm. 4. Def. of ≅  27. Use the given two-column proof to write a paragraph proof. Given: ∠1 ≅ ∠3 Prove: ∠4 ≅ ∠5 Proof: Statements Reasons 1. ∠1 ≅ ∠3 1. Given 2. ∠1 ≅ ∠4, ∠3 ≅ ∠5 2. Vert.  Thm. 3. ∠1 ≅ ∠5 4. ∠4 ≅ ∠5 3. Trans. Prop. of ≅ 4. Trans. Prop. of ≅ TEKS TAKS Practice S7S7 �� Chapter 3 Skills Practice Lesson 3-1 Identify each of the following. 1. a pair of parallel segments 2. a pair of perpendicular segments 3. a pair of skew segments Identify the transversal and classify each angle pair. 4. ∠5 and ∠3 5. ∠2 and ∠4 6. ∠5 and ∠1 Lesson 3-2 Find each angle measure. 7. m∠XYZ 8. m∠KJH 9. m∠ABC 10. m∠LMN 11. m∠PQR 12. m∠TUV Lesson 3-3 Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ ǁ m. 13. ∠2 ≅ ∠4 14. m∠8 = 5x + 36, m∠6 = 11x + 12, x = 4 Use the theorems and given information to show that p ǁ q. 15. ∠1 ≅ ∠8 16. m∠2 = 9x + 31, m∠3 = 6x + 14, x = 9 17. Write a two-column proof. Given: ∠1 and ∠5 are supplementary. Prove: ℓ ǁ m S8 S8 TEKS TAKS Practice �� Lesson 3-4 18. Name the shortest segment from point A to   BE . 19. Write and solve an inequality for x. Solve for x and y in each diagram. 20. 21. 22. Write a two-column proof. Given: ℓ ⊥ p, m ⊥ p Prove: ℓ ǁ m Lesson 3-5 Use the slope formula to determine the slope of each line. 23.  FG 24.   HJ Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. 25.  AB and   CD for A (4, 7) , B (3, 2) , C (-3, 4) , D (2, 3) 26.  EF and   GH for E (-2, 4) , F (3, 1) , G (-1, -2) , H (4, -5) 27.  JK and   LM for J (-3, 3) , K (4, -2) , L (4, 2) , M (0, -4) Lesson 3-6 Write the equation of each line in the given form. 28. the line with slope - 2 _ through (3, -1) in point-slope form 3 29. the line through (-2, 2) and (4, -1) in slope-intercept form 30. the line with x-intercept -3 and y-intercept 4 in slope-intercept form Graph each line. 31 33. y = 2 32. y + 4 = -3 (x + 2) 34. x = -1 Determine whether the lines are parallel, intersect, or coincide. 35. y = 4x + 2, 4x - y = 1 37. 2x + 5y = 1, 5x + 2y = 1 36. y =- 1 _ 2 x + 3, 2y + x = 6 38. 2x - y = 5, 2x - y = -5 TEKS TAKS Practice S9S9 �� Chapter 4 Skills Practice Lesson Lesson Lesson 4-1 2-5 2-5 Lesson 4-2 Lesson 4-3 Classify each triangle by its angle measures. 1. △ABC 2. △BCD Classify each triangle by its side lengths. 3. △EFG 4. △FGH 5. △EFH 6. Find the side lengths of △JKL. The measure of one of the acute angles of a right triangle is given. What is the measure of the other acute angle? 7. 38° 8. 27.6° Find each angle measure. 9. m∠A 10. m∠J and m∠P Given: △GHI ≅ △JKL. Identify the congruent corresponding parts. 11. ̶̶̶ GH ≅ ? ̶̶̶̶ 12. ̶̶ JL ≅ ? ̶̶̶̶ 13. ∠K ≅ ? ̶̶̶̶ Given: △LMN ≅ △PQN. Find each value. 14. x 15. m∠LMN 16. Given: ̶̶ AD is the perpendicular bisector of ̶̶ AD is the bisector of ∠BAC. ̶̶ AB ≅ ̶̶ AC , ∠B ≅ ∠C ̶̶ BC . Prove: △BAD ≅ △CAD Lesson 4-4 Use SSS to explain why the triangles in each pair are congruent. 17. △QRS ≅ △QRT 18. △UVW ≅ △WXU Show that the triangles are congruent for the given value of the variable. 19. △XYZ ≅ △ABC, 20. △DEF ≅ △GFE, x = 4 y = 8 21. Given: K is the midpoint of Prove: △HJK ≅ △LMK ̶̶ HL and ̶̶ MJ . S10 S10 TEKS TAKS Practice �� Lesson 4-5 Determine if you can use ASA to prove the triangles congruent. Explain. 22. △ACB and △ACD 23. △EFG and △HGF Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 24. △ABC ≅ △EDC 25. △FGH ≅ △FJH Lesson 4-6 Lesson 4-7 26. Given: ̶̶̶ ̶̶ LP , MN ǁ ∠N ≅ ∠L ̶̶ ̶̶̶ PN ML ≅ Prove: 27. Given: ∠1 ≅ ∠6, ∠4 ≅ ∠6 ̶̶ AE ∠1 ≅ ∠3, ̶̶ AB ≅ Prove: △ACD is isosceles. 28. Given: △ABC with vertices A (2, 4) , B (3, 1) , C (5, 2) and △DEF with vertices D (-4, -2) , E (-1, -3) , F (-2, -5) Prove: ∠BAC ≅ ∠EDF Position each figure in the coordinate plane. 29. a rectangle with length 7 units and width 3 units 30. a square with side length 3a Write a coordinate proof. 31. Given: Right △GHI has coordinates G (0, 0) , H (0, 4) , and I (6, 0) . ̶̶̶ GH , and K is the midpoint of J is the midpoint of ̶̶ GI . Prove: The area of △GJK is 1 __ 4 the area of △GHI. Assign coordinates to each vertex and write a coordinate proof. 32. Given: A is the midpoint of B is the midpoint of ̶̶̶ XW in rectangle WXYZ. ̶̶ YZ . Prove: AB = XY Lesson 4-8 Find each angle measure. 33. m∠X Find each value. 35. x 37. Given: △XYZ is isosceles. A is the midpoint of ̶̶ XZ . ̶̶ XY ≅ ̶̶ YZ 34. m∠A 36. y Prove: △YAZ is isosceles. TEKS TAKS Practice S11 S11 �� Chapter 5 Skills Practice Lesson 5-1 Find each measure. 1. CD 2. HG 3. JM 4. m∠SRT, given m∠SRU = 126° 5. PQ 6. m∠WXV 7. Write an equation in point-slope form for the perpendicular bisector of the segment Lesson 5-2 with endpoints A (1, 4) and B (-5, -2) . ̶̶ ̶̶ DG , EG , and Find each length. ̶̶ FG are the perpendicular bisectors of △ABC. 8. BG 9. AG Find the circumcenter of a triangle with the given vertices. 10. H (5, 0), J (0, 3), K (0, 0) ̶̶ QS and 12. the distance from S to ̶̶ RS are angle bisectors of △QPR. Find each measure. 13. m∠SQP ̶̶ PR 11. L (0, 0), M (-2, 0), N (0, -4) Lesson 5-3 In △DEF, DJ = 30, and FM = 12. Find each length. 15. MJ 14. DM 16. GF 17. GM Find the orthocenter of a triangle with the given vertices. 18. N (-2, 2), P (4, 2), Q (0, -2) 19. R (-2, 1), S (2, 5), T (4, 1) Lesson 5-4 20. The vertices of △WXY are W (-3, 2), X (5, 2), and Y (1, -4). A is the midpoint of and B is the midpoint of ̶̶ XY . Show that ̶̶ AB ǁ ̶̶̶ WX and AB = 1 __ 2 WX. ̶̶̶ WY , Find each measure. 21. DE 23. DG 25. m∠FHE 22. FG 24. m∠CHF 26. m∠CED S12S12 TEKS TAKS Practice �� Lesson 5-5 Write an indirect proof of each statement. 27. An isosceles triangle cannot have an obtuse base angle. 28. A right triangle cannot have three congruent sides. 29. Write the angles in order from smallest to largest. 30. Write the sides in order from shortest to longest. Tell whether a triangle can have sides with the given lengths. Explain. 32. 7, 9, 18 31. 4, 7, 8 33. 2x + 5, 4x, 3 x 2 , when x = 3 The lengths of two sides of a triangle are given. Find the range of possible lengths for the third side. 34. 4 in., 10 in. 36. 6.2 cm, 12 cm 35. 8 ft, 8 ft Lesson 5-6 Compare the given measures. 37. Compare RS and UV. 38. Compare m∠XWY 39. Find the range of and m∠ZWY. values for x. 40. Write a two-column proof. Given: m∠X > m∠Y, m∠B > m∠A Prove: AY > XB Lesson 5-7 Lesson 5-8 Find the value of x. Give your answer in simplest radical form. 43. 41. 42. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 44. 46. 45. Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 47. 4, 7.5, 8.5 48. 6, 10, 11 49. 9, 21, 25 Find the value of x. Give your answer in simplest radical form. 52. 50. 51. Find the values of x and y. Give your answers in simplest radical form. 53. 55. 54. TEKS TAKS Practice S13 S13 �� Chapter 6 Skills P
ractice Lesson Lesson Lesson 6-1 2-5 2-5 Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 1. 2. 3. Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. 4. 5. 6. 7. Find the measure of each interior angle of pentagon ABCDE. 8. Find the sum of the interior angle measures of a convex heptagon. 9. Find the measure of each interior angle of a regular 15-gon. 10. Find the value of x in polygon FGHJKL. 11. Find the measure of each exterior angle of a regular dodecagon. Lesson 6-2 MNOP is a parallelogram. Find each measure. 12. MP 13. m∠M 14. m∠N Three vertices of QRST are given. Find the coordinates of T. 15. Q (-5, 3) , R (3, 6) , S (6, 4) 16. Q (-1, 7) , R (3, 3) , S (-2, 3) Write a two-column proof. 17. Given: ABFG and HDEG are parallelograms. Prove: ∠B ≅ ∠D Lesson 6-3 18. Show that RSTU is a parallelogram 19. Show that WXYZ is a parallelogram for x = 2 and y = 3. for a = 6 and b = 11. Determine if each quadrilateral must be a parallelogram. Justify your answer. 20. 22. 21. Show that the quadrilateral with the given vertices is a parallelogram. 23. W (0, 0) , X (-3, 3) , Y (5, 5) , Z (8, 2) 24. A (-3, 1) , B (-2, 4) , C (1, 2) , D (0, -1) S14S14 TEKS TAKS Practice �� Lesson 6-4 EFGH is a rectangle. Find each measure. 25. EH 26. HF JKLM is a rhombus. Find each measure. 27. JK 28. m∠NKL Show that the diagonals of a square with the given vertices are congruent perpendicular bisectors of each other. 29. N (1, 4) , P (4, 1) , Q (1, -2) , R (-2, 1) 30. S (-2, 7) , T (2, 8) , U (3, 4) , V (-1, 3) 31. Given: WXYZ is a rectangle. Prove: ̶̶̶ WB ≅ ̶̶ YA ̶̶ XB ≅ ̶̶ AZ Lesson 6-5 Lesson 6-6 Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. ̶̶ ̶̶ 32. Given: XY ≅ XY ǁ ̶̶ XZ ⊥ Conclusion: WXYZ is a rhombus. ̶̶̶ WZ , ̶̶̶ WZ , ̶̶̶ WY 33. Given: ̶̶̶ WX ≅ ̶̶ XY Conclusion: WXYZ is a square. ̶̶ XY , Conclusion: WXYZ is a rectangle. ̶̶̶ WX ⊥ ̶̶̶ WX ⊥ ̶̶̶ WZ 34. Given: Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 35. A (1, 0) , B (2, -4) , C (6, -3) , D (5, 1) 36. E (-3, -1) , F (-4, -4) , G (2, -6) , H (3, -3) In kite TUVW, m∠TUX = 65°, and m∠UVT = 32°. Find each measure. 37. m∠TUX 39. m∠TWX 38. m∠XUV Find each measure. 40. m∠C 41. HJ, given that EG = 32.8 and FJ = 24.3 42. Find the value of x so that JKLM is isosceles. 43. Given RP = 8y - 7 and NQ = 10y - 12, find the value of y so that NPQR is isosceles. 44. Find RS. 45. Find XY. TEKS TAKS Practice S15 S15 �� Chapter 7 Skills Practice Lesson 7-1 Write a ratio expressing the slope of each line. 1. line ℓ 2. line m 3. line n 4. The ratio of the side lengths of a quadrilateral is 2 : 4 : 5 : 6, and its perimeter is 85 inches. What is the length of the shortest side? 5. The ratio of angle measures in a triangle is 3 : 10 : 12. What is the measure of each angle? Solve each proportion. = 6 _ 6. x _ 5 20 7. 21 _ 6 9 9. Given that 3x = 12y, find the ratio of x to y in simplest form. Lesson 7-2 Identify the pairs of congruent angles and corresponding sides. 10. 11. Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 12. rectangles ABCD and EFGH 13. △JKL and △MNO Lesson 7-3 Explain why the triangles are similar and write a similarity statement. 14. 15. Verify that the triangles are similar. 16. △FGH ∼ △JKH 17. △ACE ∼ △BCD Explain why the triangles are similar and then find each length. 18. △XYZ and △ABC, BC 19. △RSV and △UST, TU S16S16 TEKS TAKS Practice �� Lesson 7-4 Lesson 7-5 Find the length of each segment. 20. ̶̶ AE Verify that the given segments are parallel. 22. ̶̶ EF and ̶̶ JG Find the length of each segment. 24. ̶̶ RS and ̶̶ ST 21. ̶̶ KJ 23. ̶̶ LP and ̶̶̶ MN 25. ̶̶̶ XW and ̶̶̶ WZ The scale drawing of the playhouse is 1 in. : 10 ft. Find the actual lengths of the following walls. 26. ̶̶̶ GH ̶̶ EF ̶̶ DC 27. 28. The school courtyard is 25 ft by 40 ft. Make a scale drawing of the courtyard using the following scales. 29. 1 cm : 1 ft 31. 1 cm : 10 ft 30. 1 cm : 5 ft 32. Given that △ABC ∼ △DEF, find the perimeter P and area A of △DEF. Lesson 7-6 33. Given that △RSV ∼ △RTU, find the coordinates of S and the scale factor. 34. Given: A (-3, 3) , B (1, 7) , C (5, 5) , D (-1, 5) , E (1, 4) Prove: △ABC ∼ △ADE TEKS TAKS Practice S17 S17 �� Chapter 8 Skills Practice Lesson Lesson Lesson 8-1 2-5 2-5 Write a similarity statement comparing the three triangles in each diagram. 1. 2. 3. Find the geometric mean of each pair of numbers. If necessary, give the answers in simplest radical form. 4. 3 and 9 5. 4 and 7 6. 1 _ 2 and 5 Find x, y, and z. 7. 8. 9. Lesson 8-2 Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. 10. sin A 11. cos A 12. tan A Use a special right triangle to write each trigonometric ratio as a fraction. 13. cos 30° 14. sin 45° 15. tan 60° Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 16. sin 38° 17. cos 47° 18. tan 21° Find each length. Round to the nearest hundredth. 19. DE 20. GH 21. KL Lesson 8-3 22. tan -1 (3.5) Use your calculator to find each angle measure to the nearest degree. 23. sin -1 ( 1 _ ) 5 Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 25. 27. 26. 24. cos -1 (0.05) For each triangle, find the side lengths to the nearest hundredth and the angle measures to the nearest degree. 28. A (1, 4) , B (1, 1) , C (4, 1) 29. D (-3, 5) , E (-3, 1) , F (2, 5) S18S18 TEKS TAKS Practice �� Lesson 8-4 Lesson 8-5 Classify each angle as an angle of elevation or angle of depression. 30. ∠1 31. ∠2 32. ∠3 33. ∠4 Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. 34. cos 127° 37. tan 158° 35. tan 131° 38. sin 85° 36. sin 114° 39. cos 161° Find each measure. Round lengths to the nearest tenth and angle measure to the nearest degree. 40. AC 41. m∠E 42. m∠G 43. m∠T 44. VX 45. BC Lesson 8-6 Write each vector in component form. 46.  AB with A (2, 3) and B (5, 6) 47. the vector with initial point C (3, 6) and terminal point D (2, 4) 48.  EF 49.  GH Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 50. 〈-3, 2〉 52. 〈2, -5〉 51. 〈4, 3〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 53. A wind velocity is given by the vector 〈3, 4〉. 54. The velocity of a rocket is given by the vector 〈8, 1〉. Identify each of the following in the diagram. 55. equal vectors 56. parallel vectors Find each vector sum. 57. 〈5, 0〉 + 〈-3, 6〉 58. 〈-3, -1〉 + 〈0, -7〉 59. 〈1, 8〉 + 〈2, 3〉 60. 〈-2, -1〉 + 〈-7, 9〉 TEKS TAKS Practice S19 S19 �� Chapter 9 Skills Practice Lesson 9-1 Find each measurement. 1. the area of the parallelogram 2. the perimeter of the rectangle in which A = 15 x 2 ft 2 3. b 2 of the trapezoid in which A = 35 ft 2 4. the area of the kite 5. the base of a triangle in which h = 9 and A = 135 in 2 6. the area of a rhombus in which d 1 = (3x + 5) cm and d 2 = (7x + 4) cm Lesson 9-2 Find each measurement. 7. the circumference of ⊙C in terms of π 8. the area of ⊙D in terms of π 9. the circumference of ⊙F in which A = 49 x 2 π cm 2 10. the radius of ⊙E in which C = 36π in. Find the area of each regular polygon. Round to the nearest tenth. 11. a regular hexagon with a side length of 8 in. 12. an equilateral triangle with an apothem of 5 √  3 _ 3 cm Lesson 9-3 Find the shaded area. Round to the nearest tenth, if necessary. 15. 13. 14. Use a composite figure to estimate each shaded area. The grid has squares with side lengths of 1 in. 16. 17. S20S20 TEKS TAKS Practice �� Lesson 9-4 Estimate the area of each irregular shape. 18. 19. Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 20. A (-2, 3) , B (0, 6) , C (6, 2) , D (4, -1) 21. E (-1, 3) , F (2, 3) , G (2, -1) Find the area of each polygon with the given vertices. 22. R (-2, 3) , S (1, 5) , T (3, 1) , U (0, -2) 23. W (-4, 0) , X (4, 3) , Y (6, 1) , Z (2, -1) Lesson 9-5 Describe the effect of each change on the area of the given figure. 24. The height of the rectangle with height 10 ft and width 12 ft is multiplied by 1 _ . 2 25. The base of the parallelogram with vertices A (-2, 3) , B (3, 3) , C (0, -1) , D (-5, -1) is doubled. Describe the effect of each change on the perimeter or circumference and the area of the given figure. 26. The radius of ⊙E is multiplied by 1 _ . 4 27. The base and height of a rectangle with base 6 in. and height 5 in. are multiplied by 3. 28. A square has a side length of 7 ft. If the area is tripled, what happens to the side length? 29. A circle has a diameter of 20 m. If the area is doubled, what happens to the circumference? Lesson 9-6 A point is chosen randomly on ̶̶ 30. The point is on AC . 32. The point is not on ̶̶ BC . ̶̶ AD . Find the probability of each event. ̶̶ AB or ̶̶ BD . 31. The point is on 33. The point is on ̶̶ CD . Use th
e spinner to find the probability of each event. 34. the pointer landing on green 35. the pointer landing on blue or red 36. the pointer not landing on orange 37. the pointer not landing on red or yellow Find the probability that a point chosen randomly inside the rectangle is in each shape. Round to the nearest hundredth. 38. the equilateral triangle 39. the parallelogram 40. the circle 41. the part of the rectangle that does not include the circle, triangle, or parallelogram TEKS TAKS Practice S21 S21 �� Chapter 10 Skills Practice Lesson 10-1 Classify each figure. Name the vertices, edges, and bases. Classify each figure. Name the vertices, edges, and bases. 1. 1. 2. 3. Describe the three-dimensional figure that can be made from the given net. 4. 5. 6. Use the figure made of unit cubes for Exercises 7–11. Assume there are no hidden cubes. 7. Draw all six orthographic views. 8. Draw an isometric view. 9. Draw a one-point perspective view. 10. Draw a two-point perspective view. 11. Determine whether the drawing represents the given object. Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. 12. 13. 14. Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 15. (2, 4, 9) and (3, 7, 2) 16. (0, 0, 0) and (4, 7, -4) 17. (5, 1, 0) and (0, 3, 4) Find the lateral area and surface area of each figure. Give exact answers, using π if necessary. 18. 19. 20. Lesson 10-2 Lesson 10-3 Lesson 10-4 21. The dimensions of a cylinder with r = 9 cm and h = 12 cm are multiplied by 1 _ . 3 Describe the effect on the surface area. S22S22 TEKS TAKS Practice �� Lesson 10-5 Find the lateral area and surface area of each figure. Give exact answers, using π if necessary. 22. 23. 24. 25. The dimensions of a square pyramid with B = 64 in 2 and h = 7 in. are tripled. Describe the effect on the surface area. 26. The dimensions of a right cone with r = 14 in. and ℓ = 24 in. are multiplied by 1 _ . 2 Describe the effect on the surface area. Lesson 10-6 Find the volume of each figure. Round to the nearest tenth. 27. 28. 29. 30. The dimensions of a prism with B = 14 cm 2 and h = 8 cm are doubled. Describe the effect on the volume. 31. The dimensions of a cylinder with r = 6 cm and h = 4 cm are multiplied by 2 _ . 3 Describe the effect on the volume. Lesson 10-7 Find the volume of each figure. Round to the nearest tenth. 32. 33. 34. 35. The dimensions of a cone with r = 8 cm and ℓ = 17 cm are multiplied by 1 _ . 2 Describe the effect on the volume. 36. The dimensions of a pyramid with B = 128 m m 2 and h = 56 mm are tripled. Describe the effect on the volume. Lesson 10-8 Find the surface area and volume of each figure. Give your answers in terms of π. 37. 38. 39. 40. The radius of a sphere with r = 24 cm is multiplied by 1 _ 3 surface area and volume. . Describe the effect on the 41. The radius of a sphere with r = 15 mm is multiplied by 4. Describe the effect on the surface area and volume. TEKS TAKS Practice S23 S23 �� Chapter 11 Skills Practice Lesson 11-1 Identify each line or segment that intersects each circle. Identify each line or segment that intersects each circle. 1. 1. 2. Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. 3. 4. The segments in each figure are tangent to the circle. Find each length. 5. PQ 6. WZ Lesson 11-2 Find each measure. Round to the nearest tenth, if necessary. 7. m ⁀ FB 8. PQ 9. ⊙T ≅ ⊙W. Find m∠VWX. 10. BD Lesson 11-3 Find the area of each sector or segment. Round to the nearest tenth. 11. 12. S24S24 TEKS TAKS Practice �� Find each arc length. Give your answers in terms of π and rounded to the nearest tenth. 13. 14. Lesson 11-4 Find each measure or value. Round to the nearest tenth, if necessary. 16. x 15. m∠ABD 17. x 18. angle measures of HJKL 19. m ⁀ DF Lesson 11-5 20. m∠JMK 21. m∠RTQ 22. x 23. m∠AFE 24. m ⁀ GL Lesson 11-6 Find the value of the variable. Round to the nearest tenth, if necessary. 25. 26. 27. Lesson 11-7 Write the equation of each circle. 28. ⊙A with center A (2, -3) and radius 6 29. ⊙B that passes through (3, 4) and has center B (-2, 1) Graph each equation. 30. (x + 3) 2 + (y - 4) 2 = 1 31. x 2 + (y + 4) 2 = 16 TEKS TAKS Practice S25 S25 �� Chapter 12 Skills Practice Lesson 12-1 Copy each figure and the line of reflection. Draw the reflection of the figure Copy each figure and the line of reflection. Draw the reflection of the figure across the line. 1. 1. 2. Reflect the figure with the given vertices across the given line. 3. A (-4, 1) , B (2, 4) , C (3, -2) ; x-axis 4. D (3, 1) , E (2, 4) , F (-2, 2) , G (2, -2) ; y = x Copy each figure and the translation vector. Draw the translation of the figure along the given vector. 5. 6. Translate the figure with the given vertices along the given vector. 7. A (-2, 1) , B (4, 3) , C (2, -2) ; 〈2, 3〉 8. D (-1, 3) , E (2, 4) , F (3, 3) , G (3, -2) ; 〈2, -2〉 Copy each figure and the angle of rotation. Draw the rotation of the figure about the point P by m∠A. 9. 10. Rotate the figure with the given vertices about the origin using the given angle of rotation. 11. A (2, 3) , B (-2, 1) , C (1, -1) ; 90° 12. D (-2, 3) , E (2, 4) , F (3, 1) , G (-2, 2) ; 180° Draw the result of each composition of isometries. 13. Translate △ABC along  v and then reflect it across line ℓ. 14. Reflect △DEF across line m and then translate it along  w . Lesson 12-2 Lesson 12-3 Lesson 12-4 15. Copy the figure and draw two lines of reflection that produce an equivalent transformation. S26S26 TEKS TAKS Practice �� Lesson 12-5 Describe the symmetry of each figure. Copy the shape and draw all lines of symmetry. If there is rotational symmetry, give the angle and order. 16. 18. 17. Tell whether each figure has plane symmetry, symmetry about an axis, or neither. 19. 21. 20. Lesson 12-6 Copy the given figure and use it to create a tessellation. 22. 23. 24. Classify each tessellation as regular, semiregular, or neither. 25. 26. 27. Lesson 12-7 Copy each figure and center of dilation P. Draw the image of the figure under a dilation with the given scale factor. 28. scale factor: 3 29. scale factor: - 2 _ 3 Draw the image of the figure with the given vertices under a dilation with the given scale factor centered at the origin. 30. A (1, 3) , B (1, 5) , C (4, 3) ; scale factor 2 31. E (-2, 2) , F (2, 4) , G (4, -2) ; scale factor - 1 _ 2 TEKS TAKS Practice S27 S27 �� Chapter 1 Applications Practice Athletics Use the following information for Exercises 1–3. During gym class, a teacher notices the following. Decide if each resembles a point, segment, ray, or line. (Lesson 1-1) 1. Kyle starts running in a straight line. Suppose he does not stop running. 2. Agnes runs a quarter-mile in a straight line. 3. Jimmy stands perfectly still. Travel Use the following information for Exercises 4–6. The Perez family is driving from Austin, Texas, to Dallas, Texas. The city of Waco is the approximate midpoint between these two cities. It is 102 miles from Austin to Waco. (Lesson 1-2) 9. Entomology Because the insect is symmetrical, ∠1 ≅ ∠4 and ∠2 ≅ ∠3. Also, ∠1 and ∠2 are complementary, and ∠3 and ∠4 are complementary. If m∠1 = 48.5°, find m∠2, m∠3, and m∠4. (Lesson 1-4) � � � � Architecture Use the following information for Exercises 10 and 11. The bricks used to make a building are one-fourth as tall as they are wide, and the bricks are 2.25 inches tall. (Lesson 1-5) 4. What is the total distance from Austin to 10. What is the area of the largest face of each Dallas? brick? 5. The approximate midpoint from Waco to Dallas is Milford. What is the distance from Austin to Milford? 11. A certain exterior wall is 33 bricks long and 20 bricks tall. What is the area of the wall in square inches? 6. The Perez family averages 64 miles per hour. About how long will the entire drive take? Probability Use the following information for Exercises 7 and 8. In a carnival game, each contestant spins the wheel and wins the prize indicated by the color. (Lesson 1-3) 12. Sports A football coach has his team run sprints diagonally across a football field. If the field is 120 yards long and 160 feet wide, what is the distance they run? Write your answer to the nearest hundredth of a foot. (Lesson 1-6) 13. Crafts The picture below shows half of a stenciled design. The full design should resemble a sun. Name two transformations that can be performed on the image so that the image and its preimage form a complete picture. Be as specific as possible, referring to L and P. (Lesson 1-7) 7. Using a protractor, measure each angle on the wheel. 8. Since there are 360° in a circle, the probability of the wheel landing on a given color is the number of degrees in the angle divided by 360°. Find the probability of the wheel landing on each prize. Express your answer as a fraction in lowest terms. S28S28 TEKS TAKS Practice �� Chapter 2 Applications Practice 1. Health Mike collected the following data about the heights of twelve students in his tenth-grade class. Use the table to make a conjecture about the heights of boys and girls in the tenth grade. (Lesson 2-1) Height (in.) of Tenth-Grade Students Boys Girls 70 67 71 64 68 64 67 65 70 68 67 66 2. Government Voter Tur
nout Year 1998 1996 Voters 12,530 Presidential elections are held every four years. Elections for senators are held every two years. So in years not divisible by 4, only Senate seats are up for election. The table shows voter turnout for a small town during recent election years. Make a conjecture based on the data. (Lesson 2-1) 15,210 14,380 8,750 7,370 2000 2002 2004 3. Biology Write the converse, inverse, and contrapositive of the conditional statement “If an animal is a fish, then it swims in salt water.” Find the truth value of each. (Lesson 2-2) 4. Gardening Write the converse, inverse, and contrapositive of the conditional statement “If a plant is watered, then it will grow.” Find the truth value of each. (Lesson 2-2) 5. Sports Determine if the conjecture is valid by the Law of Detachment. (Lesson 2-3) Given: If you participate in a triathlon, then you run, swim, and bike. Margie runs, swims, and bikes. Conjecture: Margie participates in a triathlon. 6. Health Students are required to have certain immunizations before attending school to prevent the spread of disease. Write the conditional statement and converse within the biconditional “Students can attend public school if and only if they have the required immunizations.” (Lesson 2-4) 7. Weather Hurricanes are assigned category numbers to describe the amount of flooding and wind damage they are likely to cause. Write the statement “If a hurricane has sustained winds of more than 155 miles per hour, then it is Category 5” as a biconditional statement. (Lesson 2-4) 8. Athletics The equation c = 5w + 25 relates the number of workouts w to the cost c of a weight training group. If Matthew plans to spend $200 on weight training, how many workouts can he participate in? Solve the equation for w and justify each step. (Lesson 2-5) 9. Nutrition Rick has allotted himself 200 Calories for his evening snack, which consists of a glass of milk and crackers. A glass of milk has 110 Calories, and each cracker has 15 Calories. The equation s = 110 + 15c relates the number of crackers c to the total number of Calories s in Rick’s evening snack. How many crackers can Rick have? Solve the equation for c and justify each step. (Lesson 2-5) 10. Travel On a city map, the library, post office, and police station are collinear points in that order. The distance from the library to the post office is 2.3 miles. The distance from the post office to the police station is 5.1 miles. Which theorem can you use to conclude that the distance from the library to the police station is 7.4 miles? (Lesson 2-6) 11. Recreation Kyle is making a kite from the pattern below by cutting four triangles from different pieces of material. Write a paragraph proof to show that m∠3 = 90°. (Lesson 2-7) Given: ∠1 ≅ ∠2 Prove: m∠3 = 90° TEKS TAKS Practice S29 S29 �� Chapter 3 Applications Practice 1. Recreation A scuba diver leaves a flag on 5. Transportation The railroad ties in the the surface of the water to alert boaters of his location. Describe two parallel lines and a transversal in the flag. (Lesson 3-1) diagram are all parallel. m∠1 = 19x - 5 and m∠ 2 = 4x + 5y. Find x and y so that the ties are all perpendicular to the tracks. (Lesson 3-4) 2. Carpentry In the stairs shown, the horizontal treads and the vertical risers are all parallel. m∠1 = (14x + 6) ° and m∠2 = (19x - 24) °. Find x. (Lesson 3-2) 6. Art The sides of a picture frame are cut so that the opposite sides of the frame are parallel and the consecutive sides are perpendicular. Find the values of x and y in the diagram. (Lesson 3-4) 3. Transportation The train tracks shown cross the street lanes. The lanes of the street are parallel. Find x in the diagram. (Lesson 3-3) 7. Recreation At 1:00 P.M., a boat on a river passes a point that is 3 miles from a lodge. At 5:30 P.M., the boat passes a point that is 8 miles from the lodge. Graph the line that represents the boat’s distance from the lodge. Find and interpret the slope of the line. (Lesson 3-5) 8. Sports A marathon runner runs 10 miles by 3:00 P.M. and 25 miles by 4:30 P.M. Graph the line that represents her distance run. Find and interpret the slope of the line. (Lesson 3-5) 9. Business A cab company charges $8 per ride plus $0.25 per mile. Another cab company charges $5 per ride plus $0.35 per mile. For how many miles will two cab rides cost the same amount? (Lesson 3-6) 4. Sports At a track meet, the starting blocks are placed along a line that is a transversal to the lanes. m∠1 = 12x - 8, m∠2 = 8x + 12, and x = 5. Show that the lines between the lanes are parallel. (Lesson 3-3) � � � 10. Food A pizza parlor is catering a school event. Pete’s Pizza charges $85 for the first 20 students and $5 for each additional student. Polly’s Pizza charges $125 for the first 20 students and $3 for each additional student. For how many students will the pizza parlors cost the same? (Lesson 3-6) S30S30 TEKS TAKS Practice �� Chapter 4 Applications Practice 1. Camping Three poles are used to create the frame for a tent. The front of the tent is an isosceles triangle with the base is 1.5 times the length of the sides. The perimeter of the triangle is 21 ft. Find each side length. (Lesson 4-1) ̶̶ BC . The length of ̶̶ AB ≅ 2. Geography The universities in Durham, Chapel Hill, and Raleigh, North Carolina form what is known as the Research Triangle. Use the map to find the measure of the angle whose vertex is at Durham. (Lesson 4-2) �� 3. Business Oil derricks are used as supports for oil drilling equipment. Use the diagram to prove the following. (Lesson 4-3) 6. Surveying To find the distance AB across a lake, first locate point C. Then measure the distance from C to B. Locate point D the same distance from C as B, but in the opposite direction. Then measure the distance from C to A and locate point E in a similar manner. What is the distance AB across the lake? (Lesson 4-6) 7. The first step in creating a Sierpinski triangle is to connect the midpoints of the sides of a triangle as shown. (Lesson 4-7) Given: ̶̶̶ ̶̶ ̶̶ AB ≅ HB ≅ HG , ∠GAB ≅ ∠BHG, ∠AGB ≅ ∠HBG ̶̶ AG Prove: △AGB ≅ △HBG 4. Sports A kite is made up of two pairs of congruent triangles. Use SAS to explain why △ABD ≅ △CBD. (Lesson 4-4) 5. Recreation A student is estimating the height of a water slide. From a certain distance, the angle from where he is standing to a point on the highest part of the slide is 35°. From a distance 200 m closer, the same angle is 45°. a. Draw a triangle with the point at the top of the slide as one vertex, and the points where the measurements were taken as the other vertices. b. Which postulate or theorem can be used to show that this triangle is uniquely determined? (Lesson 4-5) Given: Equilateral △ABC, D is the midpoint ̶̶ AB , E is the midpoint of ̶̶ AC , and F is of the midpoint of ̶̶ BC . Prove: The area of △DEF is 1 __ 4 the area of △ABC. 8. Recreation A boat is sailing parallel to the coastline along   XY . When the boat is at X, the measure of the angle from the lighthouse W to the boat is 30°. After the boat has traveled 5 miles to Y, the angle from the lighthouse to the boat is 60°. How can you find WY? (Lesson 4-8) TEKS TAKS Practice S31 S31 �� Chapter 5 Applications Practice 1. Building The guy wires ̶̶ AB and ̶̶ CB supporting a cell phone tower are congruent and are equally spaced from the base of the tower. How do these wires ensure that the cell phone tower is perpendicular to the ground? (Lesson 5-1) 2. Safety City planners want to relocate their town’s firehouse so that it is the same distance from the three main streets of the town. Draw a sketch to show where the firehouse should be positioned. Justify your sketch. (Lesson 5-2) 3. Safety A lifeguard needs to watch three areas of a water park. Draw a sketch to show where she should stand to be the same distance from all the swimmers. Justify your sketch. (Lesson 5-2) 4. Art An artist is designing a sculpture composed of a pedestal with a triangular top. The vertices of the top are A (-4, 2) , B (2, 4) , and C (4, -3) . Where should the artist attach the pedestal so that the triangle is balanced? (Lesson 5-3) 5. Measurement City engineers plan to build a bridge across the pond shown. What will be the length of the bridge, GH? (Lesson 5-4) � �� S32S32 TEKS TAKS Practice Engineering Use the following information for Exercises 6 and 7. Playground engineers are planning a sidewalk that will connect the swings, seesaw, and slide. (Lesson 5-5) 6. If the angle at the swings is the largest, which portion of the sidewalk will be the longest? 7. The distance from the swings to the seesaw is 37 ft. Can the lengths of the other sides be 40 ft and 50 ft? Explain. 8. Geography The cities of Allenville, Baytown, College City, and Dean Park are shown on the map. Baytown and Dean Park are each 30 miles from College City. Which city is closer to Allenville: Baytown or Dean Park? (Lesson 5-6) 9. Mark is late for school. He usually goes around the park so he can walk along the water. Today he decides to cut through the park. About how many feet does he save by going through the park? (Lesson 5-7) 10. Sports A baseball diamond is a square with a side length of 90 ft. What is the distance from first base to third base? (Lesson 5-8) 11. Recreation Haley, who is 5 ft tall, is flying a kite on 100 ft of string. How high is the kite? (Lesson 5-8) �� Chapter 6 Applications Practice 1. Safety A stop sign is in the shape of a regular octagon. What is the value of x? (Lesson 6-1) Design Use the following information for Exercises 7–9. 2. Hobbies Nancy is planting a garden shaped like a regular pentagon. She bought metal edging to surround the garden and prevent
weeds. What angle should the edging form at the vertices of the garden? (Lesson 6-1) Fishing Use the following information for Exercises 3–5. The hinges for the trays in a tackle box form parallelograms to ensure that the trays stay parallel to the base of the box. In ABCD, AB = 21 in., AE = 9 in., and m∠BCD = 125°. Find each measure. (Lesson 6-2) 3. DC 4. EC 5. m∠ADC 6. Design A glide rocker uses hinged parallelograms to move the chair back and forth. In ABCD, AB = DC, and AD = BC. ̶̶ BC , The sides of the parallelogram, rotate together to move the chair. Why is ABCD always a parallelogram? (Lesson 6-3) ̶̶ AD and When extended, the legs of a folding table must form a rectangle so the tabletop is parallel to the ground. Given that JK = 48 in. and KN = 36 in., find each length. (Lesson 6-4) 7. JM 8. JN 9. NM 10. Hobbies Elise is creating a decorative page for her scrapbook. She has a piece of ribbon that is 12 inches long. She wants to outline a rhombus with the ribbon. How can Elise cut the ribbon to ensure that the final shape is a rhombus? (Lesson 6-5) 11. Carpentry Luke is cutting a rectangular window frame. The dimensions of the window are to be 3 feet by 4 feet. What should the diagonal of the frame measure so that the window is rectangular? (Lesson 6-5) 12. Hobbies Addie is making a kite with diagonals of 32 inches and 18 inches. She wants to put a ribbon around the edge of the kite. She will add an 8-foot tail to the kite, made of the same ribbon. If ribbon can be purchased in packages of 3 yards, how many packages should she buy for the entire project? (Lesson 6-6) 13. Carpentry Aaron is building a shadow box for his baseball memorabilia. The shadow box will be in the shape of a trapezoid, as shown below. The wood for the box costs $1.59 per foot. Estimate the cost of the lumber. (Lesson 6-6) TEKS TAKS Practice S33 S33 �� Chapter 7 Applications Practice 9. Geography Riverside Park has campsites available for rent. Lot A has 50 ft of street frontage and 80 ft of river frontage. Find the river frontage for lots B, C, and D. (Lesson 7-4) 10. Architecture An amphitheater is being built according to the design shown. If the total footage on the right of the rows of seats is 232.5 ft, find the length of each section. (Lesson 7-4) 11. Jake wants to know the height of the oak tree in his front yard. He measured his height as 68 inches and his shadow as 34 inches. At the same time, the tree has a shadow of 5.5 feet. How tall is the tree? (Lesson 7-5) 12. Recreation The kiddie pool and the lap pool at Centerville Park are similar rectangles. The lap pool measures 25 ft wide by 48 ft long. The kiddie pool is 8 ft long. How wide is the kiddie pool to the nearest tenth? (Lesson 7-5) 13. Melissa is enlarging her 4-by-6 photo by 150%. Find the coordinates of the enlarged photo. (Lesson 7-6) Hobbies Use the following information for Exercises 1–4. Jason and Matthew share 210 CDs. The ratio between Matthew’s CDs and Jason’s CDs is 4:3. (Lesson 7-1) 1. Write a proportion that can be used to find the number of CDs each one has. 2. How many CDs does Jason have? 3. The number of Matthew’s CDs is what fraction of the total number of CDs? 4. Jason wants to have the most CDs. What is the least number of CDs he would have to purchase to have more than Matthew? 5. Carpentry Ava’s dollhouse is a scale model of a castle. The great room of the castle has a width of 40 ft and a length of 50 ft. The width of the great room of the dollhouse is 8 in. What is the length of the great room of the dollhouse? (Lesson 7-1) 6. Travel A map is a scale model of a real city. The scale on the map is 1 in.:30 mi. Two cities are 165 mi apart. How far apart will the cities be on the map? (Lesson 7-2) 7. Recreation The sails on the sailboat below have the given dimensions. Use similar triangles to prove △ABC ∼ △DEF. (Lesson 7-3) � � � � � � �� 8. Graphics A photograph shows a smaller version of the real item. The height of the Washington Monument is approximately 555 ft. The monument in a photo is 5 in. tall. What is the scale factor of the actual monument to the monument in the photo? (Lesson 7-3) S34S34 TEKS TAKS Practice �� Chapter 8 Applications Practice 1. Diving To estimate the height of a diving platform, a spectator stands so that his lines of sight to the top and bottom of the platform form a right angle as shown. The spectator’s eyes are 5 ft above the ground. He is standing 15 ft from the diving platform. How high is the platform? (Lesson 8-1) 6. Safety A lifeguard sees a swimmer struggling in the water at an angle of depression of 15°. The stand is 10 feet tall. What is the horizontal distance from the stand to the swimmer? Round to the nearest foot. (Lesson 8-4) �� 2. Recreation A neighborhood park has a 15-foot-long space available to install a playground slide. If the maximum height of the slide is 6 ft, what are the lengths of the slide x and ladder y that should be installed? Round to the nearest tenth of a foot. (Lesson 8-1) 3. Building The escalator at the mall forms a 35° angle with the floor. The vertical distance from the bottom of the escalator to the top is 25 ft. How long is the escalator? Round to the nearest foot. (Lesson 8-2) 4. Sports A 3-foot-long skateboard ramp forms a 40° angle with the ground. How far above the ground is the end of the ramp? Round to the nearest foot. (Lesson 8-2) 5. Running A race includes a 0.25-mile hill on which runners travel from 510 ft of elevation to 570 ft of elevation. What angle does the hill form? Round to the nearest degree. (Lesson 8-3) 7. Aviation A helicopter pilot flying at an altitude of 1200 ft sees two landing pads directly in front of him. The angle of depression to the first landing pad is 40°. The angle of depression to the second pad is 28°. What is the distance between the two pads? Round to the nearest foot. (Lesson 8-4) 8. Carpentry Sean is creating a triangular frame from three wooden dowels, which are 18 in., 12 in., and 15 in. long. What are the measures of each angle of the triangle? Round to the nearest degree. (Lesson 8-5) 9. Sports To estimate the width of the sand trap on a golf course, Matthew locates three points and measures the distances shown. What is the width, XZ, of the sand trap to the nearest foot? (Lesson 8-5) 10. Recreation Jill swims due east across a river at 2 mi/h. The river is flowing north at 1.5 mi/h. What are Jill’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. (Lesson 8-6) TEKS TAKS Practice S35 S35 �� Chapter 9 Applications Practice 1. Recreation Kathy is making a kite with diagonals of lengths 30 inches and 20 inches. How many square inches of fabric will she need? (Lesson 9-1) Agriculture Use the following information for Exercises 2 and 3. An acre is 43,560 square feet. (Lesson 9-1) 2. If a one-acre piece of land is a rectangle with a base of 100 ft, what is its height? 3. If a one-acre piece of land is a square, what is the length of each side? Round to the nearest tenth. 4. The garden shown is a regular hexagon with a circular fountain at the center. What is the area of the garden? Round to the nearest square foot. (Lesson 9-2) �� 5. Food A bakery has cheesecake pans with three diameters: 18 cm, 22 cm, and 26 cm. Find the area of the bottom of each pan. Round to the nearest square centimeter. (Lesson 9-2) 6. Recreation A track for a toy car is a 2 ft by 2 ft square with a semicircle at each end. What is the distance around the track? Round to the nearest foot. (Lesson 9-3) 7. Art Jonas is painting the shape shown on his ceiling. If a quart of paint covers 75 square feet, will one quart be enough to paint the entire shape? Explain. (Lesson 9-3) Transportation Use the following information for Exercises 8 and 9. The graph shows the speed of a car versus time. The base of each square on the graph represents 10 minutes, and the height represents 10 miles per hour. (Lesson 9-4) 8. What is the area of one square on the graph? 9. Estimate the shaded area of the graph. 10. Art Rasha is cutting a mat for a poster with an area of 480 in 2 . To find the dimensions of the mat, she multiplies the dimensions of the poster by 1.2. To find the dimensions of the opening, she multiplies the dimensions of the poster by 0.9. What is the area of the remaining part of the mat? (Lesson 9-5) 11. Food A restaurant sells two sizes of pizzas. The smaller pizza has a 12-inch diameter. If the area of the larger pizza is twice the area of the smaller pizza, what is the diameter of the larger pizza? Round to the nearest inch. (Lesson 9-5) 12. Transportation A commuter train stops at a station every 3 minutes and stays at the station for 20 seconds. If you arrive at the station at a random time, what is the probability that you will have to wait more than one minute for a train? Round to the nearest hundredth. (Lesson 9-6) 13. Sports A skydiver is delivering the game ball for a baseball game. Suppose he lands at a random point on the field. What is the probability that he will not land on the pitcher’s mound? Round to the nearest hundredth. (Lesson 9-6) �� S36S36 TEKS TAKS Practice �� Chapter 10 Applications Practice 1. Food Cookie dough is rolled in the shape of a cylinder. How can the dough be sliced to make circular cookies? (Lesson 10-1) 2. Recreation The tent shown is in the shape of a pentagonal prism. If a wall is used to divide the tent into two rooms, what shapes could the wall be? (Lesson 10-1) 7. Camping The tent structure shown is in the shape of a square pyramid. How many square inches of canvas are required to cover the tent? Round to the nearest square inch. (Lesson 10-5) 3. Business Eli is creating a logo for his busine
ss by drawing his name in block capital letters using one-point perspective. Draw Eli’s logo. (Lesson 10-2) 4. Recreation Two hot air balloons were launched from the same location. The first balloon is 5 miles north, 9 miles east, and 0.5 mile above the launching point. The second balloon is 9 miles north, 5 miles east, and 0.8 mile above the launching point. How far apart are the two balloons? Round to the nearest tenth. (Lesson 10-3) 5. Manufacturing The two packages shown hold the same amount of food. Which requires a greater amount of material to produce? (Lesson 10-4) Recreation Use the following information for Exercises 8 and 9. A cylindrical pool has a 10 ft diameter. (Lesson 10-6) 8. How many gallons of water are needed to fill the pool to a depth of 4 feet? Round to the nearest gallon. (Hint: 1 gallon ≈ 0.134 cubic feet.) 9. If the pool is filled to a depth of 4 feet, how much will the water weigh? Round to the nearest pound. (Hint: 1 gallon weighs about 8.34 pounds.) 10. Hobbies The greenhouse shown is in the shape of a cube with a square pyramid on top. What is the volume of the greenhouse? (Lesson 10-7) 6. Hobbies Ashley is using the pattern shown to make cones to protect her plants from freezing. How tall can the plants be to fit in the cone? Round to the nearest tenth. (Lesson 10-5) 11. Food A snow-cone cup has a 3-inch diameter and is 4 inches tall. Another snow-cone cup has a 4-inch diameter and is 3 inches tall. Which cup will hold more? (Lesson 10-7) 12. Sports The circumference of a size 3 soccer ball is 24 in. The circumference of a size 5 soccer ball is 28 in. How many times as great is the volume of a size 5 ball as the volume of a size 3 ball? (Lesson 10-8) TEKS TAKS Practice S37 S37 �� Chapter 11 Applications Practice 1. Measurement There is a water tower near Peter’s house in the shape of a cylinder. He wants to find the diameter of the tank. Peter stands 25 feet from the tower. The distance from Peter to a point of tangency on the tower is 80 feet. What is the diameter of the tank? (Lesson 11-1) 2. Travel Pikes Peak is 14,110 feet above sea level. What is the distance from the summit to the horizon, to the nearest mile? (Hint: Earth’s radius ≈ 4000 mi) (Lesson 11-1) Art Use the diagram to find each value for Exercises 13–15. The diagram represents an engraving on a stained glass window. (Lesson 11-4) 13. x 14. y 15. m ⁀ FE Hobbies Use the circle graph to find each measure for Exercises 3–6 to the nearest degree. Eric collects baseball cards. He has 85 cards from the 1970s, 95 cards from the 1980s, and 125 cards from the 1990s. (Lesson 11-2) 3. ⁀ AB 4. ⁀ AC 5. ∠CDB 6. ∠ADC Data Use the circle graph to find each measure for Exercises 7–10 to the nearest degree. The circle graph shows the color of cars in a parking lot at the mall. (Lesson 11-2) 7. ⁀ HG 8. ⁀ CD 9. ∠AJH 10. ∠FJE 16. Astronomy Two satellites are orbiting Earth. Satellite A is 10,000 km above Earth, and satellite B is 13,000 km above Earth. How many arc degrees of Earth does each satellite see? (Lesson 11-5) �� 17. Entertainment A group of friends ate most of a pepperoni pizza. All that was left was a piece of crust. What was the diameter of the original pizza? (Lesson 11-6) Hobbies Use the following information to find each area to the nearest tenth for Exercises 11 and 12. A sprinkler system has three types of sprinkler heads: a quarter circle, a semicircle, and a full circle. The sprinkler will spray a distance of 15 feet from the sprinkler head. (Lesson 11-3) 11. What is the area of the sector that will be watered by the quarter circle sprinkler head? 12. What is the area of the sector that will be watered by the semicircle sprinkler head? S38S38 TEKS TAKS Practice 18. Safety Three small towns have agreed to share a new fire station. To make sure each town has equal response time, the station should be the same distance from each town. The three towns are located on a coordinate plane at (0, 0) , (6, 0) , and (0, 8) . At which coordinates should the station be built? (Lesson 11-7) �� Chapter 12 Applications Practice 1. Transportation Two towns are located on the same side of a river. Two roads are being built to meet at the same point P on the river. Draw a diagram that shows where P should be located in order to make the total length of the roads as short as possible. (Lesson 12-1) Agriculture Use the following information for Exercises 6–8. Cattle ranchers brand their cattle to show ownership. Three different brands are shown. (Lesson 12-5) 2. Fashion A piece of fabric used for a scarf has a repeating pattern of trapezoids. To create the pattern, translate the trapezoid with vertices (-1, 3) , (3, 3) , (4, 1) , (-2, 1) along the vector 〈0, -2〉. Repeat to generate a pattern. What are the vertices of the third trapezoid in the pattern? (Lesson 12-2) 3. Computers A screen saver moves an icon around a screen. The icon starts at (20, 0) , and then it is rotated about the origin by 50°. Give the icon’s next position. Round each coordinate to the nearest tenth. (Lesson 12-3) 4. Recreation A hole at a miniature golf course has a barrier between the tee T and the hole H. Copy the figure and draw a diagram that shows how to make a hole in one. (Lesson 12-3) 5. Sports A team’s Web site shows a baseball moving across the screen. The ball is reflected over line ℓ and is then reflected over line m. Describe a single transformation that moves the ball from its starting point to its final position. (Lesson 12-4) 6. Which brands have rotational symmetry? 7. Which brands have line symmetry? 8. Which capital letters could be used to create a brand with rotational symmetry? Interior Design Use the following information for Exercises 9–11. Three kitchen backsplash tile patterns are shown. Identify the symmetry in each pattern. (Lesson 12-6) 9. 10. 11. 12. Hobbies Reid has a baseball card that is 2.5-by-3.5 inches. He wants to enlarge it to poster size using a scale factor of 8. What size poster frame should he buy? (Lesson 12-7) 13. Hobbies A 40 in. by 30 in. piece of art is being made into a 1 in. by 3 __ 4 in. postage stamp. What scale factor should be used to reduce the art? (Lesson 12-7) TEKS TAKS Practice S39 S39 �� Problem-Solving Handbook Draw a Diagram When a problem involves objects, distances, or places, drawing a diagram can make the problem clearer. You can draw a diagram to help understand and solve the problem. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E During a team-building activity, five people stand in a circle. Pieces of ribbon will be used to connect each person to each of the other four people in the circle. How many pieces of ribbon are needed to connect all five people in this way? Understand the Problem List the important information. • There are five people standing in a circle. • Each person should be connected to each of the other four people with a piece of ribbon. The answer is the number of pieces of ribbon needed to connect all five people. Make a Plan Draw a diagram to represent the information in the problem. Solve Draw a circle. Add five points to the circle to represent the five people in the problem. Then draw segments to connect each point to each of the other four points. Count the number of segments in the final diagram. The total number of segments is the answer to the problem. It takes 10 pieces of ribbon to connect each person to each of the other four people. Look Back Check that the diagram is drawn correctly and that you counted the number of line segments accurately. PRACTICE 1. A delivery truck driver travels 15 miles south to deliver his first package. He then goes 9 miles east and 6 miles north to deliver his next package. From there, the driver travels 12 miles east to make his last delivery. How far is the driver from his starting point? Round to the nearest tenth of a mile. S40 S40 Problem-Solving Handbook 1234 Make a Model When a problem involves manipulating objects, you can use those or similar objects to make a model. This can help you to understand the problem and find the solution. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E During a geometry class, Zach cuts out a parallelogram with base 12 cm and height 6 cm. Catherine cuts out a rectangle with the same base and height. Show that the two shapes have the same area. Understand the Problem List the important information. • There are two geometric shapes, a parallelogram and a rectangle. • The base and the height of the two shapes are the same. To solve the problem, you need to show that the areas of the two shapes are equal. Make a Plan You can make a model of the figures by cutting them out of paper or cardboard. Then compare the areas by placing one on top of the other. Solve If the shaded area of the parallelogram is cut and moved to the opposite side, the figure becomes a rectangle. Place the two shapes on top of each other to compare the area. The shapes have the same base and height, so these shapes have the same area. Look Back Check that your models have the correct dimensions. Use the formulas for the area of a rectangle and a parallelogram to confirm that the shapes have the same area. PRACTICE 1. Find the dimensions of a rectangular prism made up of 16 1-inch cubes. 2. Two triangles are formed by cutting a rectangle along its diagonal. What possible shapes can be formed by arranging these triang
les? Problem-Solving Handbook S41 S41 12��3��4 Guess and Test For complex problems, you can use clues to make guesses and narrow your choices for the solution. Test whether your guess solves the problem, and then continue guessing until you find the solution. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E Edgar is designing a party invitation in the shape of a right triangle. If all three side measures are to be whole numbers of inches, what is the smallest possible perimeter for the birthday card? Understand the Problem List the important information. • The invitation is to be a right triangle. • The legs and hypotenuse must be whole numbers. To solve the problem, you need to find the smallest possible perimeter for the right triangle. Make a Plan You can guess and test, starting with the smallest possible whole numbers. Solve Let a and b be the legs of the right triangle, and let c be the hypotenuse. So the relationship a 2 + b 2 = c 2 must hold. Start by using (1, 1) for (a, b) and solve for c 2 . Since c must be a whole number, continue to guess and test until c 2 is a perfect square. Guess (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) Test = 10 ✘ 1 2 + 4 2 = 17 ✘ 1 2 + 5 2 = 26 ✘ Guess (2, 2) (2, 3) (2, 4) (2, 5) Test = 13 ✘ 2 2 + 4 2 = 20 ✘ 2 2 + 5 2 = 29 ✘ Guess (3, 3) (3, 4) Test 3 2 + 3 2 = 18 ✘ 3 2 + 4 2 = 25 ✓ Based on the tables, 5 is the smallest possible whole number for c, 3 for a, and 4 for b. So the smallest possible perimeter for the card is 3 in. + 4 in. + 5 in. = 12 in. Look Back Since 3 2 + 4 2 = 5 2 , these are reasonable dimensions for the card. The problem asks for the perimeter, which is 12 inches. PRACTICE 1. The sum of Cary’s age and his brother’s age is 34. The difference between their ages is 4. How old are Cary and his brother? 2. Adult tickets for a theater performance cost $8 and children’s tickets cost $3. A group with twice as many adults as children attends the performance and spends $133 on tickets. How many people are in the group? S42 S42 Problem-Solving Handbook 1234 Work Backward Some problems involve a series of events, giving you information about the last event, and then ask you to solve something related to the initial situation. You can work backward to solve these problems. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E Sandy is creating a pattern made from isosceles right triangles as shown below. If the hypotenuse of the fifth triangle is 4 in., what are the dimensions of the smallest triangle? Understand the Problem List the important information. • Each triangle is an isosceles right triangle, so each triangle’s legs are congruent. • The hypotenuse of one triangle is equal to the leg length of the next triangle. • The fifth triangle’s hypotenuse is 4 in. You must work backward to find the dimensions of the first triangle. Make a Plan Let h be the hypotenuse and s be the leg length of each triangle. Start with a hypotenuse length of 4 and work backward using the Pythagorean Theorem, which states that . Solve Triangle 5: h = 4 , so s = √  8 Triangle 4 The hypotenuse of Triangle 4 equals the leg length of = 2 s 2 , so s = 2. Triangle 5. ( √  8 ) 2 Triangle 3: h = 2 , so s = √  2 . Triangle 2 , so s = 1. Triangle 1: h = 1; s = √  0.5 1 2 = 2 s 2 , so s = √  0.5 . The first triangle should have a leg length of √  0.5 and a hypotenuse of 1. Look Back Recreate the diagram starting with the dimensions you found for the first triangle, and confirm that the fifth triangle has a hypotenuse of 4 inches. PRACTICE 1. In a trivia game, each question is worth twice as many points as the one before it. Chelsea answers 5 questions and earns 1550 points. How many points was her first question worth? 2. Sheryl has 4 siblings. She is 4 years younger than her sister Meagan. Meagan is twice as old as Tina. Jack is 3 years older than Tina, and Tina is 1 year older than Bryan, who is 9. How old is Sheryl? Problem-Solving Handbook S43 S43 ��1234 Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Find a Pattern In some problems, there is a relationship between different pieces of information. You can find a pattern to help solve these problems. E X A M P L E Frank plants turnips in rows and columns. Each year, he increases the size of his turnip patch by adding one row and one column, as shown in the diagram. How many turnip plants will Frank have after year 5? Understand the Problem List the important information. • In year 1, Frank has 3 turnip plants. • In year 2, he has 8 turnip plants. • In year 3, he has 15 turnip plants. The answer will be the number of turnip plants in year 5. Make a Plan Find the pattern based on the diagram. Solve Make a table of the given information and find a pattern. Number of Turnip Plants Possible Pattern Year 1 Year 2 Year 3 3 8 15 0 + 3 3 + 5 8 + 7 The pattern seems to be the number of turnip plants in the previous year plus the next odd number. So in year 4, Frank will have 15 + 9 = 24 turnip plants, and in year 5, he will have 24 + 11 = 35 turnip plants. Look Back By thinking of the number of plants as the product of the number of rows and columns, you might notice another pattern, n (n + 2) , where n is the year number. Use this to confirm your answer. Year 4: 4 (4 + 2) = 24 turnip plants Year 5: 5 (5 + 2) = 35 turnip plants PRACTICE 1. Use the key GDB = DAY to decode the sentence DQ DSSOH D GDB NHHSV WKH GRFWRU DZDB. 2. Describe the pattern 15, 22, 29, 36, 43, ... and find the next two numbers. S44 S44 Problem-Solving Handbook ��1234 Make a Table To solve a problem that involves a relationship between two sets of numbers, you can make a table to organize and analyze the data. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E Roy’s Geometry class is playing a game to practice identifying shapes. There are eight shapes in the game: an acute triangle, a right triangle, a square, a rectangle, a rhombus, a parallelogram, a kite, and an isosceles trapezoid. On Roy’s turn, the teacher reads the following clues: The shape is a quadrilateral with four right angles in which all sides are not congruent. Which shape should Roy select? Understand the Problem List the important information. • The possible shapes are an acute triangle, a right triangle, a square, a rectangle, a rhombus, a parallelogram, a kite, and an isosceles trapezoid. • Roy’s shape is a quadrilateral with four right angles. • Roy’s shape does not have four congruent sides. The answer will be the shape that matches Roy’s clues. Make a Plan Make a table and use the given information to identify Roy’s shape. Solve Use the given clues to complete a table and identify Roy’s shape. Shape Quadrilateral? 4 Right Angles? Sides Not Congruent? Acute triangle Right triangle Square Rectangle Rhombus Parallelogram Kite Isosceles trapezoid The rectangle is the only shape that satisfies the given clues. Look Back Make sure that your answer satisfies the given clues. N N N Y N N Y Y PRACTICE 1. Katie gets the following clues: The shape has at least one right angle, has no parallel sides, and is not the kite. Which shape should Katie select? 2. Mary gets the following clues: The shape has no congruent sides. How many possible shapes might Mary select? Problem-Solving Handbook S45 S45 1234 Solve a Simpler Problem A problem with many steps or involving very large numbers can be overwhelming. Sometimes it helps to solve a simpler problem first, or to break the complex problem into multiple simpler ones. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E Tom plans to repaint his patio, which has the measurements shown below. What is the total area that Tom needs to paint? Understand the Problem List the important information. • AH = FG = 4 ft • DE = 3 ft • CD = 20 ft • GH = EF = 5 ft The answer will be the total area of the patio. Make a Plan To simplify the problem, divide the patio into basic geometric shapes and add their areas together. Solve Find the area of the patio as if it were a complete rectangle, and then subtract the area of the smaller rectangle that is not part of the patio. Step 1: Find the area of each rectangle. Larger rectangle Smaller rectangle Length CD = 20 ft AH + FG + DE = 11 ft GH = 5 ft FG = 4 ft Width Area ℓw = (20) (11) = 220 ft 2 ℓw = (5) (4) = 20 ft 2 Step 2: Subtract the areas to find the area of the patio. Area of patio = 220 - 20 = 200 ft 2 Tom needs to paint 200 square feet. Look Back Divide the patio into a different arrangement of smaller shapes to check your answer. For example, by dividing the patio into three rectangles stacked on top of each other, you find that (4) (20) + (4) (15) + (3) (20) = 200 ft 2 , which confirms the first answer. PRACTICE 1. How much paint does Rose need to repaint her patio? 2. Rose plans to add a decorative railing around the outer edges of her patio. The railing will cover every edge except the 30-foot side of the patio that joins her house. About how many feet of railing does Rose need? Round to the nearest foot. S46 S46 Problem-Solving Handbook ��1��234 Use Logical Reasoning Some problems provide clues and facts that you must use to find the solution. To use logical reasoning, ide
ntify these facts and draw conclusions from them. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E Dawn, Chloe, and Tyra finish first through third in a cross-country race. The girls wear the numbers 7, 8, and 12. Dawn does not wear an even number. The one who wears number 8 comes in first. Chloe comes in third. Who wears which number, and in what place did each runner finish? Understand the Problem List the important information. • Dawn wears an odd number. • The girl who wears number 8 comes in first place. • Chloe comes in third place. The answer will be a list of who wears which number and each girl’s finishing position. Make a Plan Start with the clues given in the problem. Use logical reasoning to determine each girl’s number and finishing position. Solve Make a table. Read the clues one at a time, and mark the table appropriately. • Dawn wears an odd number, so she must wear number 7. No other girl can wear number 7. • The girl who wears number 8 comes in first place. No other number is the first-place winner. Also, since Dawn wears number 7, she didn’t come in first. • Chloe comes in third. By process of elimination, Dawn must have come in second, and Tyra came in first. So Tyra wears number 8, and thus Chloe wears number 12. 7 Dawn ✓ Chloe Tyra 1st 2nd 3rd ✘ ✘ ✘ ✓ ✘ 8 ✘ ✘ ✓ ✓ ✘ ✘ 12 1st 2nd 3rd ✘ ✘ ✓ ✓ ✘ ✘ ✘ ✓ ✘ ✘ ✓ ✘ ✘ ✘ ✓ Look Back Compare your answer to the facts given in the problem. Make sure none of your conclusions conflict with the given clues. PRACTICE 1. Mike, Jack, and Ann each wear a different type of top in three different colors. The tops are a button-down shirt, a pullover, and a sweater. The colors are blue, yellow, and red. Mike wears a blue shirt, and Jack wears a button-down. The yellow top is a pullover. Who wears the sweater and who wears the red top? 2. The Warriors, Jaguars, and Cougars each have a different-colored shape on their team shirt. The colors are green, purple, and red, and the shapes are a triangle, a rectangle, and a hexagon. The Warriors’ shape has the most sides, the color of the Jaguars’ shape is green, and the rectangle is purple. Which team has which shape and in which color? Problem-Solving Handbook S47 S47 1234 Use a Venn Diagram A Venn diagram can be useful when you solve a problem involving relationships among sets or groups. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E In a class of 15 students, ten play on at least one of the school sports teams—the basketball team or the baseball team. Five of them are on the basketball team. Three students are on both the basketball team and the baseball team. How many of the students play on the baseball team? Understand the Problem List the important information. • 5 students play on the basketball team. • 3 students play on both teams. The answer is the number of students who play on the baseball team. Make a Plan Organize the information by drawing a Venn diagram. Solve Draw and label the Venn diagram. • 3 people will be in the overlapping area. • Since 5 people play on the basketball team, and 3 of them are also on the baseball team, only 2 people play on only the basketball team. There are 10 student players in all, and five are already accounted for. Therefore, the remaining five play only baseball. Adding the three students who also play basketball, a total of eight students in the class play on the baseball team. Look Back Check your Venn diagram to make sure it is an accurate representation of the information given in the problem. Confirm that the numbers in each of the labeled sections add up to the total number of students in the problem. PRACTICE 1. At Lucy’s Home-Style Restaurant, four of the meals include a side salad, six include only soup as a side, and two meals come with both salad and soup as sides. If all meals come with at least one side, how many different meals are on Lucy’s menu? 2. A cupboard contains 12 cups, and each cup has a lid, a handle, or both. There are seven cups with handles, and three cups with both a lid and a handle. How many cups have only a lid? S48 S48 Problem-Solving Handbook ��1234 Make an Organized List If a problem involves multiple outcomes, it may be useful to make an organized list to record the data and count the different outcomes. Problem-Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List E X A M P L E Sally randomly selects two shapes from a bag that contains five different cut-outs: a triangle, a square, a rectangle, a pentagon, and a hexagon. The sum of the number of sides of the two shapes is eight. What combinations of shapes might Sally have selected? Understand the Problem List the important information. • The possible shapes are a triangle, a square, a rectangle, a pentagon, and a hexagon. • The sum of the number of the sides is 9. The answer will be the two shapes Sally selected. Make a Plan Make an organized list of the possible combinations of shapes. Then list the number of sides and the sum of the sides. Solve List the possible combinations of shapes, and find the sum of the shapes’ sides. Triangle (3) Square (4) ✘ ✘ Rectangle (4) Pentagon (5) Hexagon (6) Total number of sides ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ 7 7 8 ✘ 9 8 9 10 9 10 11 There are two combinations of shapes that have a total number of eight sides: the triangle and pentagon and the square and rectangle. Look Back Make sure all possible combinations are shown in the table. Check that the total number of sides for both combinations (triangle and pentagon, and square and rectangle) is 8. PRACTICE 1. How many ways can you make $0.30 by using quarters, dimes, nickels, and pennies? 2. Pete’s Pizza Palace has 5 choices of meat, 4 choices of vegetables, and 2 choices of cheese. You want to order a pizza with one of each. How many combinations can you order? Problem-Solving Handbook S49 S49 1234 Skills Bank Operations with Real Numbers The four basic operations with real numbers are addition, subtraction, multiplication, and division. E X A M P L E Simplify each expression.5 · 3 3.5 · 3 = 10.5 B 0.5 - 4 0.5 - 4 = -3. PRACTICE Simplify each expression. 1. 3 + 4 2. 9 · 9 5. 0.1 ⋅ 0.1 9. -3 · 7 6. 0.5 + 2 3. 3 ÷ 3 7. 4 - 0.3 4. 10 - 8 8. 8 ÷ 2 10. 6.3 - 8.1 11. -12 ÷ (-3) 12. 17.3 + 12.9 Order of Operations When simplifying expressions, follow the order of operations. 1. Simplify within parentheses. 2. Evaluate exponents and roots. 3. Multiply and divide from left to right. 4. Add and subtract from left to right. E X A M P L E Simplify the expression (3 + 2 2 ) · (3 - 1) 2 . (3 + 2 2 ) · (3 - 1) 2 (3 + 4) · (3 - 1) 2 Simplify within parentheses. There is an exponent within the first set of parentheses, so simplify it first. 7 · 2 2 7 · 4 28 Simplify within parentheses. Evaluate the exponent. Multiply. PRACTICE Simplify each expression. 1. 3 · (4 + 1) 2 2. (1 - 3) + 4 · 5 4. 6 - 5 + 2 2 · (9 - 7) 3. S50 S50 Skills Bank Skills Bank Properties Below are the basic properties of addition and multiplication, where a, b, and c are real numbers. Addition Multiplication Closure a + b is a real number. Closure a · b is a real number. Commutative a + b = b + a Commutative a · b = b · a Associative (a + b) + c = a + (b + c) Associative (a · b) · c = a · (b · c) Identity Property of Zero a + 0 = a and 0 + a = a Identity Property of One Multiplication Property of Zero a · 1 = a and and 0 · a = 0 Distributive a · (b + c) = a · b + a · c Transitive If a = b and b = c, then a = c. Other Real Number Properties E X A M P L E 1 Name the property shown. A 2 · (3 - 3) = 0 Multiplication Property of Zero B (9 + 3) + 2 = 9 + (3 + 2) Associative Property of Addition E X A M P L E 2 Give an example of each property, using real numbers. A Closure Property of Multiplication 1.8 · 2.4 is a real number. B Commutative Property of Addition 17 + 84 = 84 + 17 PRACTICE Name the property shown, where a, b, and c are real numbers. 1. If 3 + 8 = x and x = y, then 3 + 8 = y. 2. a + b is a real number. 3. 0 · 9 = 0 5. 10 · (b + c) = 10b + 10c 4. 3c · 2a = 2a · 3c 6. 3a is a real number. 7. (2a + 3b) + 2c = 2a + (3b + 2c) 8. 1 · 2a = 2a Give an example of each property, using real numbers. 9. Identity Property of Addition 10. Distributive Property 11. Commutative Property of Multiplication 12. Closure Property of Multiplication 13. Closure Property of Addition 14. Transitive Property Skills Bank S51 S51 Estimation, Rounding, and Reasonableness Estimation involves rounding numbers. To round a number to a given place value, look at the digit to the right of that place value. If it is greater than or equal to 5, round up. If it is less than 5, round down. E X A M P L E 1 Round each number to the given place value. A 1.2941 to the nearest tenth 9 > 5; round up. 1.2941 B 3.14159 to the nearest hundredth 3.14159 1 < 5; round down. 1.3 100 ___ ̶ 3 33.33 33.333 C to the nearest unit 3 Convert to a decimal. 3 < 5; round down. 3.14 D √  5 to the nearest tenth 2.2360 Convert to a decimal. 2.2360 3 < 5; round down. 33 2.2 E X A M P L E 2 Estimate each sum by rounding. A 12.75 + 15.94 13 + 16 29 Add. Round each number. B 182 + 208 + 319 180 + 210 + 320 Round each number. 710 Add. E X A M P L E 3 Tell whether an estimate is sufficient or an exact answer is needed. A The distance from San Antonio to Austin is about 80 miles. The distance from Austin to Dallas is about 190 miles. If you drive from San Antonio to Austin then to Dallas, about how far did you drive? The problems asks “about how far,” so an estimate is sufficient. B Kim buys two shirts f
or $12.95 each. Sales tax is 8.25%. How much money does she need? The problem asks for the amount of money, so an exact answer is needed. PRACTICE Round each number to the given place value. 1. 285,618 to the nearest hundred 2. 9.7 to the nearest unit 3. 49.249 to the nearest tenth 4. 873.59 to the nearest ten Estimate each sum by rounding. 5. 73.98 + 180.76 6. 251 + 489 7. 45,792 + 13,819 8. 0.034 + 0.015 9. 27.1 + 43.8 10. 862 + 740 Tell whether an estimate is sufficient or an exact answer is needed. 11. Eric buys dinner for $10.75 and wants to leave about an 18% tip. How much money does he need? 12. Find the height of a triangle with A = 24 in 2 and b =16 in. 13. Ginny is planting grass seed in a 6 ft by 10 ft rectangular patch of lawn. A pound of seeds covers about 1000 ft 2 . About how much seed will she use? S52 S52 Skills Bank Classify Real Numbers A set is a group of items. Numbers can be organized into sets. Set Examples Venn Diagram The natural numbers are the counting numbers. The whole numbers are the natural numbers plus 0. The integers are the whole numbers and their opposites. Rational numbers can be written as a ratio of two integers. Irrational numbers cannot be written as a ratio of two integers. The real numbers are the rational numbers plus the irrational numbers. {1, 2, 3, 4, …} {0, 1, 2, 3, …} {…, -1, 0, 1, 2, …}   1 _ ⎬ ⎨ , -3.4, 0 Write all of the names that apply to each number. A -79 real number, rational number, integer B √  13 real number, irrational number PRACTICE Write all of the names that apply to each number. ̶ 2. 0. 1. 11 3 3. π 4. -4.6 5. 0 Exponents Exponents are used to describe repeated multiplication. In the expression b n , b is the base and n is the exponent . A negative exponent is used to represent the reciprocal of the base with the opposite exponent. E X A M P L E Evaluate. A 2 5 2 5 = (2) (2) (2) (2) (2) = 32 PRACTICE Evaluate. 3 6 1. 2. 2 10 B 3 -2 3 -3) (3) = 1 _ 9 3. 5 -3 4. 6 4 5. 7 -2 6. 3 4 Skills Bank S53 S53 �� Properties of Exponents The following properties can be used to simplify expressions with exponents. WORDS NUMBERS ALGEBRA The quotient of two nonzero powers with the same base is the base raised to the difference of the exponents. The product of two powers with the same base equals the base raised to the sum of the exponents If a ≠ 0, then 11 If a ≠ 0, then Any nonzero number raised to the zero power is 1 If a ≠ 0, then a 0 = 1. E X A M P L E 1 Simplify PRACTICE Simplify. 1. 3 b _ 3 2 3. ( b m ) ( b -m ) __ n 0 4. ( xyz 3 ) ( z 2 ) Powers of 10 and Scientific Notation Scientific notation is used to write very large numbers, such as the speed of light (about 300,000,000 m/s) and very small numbers, such as the average diameter of an atom (0.000000030 cm). In scientific notation, the speed of light is 3.0 × 10 8 m/s, and the average diameter of an atom is 3.0 × 10 -8 cm. E X A M P L E 1 Write each number in standard notation. A 2.99 × 10 4 B 3.04 × 10 -6 2.99 × 10,000 29,900 10 4 = 10,000 Move the decimal 3.04 × 0.000001 0.00000304 10 -6 = 0.000001 Move the decimal point 4 places right. point 6 places left. PRACTICE Write each number in standard notation. 1. 10 3 2. 10 8 4. 9.04 × 10 2 5. 9.0 × 10 -4 3. 10 -4 6. 1.0 × 10 0 S54 S54 Skills Bank Square Roots The square root of a number is one of the two equal factors of that number. Every positive number has two square roots, one positive and one negative. E X A M P L E Find the two square roots of each number. A 81 √  81 = 9 9 is a solution, since 9 · 9 = 81. - √  81 = -9 -9 is a solution, since (-9) · (-9) = 81. B 121 121 = 11 √  - √  121 = -11 11 is a solution, since 11 · 11 = 121. -11 is a solution, since (-11) · (-11) = 121. PRACTICE Find the two square roots of each number. 1. 64 2. 49 3. 225 4. 1 Simplifying Square Roots A square root is in simplest form when the radicand contains no perfect squares and no fractions and there are no square roots in a denominator. The following properties are used to simplify square roots. Multiplication Property: √  Division Property Simplify √ _  8 . 18  8 _ √ 18 = √  8 _ √  18 = √  4 · 2 _ √  Division Property Factor. Multiplication Property Simplify. Simplify. PRACTICE Simplify each square root. 1. √  640 2. √  936 3. √  242 Skills Bank S55 S55 The Coordinate Plane Recall that to locate a point on a number line with a given coordinate, you move left or right from 0. The coordinate plane is formed by two perpendicular number lines, the x-axis and the y-axis , that intersect at the origin , (0, 0) . The location of a point is described by an ordered pair , (x, y) , where x is the distance from the y-axis and y is the distance from the x-axis. The coordinate plane is divided into four quadrants . Quadrant I: x and y are both positive. Quadrant II: x is negative, and y is positive. Quadrant III: x and y are both negative. Quadrant IV: x is positive, and y is negative. E X A M P L E 1 Graph each point. A (3, -1) B (-2, 3) Start at the origin. Move right 3 and then down 1. Start at the origin. Move left 2 and then up 3. E X A M P L E 2 In which quadrant is each ordered pair located? A (1, 1) Quadrant I x and y are both positive. B (3, -3) Quadrant IV x is positive, and y is negative. PRACTICE Graph each point. 1. (5, 1) 2. (3, -1) 3. (-2, 0) 4. (-4, -3) In which quadrant is each ordered pair located? 5. 6. (-2, -6) (2, 6) 7. (-5, 1) 8. (9, -2) S56 S56 Skills Bank �� Connecting Words with Algebra Word phrases or sentences can be written as algebraic expressions. Symbol Word Phrases Algebraic Expressions + - × or · ÷ • a number plus 3 • 3 more than a number • a number minus 2 • the difference of a number and 2 • 3 times a number • the product of 3 and a number • a number divided by 2 • the quotient of a number and 2 n + 3 m - 2 3x or 3 · x k _ or Write an algebraic expression for each word phrase. A a pie cut into 8 equal slices B 3 sheets of paper added to a stack p ÷ 8 s + 3 PRACTICE Write an algebraic expression for each word phrase. 1. 3 lb more than an apple 2. 10 times as heavy as a horse 3. 3 years less than 9 times Gwen’s age Variables and Expressions A variable is a letter that represents a value that can change or vary. An algebraic expression has one or more variables. To evaluate an algebraic expression, substitute the given value for each variable, and simplify the expression. E X A M P L E Evaluate each expression for the given values of the variables. A 3n for n = 2 B a + b for a = 2 and b = 3 3n 3 (2) 6 Substitute. Simplify. a + b (2) + (3) 5 Substitute. Simplify. PRACTICE Evaluate each expression for the given values of the variables. 1. 2k for k = 6 2. 3 - n for n = 4 3. 3x for x = 0 4. xy for x = 2 and y = 3 5. i + 2 for i = 1.7 6. -1k for k = -1 7. 2x for x = 0.5 8. 4n + 7 for n = 13 9. u 2 v for u = 3 and v = 7 Skills Bank S57 S57 Solving Linear Equations The following properties can be used to solve linear equations in one variable. PROPERTY WORDS NUMBERS ALGEBRA Addition Property of Equality Subtraction Property of Equality You can add the same number to both sides of an equation, and the statement will still be true. You can subtract the same number from both sides of an equation, and the statement will still be true. Multiplication Property of Equality You can multiply both sides of an equation by the same number, and the statement will still be true. Division Property of Equality You can divide both sides of an equation by the same nonzero number, and the statement will still be true3) = 6 (3) 18 = 18 ac = bc (c ≠ 0 Solve. A x + 5 = 11 x + 5 = 11 - 5 - 5 ̶̶̶ ̶̶̶̶ x = 6 C 6y - 4 = 38 6y - 4 = 38 + 4 + 4 ̶̶̶ ̶̶̶̶̶ 6y = 42 6y = 42 _ _ 6 6 y = 7 PRACTICE Solve. 1. x + 7 = 2 Subtract 5 from both sides. B D _ u = 10 2 2 · u _ = 2 · 10 2 u = 20 _ 3 + n 8 = -2 Multiply both sides by 2. Add 4 to both sides. Divide both sides by 6 (-2) 3 + n = -16 - 3 ̶̶̶̶̶ - 3 ̶̶̶ n = -19 Multiply both sides by 8. Subtract 3 from both sides. 2. 2u = 6 3. n _ 3 = 21 4. 13 = x - 16 5. 1.5k = 27 6. 18 + p = 16 8. b - 2.7 = 3.4 9. 2w + 7 = 18 11. 4z - 8 = 18 12. a-2 _ = 1 7 7. 15 = h _ 30 10. d _ 5 + 4 = 17 S58 S58 Skills Bank Solving Equations for a Variable Equations with more than one variable are sometimes called literal equations. Sometimes it is necessary to solve literal equations for one variable in terms of the others. This is also called isolating the variable. E X A M P L E Solve for y. 3y + 2x = y + 2 3y + 2x = -y ̶̶̶̶̶̶ 2y + 2x = - 2x ̶̶̶̶̶̶ y + 2 - y ̶̶̶̶̶ 2 2y - 2x ̶̶̶̶̶̶ = - 2x + 2 2y 2x + 2 _ _ 2 2 y = -x + 1 = - To get the y-terms together, subtract y from both sides. Subtract 2x from both sides. Divide both sides by 2. PRACTICE Solve for y. 1. 3x = 2y - 3 2. y - x _ 2 = 1 3. x + y = 4y + 3 - 2x Writing and Graphing Inequalities An inequality compares two quantities by using one of these symbols: < > ≤ ≥ is less than is greater than is less than or equal to is greater than or equal to E X A M P L E Write each expression as an inequality. Graph the inequality. A q is less than or equal to 3 q ≤ 3 Use ≤ for “is less than or equal to.” Use a solid circle for ≤ or ≥. Shade the side of the line that contains the solutions. B s is greater than 18 s > 18 Use > for “is greater than.” Use an empty circle for < or >. Shade the side of the line that contains the solutions. PRACTICE Write each expression as an inequality. Graph the inequality. 1. u is less than 0 2. n is greater than or equal to 15 3. x is less than 3 4. b is greater than 5 5. y is less than or equal to 4 6. m is greater than -3 Skills Bank S59 S59 �� Solving Linear Inequalities The following properties are used to solve linear inequalities. PROPERTY WORDS NUMBERS ALGEBRA Addition Property of Inequality You can add the sa
me number to both sides of an inequality, and the statement will still be true Subtraction Property of Inequality You can subtract the same number from both sides of an inequality, and the statement will still be true. Multiplication and Division Properties of Inequality (by a positive number) You can multiply or divide both sides of an inequality by the same positive number, and the statement will still be true. Multiplication and Division Properties of Inequality (by a negative number) If you divide both sides of an inequality by the same negative number, you must reverse the inequality symbol for the statement to still be true. 5 < 10 9 < 12 a < b 9 - 5 < 12 - < 12 7 · 3< 12 · 3 21 < 36 4 < 12 < 12 ___ 4 ___ -4 -4 -1 > -3 If a < b and c > 0, then ac < bc. If a < b and c < 0, then a __ c > b __ c . These properties are also true for inequalities that use the symbols, >, ≥, and ≤. E X A M P L E Solve and graph. A 8m < 96 8m < 96 < 96 _ 8m _ 8 8 m < 12 B 3 - 2x ≤ 7 Divide both sides by 8. - 3 ̶̶̶̶̶̶ 3 - 2x ≤ 7 - 3 ̶̶̶ -2x ≤ 4 -2x _ ≥ 4 _ -2 -2 x ≥ -2 PRACTICE Solve and graph. 1. 3x + 2 > 11 4. -n < 3 7. z + 7 > 5 S60 S60 Skills Bank Subtract 3 from both sides. Divide both sides by -2. Reverse the direction of the inequality. 2. -4 < 8 - 3y 5. 2g + 13 ≥ -1 8. k _ 3 + 4 < 12 3. 6x ≥ 5x + 4 6. -4a ≥ 18 - w _ 9. 3 _ 4 4 ≥ 7 �� Absolute Value The absolute value of a number is its distance from zero on the number line. The absolute value of a number a is represented by ⎜a⎟ . E X A M P L E Simplify. A ⎜-4⎟ ⎜-4⎟ = 4 B ⎜3 - 9⎟ ⎜3 - 9⎟ = ⎜-6⎟ = 6 PRACTICE Simplify. 1. ⎜7⎟ 2. ⎜-1⎟ 5. ⎜-9 + 2⎟ 6. ⎜-3 + 4⎟ 3. ⎜10 - 15⎟ 7. ⎜-20 + 4⎟ 4. ⎜-4 - 4⎟ 8. ⎜-2 - 4⎟ - ⎜-5⎟ Relations and Functions A relation is a rule that relates two quantities. A function is a relation in which each input value corresponds to exactly one output value. The domain is the set of all possible input values, and the range is the set of all possible output values. E X A M P L E Determine whether each relation is a function.   (1, 2) , (2, 4) , (3, 6) , (4, 8) A S: ⎨ ⎬   yes Each x-value in the set corresponds to exactly one y-value. B ⎜y⎟ = x no C y = x 2 yes ⎜2⎟ = 2 and ⎜-2⎟ = 2, so (2, 2) and (2, -2) satisfy the equation. These two points have the same x-value but different y-values, so the relation is not a function. For each value of x, there is only one value of x 2 , so each x-value corresponds to exactly one y-value. PRACTICE Determine whether each relation is a function. 1. y = 3x + 4 4. 10y = -x 2. y 2 = x 5. 9y = 3   (0, 0) , (1, 2) , (0, 2) 3. S: ⎬ ⎨     (-5, 1) , (-4, 1) , (-3, 1) 6. S: ⎬ ⎨   Skills Bank S61 S61 �� Inverse Functions A function is a rule that relates two quantities, the input and the output. Each input value corresponds to only one output value. The inverse of a function is a rule that reverses the function. To find the rule for the inverse, switch the x- and y-values. The rule for the inverse is also a function if each input corresponds to only one output. �� E X A M P L E Find the inverse of each function, and state whether the inverse is also a function. ��   (0, 1) , (1, 2) , (3, 7) , (9, 9) A S: ⎬ ⎨   The rule for the inverse is   (1, 0) , (2, 1) , (7, 3) , (9, 9) ⎨ . ⎬   It is a function, because each x-value corresponds to only one y-value. PRACTICE B y = x 2 The rule for the inverse is x = y 2 . The inverse is not a function, because each x-value corresponds to two y-values, √  x and - √  x . Find the inverse of each function, and state whether the inverse is also a function.   (0, 0) , (3, 1) , (2, 2) , (5, 7) 1. K: ⎬ ⎨   2. y = 5x 3. y = ⎜x⎟ Direct Variation Direct variation is a relationship between two quantities in which one quantity is a constant multiple of the other. The constant is called the constant of variation . The relationship “y varies directly as x, where k is the constant of variation” is written as y = kx. E X A M P L E Find the constant of variation. A y varies directly as x, and y = 7 B t varies directly as c, and t = 1 when x = 3. y = kx 7 = k (3) 7 _ = k 3 Substitute 7 for y and 3 for x. when c = 0.1. t = kc (1) = k (0.1) Substitute 1 for t and 0.1 for c. Solve for k. 10 = k Solve for k. PRACTICE Find the constant of variation. 1. m varies directly as n, and m = 12 when n = 3. 2. y varies directly as x, and y = 24 when x = 8. 3. s varies directly as t, and s = 9 when t = 36. S62 S62 Skills Bank Functional Relationships in Formulas A formula is an equation that is solved for one variable in terms of the others. By rearranging the terms in a formula, you can see how each variable depends on the others. E X A M P L E Solve the formula P = 2ℓ + 2w for ℓ. If the width stays the same, what happens to the length of a rectangle as its perimeter increases? P = 2ℓ + 2w P - 2w = 2ℓ 1 _ P - w = ℓ 2 Subtract 2w from both sides. Divide both sides by 2. As the perimeter of a rectangle increases, the length also increases. PRACTICE 1. Solve the formula A = 1 __ 2 ( b 1 + b 2 ) h for h. If the base lengths stay the same, what happens to the height h of a trapezoid as its area A increases? 2. Solve the formula I = Prt for t. If the interest rate and amount stay the same, what happens to the time t as the principal amount P decreases? Transformations of Functions The basic transformations of the parent function y = f (x) are given below. Transformation Transformation Function Reflection Across the x-axis y = -ƒ (x) Reflection Across the y-axis y = ƒ (-x) Vertical Translation y = ƒ (x) + k If k > 0, translate k units up. If K < 0, translate k units down. Horizontal Translation y = ƒ (x - h) If h > 0, translate h units right. If h < 0, translate h units left. E X A M P L E Describe the transformation given by the equation y = (x - 3) 2 . Step 1 Identify the parent function. The parent function is y = x 2 . Step 2 Identify the transformation. The equation represents a horizontal translation with h = 3. So the transformation is a horizontal translation 3 units to the right. PRACTICE Describe the transformation given by each equation. 1. y = x - 6 2. y = √  -x 3. y = (-x) 2 - 1 4. y - 2 = x 2 Skills Bank S63 S63 Polynomials A monomial is a number or a product of numbers and variables with whole-number exponents. A polynomial is a monomial or the sum or difference of monomials. The degree of a polynomial is the highest power of the variables in all the terms. Polynomial Number of Terms Example Monomial Binomial Trinomial 1 2 3 3 a 2 x 2y - 7 x 2 2n + 3m - Classify each expression as a monomial, a binomial, a trinomial, or not a polynomial. A 3x + 2 y 6 + 7 q 2 e trinomial Polynomial with 3 terms B -5z monomial _ 5 c 2 d C 1.5b - 2 binomial D 1 _ x Polynomial with 1 term Polynomial with 2 terms not a polynomial A variable is in the denominator. E X A M P L E 2 Find the degree of each polynomial. A 7 v 3 - 8 w 2 - 9u 7v 3 Degree 3 The degree is 3. B -5z + 1 -5z 1 Degree 1 The degree is 1. -8w 2 Degree 2 -9u 1 Degree 1 +1 0 Degree 0 PRACTICE Classify each expression as a monomial, a binomial, a trinomial, or not a polynomial. 1. e 2 r 3 2. x -2 3. b 2 - 4ac 5. 5 x 2 - 2x + 7 6. 4 7. y - 3 Find the degree of each polynomial. 9. x 5 - x 3 + 3 10. 9 a 3 - 10 a 2 + a 4 11. 2 - 4x 4. 12 S64 S64 Skills Bank Quadratic Functions A quadratic function is a function that can be written in the form y = a x 2 + bx + c, where a ≠ 0. The graph of a quadratic function is a parabola , an almost U-shaped graph. Graph of a Quadratic Function y = a x 2 + bx + c • If a > 0, the parabola opens upward. • If a < 0, the parabola opens downward. • The axis of symmetry of the parabola is the vertical line x = - b _ . 2a • The vertex of the parabola is (- b _ , y) . 2a E X A M P L E Tell whether the graph of each quadratic function opens upward or downward. Write an equation for the axis of symmetry, and find the coordinates of the vertex. A y = -2 x 2 + 4x + 2 Step 1 Tell whether the graph opens upward or downward. Since a = -2 < 0, the graph opens downward. Step 2 Find the x-coordinate of the vertex. - b _ 2a = - 4 _ 2 (-2 ) = 1 Substitute -2 for a and 4 for b. The equation for the axis of symmetry is x = 1. Step 3 Find the y-value when x =1. y = -2 (1 ) 2 + 4 (1 ) + 2 = 4 Substitute 1 for x in the original equation. The coordinates of the vertex are (1, 4 Step 1 Tell whether the graph opens upward or downward. Since a = 3 > 0, the graph opens upward. Step 2 Find the x-coordinate of the vertex. - b _ 2a = - 0 _ 2 (3 ) = 0 Substitute 3 for a and 0 for b. The equation for the axis of symmetry is x = 0. Step 3 Find the y-value when x = 0. y = 3 (0 ) 2 - 1 = -1 Substitute 0 for x in the original equation. The coordinates of the vertex are (0, -1) . PRACTICE Tell whether the graph of each quadratic function opens upward or downward. Find the coordinates of the vertex, and write an equation for the axis of symmetry. 1. y = -4 x 2 - 8x 2. y = - x 2 - 4x + 2 3. y = 2x 2 + 4 4. y = 3 x 2 - 6x + 8 5. y = - 5x 2 + 10 6. y = 0.5x 2 + x + 2 Skills Bank S65 S65 �� Factoring to Solve Quadratic Equations One method of solving quadratic equations is to apply the Zero Product Property, which states that if ab = 0, then a = 0 or b = 0. First write the quadratic equation in standard form and then factor it. E X A M P L E Solve the quadratic equation 2 x 2 - 5x + 6 = 9 by factoring. 2 x 2 - 5x -3 = 0 (2x + 1) (x - 3) = 0 Write the equation in standard form. Factor the trinomial. 2x + 1 = 0 or x - 3 = 0 Use the Zero Product Property. x = - 1 _ 2 or x = 3 Solve each equation. PRACTICE Solve each quadratic equation by factoring. 1. x 2 - 3x + 2 = 0 3. 4 x 2 - 8x = -4 5. 4 x 2 + 8x = 32 2. x 2 + 4 = -4x 4. x 2 = 9 6. 3x + 4 = x 2 The Quadratic Formula For a quadratic equation in the form a x 2 + bx + c = 0, you can use the Quadratic Formula , x = -b ± √  b 2 - 4ac __ 2a , to solve for
x. E X A M P L E Use the Quadratic Formula to solve each equation. A 2 x 2 + 3 = 7x 2 x 2 - 7x + 3 = 0 Write the equation in standard form. a = 2, b = -7, c = 3 Find a, b, and c. B x 2 - 4x = -6 1x 2 - 4x + 6 = 0 a = 1, b = -4, c = 6 x = - (-7 ) ± √  (-7 ) 2 - 4 (2 ) (3 ) ___ 2 (2 ) Substitute into the Quadratic Formula. x = - (-4 ) ± √  (-4 ) 2 - 4 (1 ) (6 ) ___ 2 (1 ) x = x = 7 ± √  25 _ 4 7 + 5 _ 4 or x = 7 - 5 _ 4 Simplify. Simplify. x = 4 ± √  -8 _ 2 Since you cannot take the square root of a negative number, there is no solution. x = 3 or x = 1 _ 2 Write the solution. PRACTICE Use the Quadratic Formula to solve each equation. 1. x 2 + 2x = -1 2. 3 x 2 + 2x = 4 4. 4 x 2 + 8x - 12 = 0 5. 2 x 2 + 5x = 3 3. 7 x 2 + 3x = -5 6. 2 x 2 - 7x = 12 S66 S66 Skills Bank Solving Systems of Equations To solve a system of equations, you can use either the substitution method or the elimination method. E X A M P L E ⎧ 3x + 2y = 0 ⎨ Solve the following system of equations. 2x + y = 3 ⎩ Method 1 Use substitution. y = -2x + 3 3x + 2 (-2x + 3 ) = 0 3x - 4x + 6 = 0 -2 (6 ) + 3 = -9 (6, -9 ) Method 2 Use elimination. -2 (2x + y) = -2 (3) -4x - 2y = -6 3x + 2y = 0 -4x - 2y = -6 -x = -6 x = 6 y = -2 (6 ) + 3 = -9 (6, -9 ) PRACTICE Solve each system of equations. 1. ⎧ ⎪ ⎨ ⎪ ⎩ 3x - 3y = 4 x + y = 10 _ 3 5. ⎧ 4x - 6y = 1 ⎨ 3y - x = 2 ⎩ 7. ⎧ 3x - y = 6 ⎨ ⎩ y = 2x + 3 9. ⎧ -3x + 2y = 31 ⎨ x = 0.5y + 6 ⎩ Solve the second equation for y. Substitute -2x + 3 for y into the first equation. Distribute 2. Simplify. Solve for x. Substitute 6 for x into the second equation and simplify. Simplify. Write the solution as an ordered pair. Multiply each term in the second equation by -2 to get opposite y-coefficients. Simplify. Write the system using the new equation so that like terms are aligned. Add like terms on both sides of the equations. Solve for x. Substitute 6 for x into the second equation and simplify. Simplify. Write the solution as an ordered pair. ⎧ 3x + y = 1 2. ⎨ x + y = -3 ⎩ ⎧ ⎪ 4 2x - 3y = 2 ⎧ x - y = 0 6. ⎨ 2x + 3y = 0 ⎩ ⎧ 2x + 5y = 14 8. ⎨ y = 5 ⎩ ⎧ 3x + y = 4 10. ⎨ x - 2y = 6 ⎩ Skills Bank S67 S67 Solving Systems of Linear Inequalities A system of linear inequalities is a set of two or more inequalities with two or more variables. The solution to a system of inequalities consists of all the ordered pairs that satisfy all the inequalities in the system ⎨ Solve the following system of linear inequalities. + x > -1 ⎪ ⎩ 2 y > -2x - 2 Write the second inequality in slope-intercept form. Graph the solution of each inequality. The solutions of the system are represented by the overlapping shaded regions. PRACTICE Solve each system of linear inequalities. 1. ⎧ y ≤ 2x ⎨ y ≤ x ⎩ ⎧ y ≤ 2x + 1 2. ⎨ y ≤ x - 1 ⎩ Solving Radical Equations A radical equation is an equation that has a variable within a radical, such as a square root. To solve a square-root equation, square both sides and solve the resulting equation. E X A M P L E ( √  x - 9 ) 2 = (1) 2 Square both sides. Solve the equation √  x - 9 = 1. Check your answer. x - 9 = 1 Simplify. x = 10 10 - 9 = √  1 = 1 ✔ Solve for x. Check √  PRACTICE Solve each equation. Check your answer. 1. √  x + 1 = 4 3. √  1 - x = 3 5. √  7 + x = 0 7. √  3 - 2x = 3 S68 S68 Skills Bank 2. √  2x - 1 = 5 4. √  -6 - 5x = 2 6. √  4x + 4 = 2 8. √  60 - 2x = 8 �� Matrix Operations A matrix is a rectangular array of numbers enclosed in brackets. The entries in a matrix are arranged in rows and columns. Operations such as addition, subtraction, and multiplication by a constant can be performed on matrices. E X A M P L E Simplify. A ⎡ 1 ⎢ 2 -3 ⎣ ⎡ 1 ⎢ 2 -3 ⎣ -2 0 1 -2 0 1 ⎤ 2 ⎥ 3 -4 ⎦ ⎤ 2 ⎥ 3 -3 + 3 ⎥ 11 5 ⎦ Add corresponding entries. B ⎡ 1 ⎢ 2 -3 ⎣ -2 0 1 ⎤ 2 ⎥ 3 -6 ⎣ -6 -5 -5 ⎤ -5 ⎥ -5 -13 ⎦ Subtract corresponding entries3 - 3 -3 ⎣ -2 0 1 ⎤ 2 ⎥ 3 -4 ⎦ 2 (1) 2 (-2) 2 (2) 2 (0) ⎡ ⎢ ⎣ 2 (-3) 2 (1) 2 (-4) ⎦ ⎤ 2 (2) 2 (3) = ⎥ ⎡ ⎢ 2 4 -6 ⎣ -4 0 2 ⎤ 4 6 -8 ⎦ ⎥ Multiply each entry by 2. PRACTICE Simplify. ⎡ 1 1. ⎢ 0 ⎣ ⎡ ⎤ 0 -1 ⎦ 4. ⎡ 1 ⎢ 0 ⎣ ⎡ ⎤ -1 -3 ⎦ ⎡ 0 7.5 0 ⎣ ⎦ ⎤ -9 ⎥ -1 ⎦ 2. 2 ⎢ 7 ⎣ ⎤ 2 ⎥ -1 ⎦ 8. ⎡ 8 ⎢ 0 ⎣ ⎡ ⎤ -2 -1 - ⎢ ⎥ 4 -3 ⎣ ⎦ ⎤ 1 ⎥ -7 ⎦ ⎡ 3 3. -. 1 _ ⎢ 2 0 ⎣ ⎤ -8 ⎥ 2 ⎦ Skills Bank S69 S69 Structure of Measurement Systems The metric system of measurement is used worldwide. In the United States, we most commonly use the customary system of measurement but still use the metric system in most science applications. Customary System 12 in. = 1 ft 8 oz = 1 c 16 oz = 1 lb Length Capacity Weight/Mass 3 ft = 1 yd 5280 ft = 1 mi 2 c = 1 pt 2 pt = 1 qt 4 qt = 1 gal 2000 lb = 1 ton Metric System 1000 mm = 1 m 1000 mL = 1 L 1000 mg = 1 g 100 cm = 1 m 1000 L = 1 kL 1000 g = 1 kg 1000 m = 1 km E X A M P L E Complete each conversion. A 1560 mL = L B 2 mi = yd 1560 mL × 1L _ 1000 mL = 1.56 L 2 mi × 5280 ft _ × 1 mi 1 yd _ 3 ft = 3520 yd PRACTICE Complete each conversion. 1. 2.8 m = cm 2. 128 oz = lb 4. 87 ft = yd 5. 2.6 kL = mL 3. 4 1 _ 2 gal = c 6. 108 mg = g Rates and Derived Measurements A rate is the ratio of the change in one measurement to the change in another measurement, usually time. The units of a rate are derived units , or the ratio of two different units, such as miles per hour (mi/h) and kilograms per meter (kg/m car travels 1 km every 5 min. What is the speed of the car in meters per second? The speed of the car is the ratio of the change in distance to the change in time. 1 km _ 5 min × 1000 m _ 1 km Convert km to m and min to s. × 1 min _ 60 s ≈ 3.33 m/s PRACTICE 1. The mass of a small meteor is decreasing at a rate of 6000 g every 2 min. What is the rate of decrease in kilograms per second? 2. The temperature increases at a rate of 2°F every half hour. What is the rate of increase in degrees Fahrenheit per minute? Round to the nearest thousandth. 3. An athlete runs at a rate of 9.5 m/s. What is the runner’s rate in kilometers per hour? S70 S70 Skills Bank Unit Conversions To convert a measurement in one system to a measurement in another system, multiply by a conversion factor , such as 1 cm ≈ 0.394 in., written as a fraction. If you are converting ______ from inches to centimeters, “cm” goes in the numerator — 1 cm 0.394 in. ______ from centimeters to inches, “in.” goes in the numerator — 0.394 in. . 1 cm . If you are converting Common Conversion Factors Length Capacity Mass/Weight Temperature Metric to Customary 1 cm ≈ 0.394 in. 1 L ≈ 4.227 c 1 g ≈ 0.0353 oz 1 m ≈ 3.281 ft 1 L ≈ 1.057 qt 1 kg ≈ 2.205 lb F = 9 _ 5 C + 32 1 m ≈ 1.094 yd 1 L ≈ 0.264 gal 1 km ≈ 0.621 mi 1 mL ≈ 0.034 oz Customary to Metric 1 in. ≈ 2.540 cm 1 c ≈ 0.237 L 1 oz ≈ 28.350 g 1 ft ≈ 0.305 m 1 qt ≈ 0.946 L 1 lb ≈ 0.454 kg C = 5 _ 9 (F - 32) 1 yd ≈ 0.914 m 1 gal ≈ 3.785 L 1 mi ≈ 1.609 km 1 oz ≈ 29.574 mL E X A M P L E Complete each conversion. If necessary, round to the nearest hundredth. ° F A 20° C = F = 9 _ (20 ) + 32 5 F = 68 20° C is equivalent to 68° F. Simplify. Substitute 20 for C. B 25 lb ≈ g 25 lb × 0.454 kg _ × 1 lb 1000 g _ 1 kg ≈ 11,350 g C 32 ft 2 ≈ m 2 32 ft 2 × 0.305 m _ × 0.305 m _ 1 ft ≈ 2.98 m 2 1 ft Use the conversion factor for pounds to kilograms. Then convert kilograms to grams. The units are squared, so apply the conversion factor for feet to meters twice. PRACTICE Complete each conversion. If necessary, round to the nearest hundredth. 1. 40° C = 3. 2 3 _ 4 5. 18 . 12,300 mg ≈ lb 9. 64 gal ≈ kL 11. 98° F ≈ ° C 2. 15 in. ≈ cm 4. 86°F ≈ ° C 6. 18 kg ≈ lb 8. 150 mL ≈ c 10. 98 lb ≈ kg 12. 100 yd ≈ m Skills Bank S71 S71 Accuracy, Precision, and Tolerance The accuracy of a measurement is the numerical measure of how close the measured value is to the actual value of the quantity. A measurement of 133 cm for an actual object with a length of 132.7 cm is accurate to the nearest centimeter. The precision of a measurement is the level of detail determined by the number of decimal places to which the measurement is taken. The measurements 27 cm and 39.48 m, or 3948 cm, are both precise to the nearest centimeter. Tolerance is the range of values within which a measurement lies. In the measurement 16.3 ± 0.1 mm, the ± 0.1 mm is the tolerance of the measurement. So if the measurement is accurate, the actual value of the object being measured is from 16.2 mm to 16.4 mm. E X A M P L E The weight of an object is 3.72 lb. Three measurements of the object were recorded: 3.75 ± 0.02 lb, 3.718 ± 0.002 lb, and 3.73 ± 0.001 lb. Which measurement is the most accurate? Which is the most precise? Which has the smallest tolerance? 3.75 lb ± 0.02 lb 3.718 lb ± 0.002 lb 3.73 lb ± 0.001 lb Ranges from 3.73 lb to 3.77 lb Ranges from 3.716 lb to 3.720 lb Ranges from 3.729 lb to 3.731 lb Step 1: Find the most accurate measurement. The most accurate measurement is the measurement closest to the actual weight of 3.72 lb. 3.718 ± 0.002 lb Step 2: Find the most precise measurement. The most precise measurement is the measurement (not the tolerance) with the most decimal places. 3.718 ± 0.002 lb Step 3: Find the measurement with the smallest tolerance. The most tolerant measurement is the one with the smallest ± value. 3.73 ± 0.001 lb PRACTICE In each problem, the first value is the actual measure of an object, followed by multiple recorded measurements. Which measurement is the most accurate? Which is the most precise? Which has the smallest tolerance? 1. 1.00 in. {1.0 ± 0.1 in., 1.01 ± 0.01 in., 2 ± 1 in., 1.001 ± 0.1 in.} 2. 2.50 s {2.5 ± 0.1 s, 2.5100 ± 0.0001 s, 2.515 ± 2 s, 2.51 ± 0.01 s} 3. 11.51 m {10.5 ± 0.5 m, 11.51 ± 5 m, 11.500001 ± 0.9 m, 22 ± 12 m} 4. 0.50102041 g {0.5010204 ± 0.5 g, 22.51 ± 0.00000001 g, 51.5120843447 ± 50 g, 0.50102041 ± 5 g} S72 S72 Skills Bank Relative and Absolute Error The absolute error of a measurement is the difference between the measured value and the actual value of the quantity being measured. Absolute error can be misleading when very large or very small numbers are being measured. You can avoid this by
using the relative error , which is the absolute error divided by the actual value. Relative error has no units. When expressed as a percent, this is the percent error of a measurement. E X A M P L E Find the absolute, relative, and percent errors. The first number is the actual value, and the second is the measured value. A 5.1 m, 5.0 m B 5.1 ft, 5.5 ft absolute error = 5.0 m - 5.1 m = -0.1 m relative error = 5.0 m - 5.1 m __ 5.1 m = -0.0196 absolute error = 5.5 ft - 5.1 ft = 0.4 ft relative error = 5.5 ft-5.1 ft __ = 0.0784 5.1 ft percent error = -1.96% percent error = 7.84% PRACTICE Find the absolute, relative, and percent errors. The first value is the actual value, and the second is the measured value. 1. 1.23, 1.00 3. 5.55, 6.00 2. 123, 100 Significant Digits All the digits in a measurement that are known to be exact are called significant digits . Rule Example Number of Significant Digits Significant Digits All nonzero digits Zeros between nonzero digits Zeros after the last nonzero digit that are to the right of the decimal point 14.28 8.002 0.030 4 4 2 14.28 8.002 0.030 Zeros at the end of a whole number are assumed to be nonsignificant. So the measurement 700 has 1 significant digit—the 7. E X A M P L E Determine the number of significant digits in each measurement. A 815 lb B 2 × 10 -2 L C 750 kg D 15.08 m 3 significant digits 1 significant digit 2 significant digits 4 significant digits PRACTICE Determine the number of significant digits in each measurement. 1. 1203 lb 3. 6.003 mi 2. 3.0 cm 4. 5.000 kg 5. 1000 in. 9. 91.0 s 6. 1 × 10 3 L 7. 03.101 g 8. 1.60200 km 10. 2.00 × 10 4 cm 11. 0.10 m 12. 0.00105 lb Skills Bank S73 S73 Choose Appropriate Units When measuring a quantity, it is important to choose the appropriate units so that the measurement will have a reasonable magnitude. E X A M P L E Name the appropriate unit to measure the mass of an elephant (milligram, gram, kilogram, metric tons). The average mass of an elephant is around 5 metric tons, 5000 kg, 5,000,000 g, or 5,000,000,000 mg. Since 5 metric tons has the most reasonable magnitude, the mass of an elephant should be weighed in metric tons. PRACTICE Select the appropriate unit for each measurement. 1. the height of a classroom (millimeter, centimeter, meter, kilometer) 2. the distance from Earth to the Sun (inches, feet, yards, miles) 3. the length of a decade (seconds, minutes, hours, years) Nonstandard Units There are several nonstandard unit systems. pH , a measure of the concentration of hydrogen ions in a solution, ranges from 0 to 14. Pure water has a pH of 7, which is considered neutral. A pH less than 7 is acidic, and a pH greater than 7 is basic, or alkaline. The Richter scale measures the magnitude of earthquakes. The pH scale and the Richter scale are related by powers of 10. An increase of 1 unit on the scale means an increase by a factor of 10 in the quantity. For example, an earthquake that measures 6.0 on the Richter scale is 10 times as great as one that measures 5.0. E X A M P L E 1 Solutions A and B have the same volume. The pH of solution A is 4, and the pH of solution B is 5. How much more acidic is solution A than solution B? Since 5 - 4 = 1, and 10 1 = 10, solution A is 10 times more acidic than solution B. E X A M P L E 2 Earthquake A had a magnitude of 2.7 on the Richter scale. Earthquake B had a magnitude of 4.7. How much stronger was earthquake B than earthquake A? Since 4 - 2 = 2, and 10 2 = 100, earthquake B was 100 times stronger than earthquake A. PRACTICE 1. Solutions P and Q have the same volume. The pH of solutions P and Q are 1 and 7, respectively. How much more basic is solution Q than solution P? 2. Earthquake R has a magnitude of 5. How much stronger is earthquake R than earthquake S, with magnitude 2? S74 S74 Skills Bank Use Tools for Measurement Length can be measured with tools such as a ruler, a measuring tape, or a micrometer. Weight can be measured with various types of balance scales. Time can be measured with a calendar, a clock, or a stopwatch. Electronic timing devices that measure to a precision of 0.01 s or 0.001 s are used in track events. E X A M P L E What is the width of the bolt? Read the scale on the micrometer. The main scale reads 8.5 mm. The fine scale reads 0.120 mm. The width of the bolt is therefore 8.620 mm. PRACTICE 1. Use a ruler to measure the dimensions of your Geometry book. 2. Use a stopwatch to measure the time it takes to climb a set of stairs. 3. Use a tape measure to find the height of your classroom doorway. Choose Appropriate Measuring Tools To choose an appropriate measuring tool, consider the following criteria: • How large is the quantity being measured? • How much precision is needed for the measurement? E X A M P L E What instrument would you use to measure the width of a textbook to the nearest sixteenth of an inch: a micrometer, a ruler, or a measuring tape? A measuring tape may be precise only to the nearest quarter of an inch. A micrometer is not large enough to measure a textbook. So a ruler is the most appropriate tool. PRACTICE What instrument would you use to measure each quantity? 1. the duration of a feature film to the nearest minute 2. the length and width of a room to the nearest inch 3. the weight of a feather to the nearest microgram 4. the capacity of a drinking glass to the nearest ounce Skills Bank S75 S75 Measures of Central Tendency A measure of central tendency describes how data clusters around a value in a statistical distribution. The mean is the average value of the data. The median is the middle value when the data is in numerical order or the average of the two middle terms if there is an even number of terms. The mode is the value that occurs most often. There can be one, several, or no modes. E X A M P L E Find the mean, median, and mode of the following data set. {3, 5, 1, 9, 0, 1, 5, 6, 0, 3, 5, 7, 8, 1, 3, 5, 8, 3, 2, 7, 1, 6, 8, 4, 3, 2, 7, 3, 6, 8} Step 1: Find the mean. Add all the data values, and then divide the sum by the number of terms in the set. _ 130 30 mean ≈ 4.33 The sum of the data values is 130, and there are 30 values. = 13 _ 3 Step 2: Find the median. Order the set from the least value to the greatest value. {0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9} To find the median, locate the middle term. Since there are 30 terms, the median will be the average of the 15th and 16th terms. {0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9} 15 terms 4 + 5 = 9 _ _ 2 2 median = 4.5 Step 3: Find the mode. 15 terms The 15th term is 4. The 16th term is 5. Find the average. Data How Often occurs most often. mode = 3 PRACTICE Find the mean, median, and mode of each data set. Round to the nearest hundredth if necessary. 1. {1, 2, 2, 1, 2, 1, 2, 1} 2. {-1, 0, 4, 3, 2, 0, -3, -4, -1, 0} 3. {0, 1, 2, 1, 0, 2, 3, 7, 2} 4. {3, 7, 12, 8, 3, 7, 1, 6, 3, 7, 1, 9, 3, 100, 2} 5. {100, 111, 112, 100, 99, 104, 103, 108} 6. {4, 34, 72, 675, 12, 56, 2, 67, 12, 5, 387, 12, 4, 23, 5, 72, 56, 23, 56, 45, 2, 6} 7. {43, 23, 31, 53, 97, 79, 57, 11, 13, 11, 43, 61, 91, 87, 83, 73, 37, 41, 29} 8. {34, 45, 12, 6, 12, 45, 23, 6, 12, 78, 45, 67, 33} S76 S76 Skills Bank Probability Probability is a measure of how likely an event is to occur. It is calculated by taking the number of outcomes for which the event occurs and dividing by the total number of possible outcomes. Probability can be expressed as a fraction or as a decimal between 0 and 1, or as a percentage between 0% and 100%. The higher the probability, the more likely the event will occur. E X A M P L E Find the probability of rolling each sum with two number cubes. The two number cubes are independent. In order to roll a 2, both number cubes need to show 1. So there is one possible way to roll a 2. To roll a 3, the first cube can show 1 and the second cube can show 2, or the first cube can show 2 and the second cube can show 1. So there are two possible ways to roll a 3. Repeating this method gives the following table. Sum Number of Outcomes Probability (fraction) Probability (decimal) Probability (percent) 2 3 4 5 6 7 8 9 10 11 12 Total: 36 1 _ 36 = 1 _ 2 _ 18 36 3 _ = 1 _ 12 36 = 1 _ 4 _ 9 36 5 _ 36 6 _ = 1 _ 6 36 5 _ 36 = 1 _ 4 _ 36 9 3 _ = 1 _ 12 36 = 1 _ 2 _ 18 36 1 _ 36 36 _ 36 = 1 0.0278 0.0556 0.0833 0.1111 0.1389 0.1667 0.1389 0.1111 0.0833 0.0556 0.0278 1 2.8% 5.6% 8.3% 11.1% 13.9% 16.7% 13.9% 11.1% 8.3% 5.6% 2.8% 100% PRACTICE Find the probability of each event. Write your answer as a percent rounded to the nearest tenth if necessary. 1. Dave puts 2 red marbles, 3 blue marbles, and 5 green marbles into a bag. Samir then randomly pulls a marble out of the bag and replaces it. What is the probability that Samir chooses a red marble? a blue marble? a green marble? 2. What is the probability of drawing a heart from a standard deck of 52 cards? a jack, queen, or king? an ace? 3. Shireen asks Kate to guess the number she has chosen between 1 and 100. What is the probability that Kate guesses the number correctly? Skills Bank S77 S77 Organizing and Describing Data One way to organize and display data is by using a table. The table can then help you to understand the meaning of the data and identify any relationships within the data. E X A M P L E Organize the given data in a table. Then describe the data. Rick has been tracking his recent test results. He scored 81% on a Language Arts test, 67% on a History test, 78% on a Science test, 82% on a Spanish test, and 90% on a Geometry test. Test Percent History Science Language Arts Spanish Geometry 67% 78% 81% 82% 90% List each test and its percent score. The table shows that Rick did very well on his Geometry test, but needs to improve on History. All his test scores except for History are above 75%. PRACTICE 1. Organize the given temperatures in a table. Then describe the data. The temperature was 32°F from 6 to 8 A.M., 40°F from 8 to 10 A.M., 56°F from 1
0 A.M. to 12 P.M., 72°F from 12 to 2 P.M., 65°F from 2 to 4 P.M., 54°F from 4 to 6 P.M., and 40°F from 6 to 8 P.M. Displaying Data Two other ways to display data are to use a bar graph and a histogram. A bar graph is used when the data values are disjoint, and the data represent categories that are not directly related to each other. The bars do not touch. In a histogram , the data categories are usually numerical intervals such as 0–9, 10–19, and so on. The bars in a histogram do touch. E X A M P L E Use the data in the example above to make a vertical bar graph of Rick’s test scores. Choose an appropriate scale and interval. Draw a bar for each test. Title and label the graph. PRACTICE 1. Use the data in Practice Problem 1 above to make a histogram of the temperatures. S78 S78 Skills Bank �� Scatter Plots and Trend Lines A scatter plot is a graph of ordered pairs of data. A scatter plot can help you see any clusters or trends in the data. A trend line is a straight line that accurately expresses the trend of the data on a scatter plot. When drawing a trend line, try to follow clusters of data and the overall pattern of the plot teacher asked 15 students to record the time they spent studying for their geometry test. The results are displayed in the table. Make a scatter plot of the data in the table. Then draw a trend line for the data. Explain the meaning of the trend line. Time (min) Test Score Graph the ordered pairs (time, score) on a graph using a reasonable scale. Draw a trend line. 0 60 180 160 100 30 15 45 15 0 60 120 90 45 30 50 60 95 100 90 70 60 80 50 60 75 85 80 70 65 The trend line indicates that a student’s scores improved as study time increased. PRACTICE Make a scatter plot of each data set. Then draw a trend line for the data. Explain the meaning of the trend line. 1. A teacher asked nine students to record the number of minutes they spent watching TV the day before a test. The results are displayed in the table. Time Watching TV (min) 0 60 20 45 10 30 60 40 15 Test Score 28 15 20 18 27 29 16 17 24 2. A study was performed to see how age relates to the number of pizza slices someone eats. The results of the study are in the table. Age Pizza Slices 12 4 6 2 18 20 11 12 15 10 10 5 5 3 10 14 17 15 20 6 5 8 6 9 5 4 9 6 14 6 Skills Bank S79 S79 �� Quartiles and Box-and-Whisker Plots A box-and-whisker plot is a graph showing the lower extreme (the least value), the upper extreme (the greatest value), the median, the lower quartile (the median of the lower half of the data), and the upper quartile (the median of the upper half of the data). E X A M P L E 1 Make a box-and-whisker plot of the test scores below. Test Score 9 6 3 8 10 4 7 3 9 7 3, 3, 4, 6, 7, 7, 8, 9, 9, 10 3, 3, 4, 6, 7, 7, 8, 9, 9, 10 3, 3, 4, 6, 7, 7, 8, 9, 9, 10 Order the data from least to greatest. Identify the lower and upper extremes. Identify the median. 3, 3, 4, 6, 7, 7, 8, 9, 9, 10 Identify the lower quartile and the upper quartile. Make a box using the median and upper and lower quartiles. Place a bar at the upper and lower extremes. Connect the bars to the box with segments called whiskers. PRACTICE 1. Make a box-and-whisker plot of the data set. 12 18 10 17 18 15 17 13 7 14 19 Circle Graphs Circle graphs are used to represent data as percentages of the total. To draw a circle graph, convert the data to percentages, and then make a section of the circle for each category. E X A M P L E 1 Make a circle graph of the data from the study in the table. Team Fans Cougars Panthers Jaguars Tigers Lions 375 375 125 75 50 Percent of fans Degrees of Circle ____ 375 1000 = 37.5% ____ 375 1000 = 37.5% ____ 125 1000 = 12.5% 75 ____ 1000 = 7.5% 50 ____ 1000 = 5% 0.375 (360°) = 135° 0.375 (360°) = 135° 0.125 (360°) = 45° 0.075 (360°) = 27° 0.05 (360°) = 18° Draw a circle, and use the angle degrees to make the sections. PRACTICE 1. Make a circle graph of the data in the table. Favorite Food Number of Students Salad Pizza Smoothie Soup 5 10 4 1 S80 S80 Skills Bank �� Misleading Graphs and Statistics Graphs can be misleading if the related statistics are distorted. E X A M P L E Explain why this graph is misleading. Gary’s score appears to be about three times that of Matt’s. Matt appears to do twice as well as Peter. This is because the scale on the graph begins well above zero. In fact, Gary is only 0.6% above Matt, and Peter is 0.2% below Matt. PRACTICE Explain why each graph is misleading. 1. 2. Venn Diagrams Venn diagrams are used to show relationships between two or more sets of numbers or objects. They show which elements are common between sets. E X A M P L E Draw a Venn diagram to show the relationship between the factors of 24 and factors of 32. Step 1: Find the elements of each set. A: {1, 2, 3, 4, 6, 8, 12, 24} Step 2: Draw a Venn diagram. B: {1, 2, 4, 8, 16, 32} Draw two overlapping ovals, one for each set. Write factors that are in both sets in the overlapping region. Write factors in one set only in the non-overlapping parts. PRACTICE Draw a Venn diagram to show the relationship between the following sets. 1. factors of 9 and factors of 8 2. factors of 36 and factors of 30 3. factors of 60 and factors of 72 Skills Bank S81 S81 �� Postulates, Theorems, and Corollaries Chapter 1 Post. 1-1-1 Through any two points there is exactly one line. (p. 7) Post. 1-1-2 Through any three noncollinear points there is exactly one plane containing them. (p. 7) Post. 1-1-3 If two points lie in a plane, then the line containing those points lies in the plane. (p. 7) Post. 1-1-4 If two lines intersect, then they intersect in exactly one point. (p. 8) Post. 1-1-5 If two planes intersect, then they intersect in exactly one line. (p. 8) Post. 1-2-1 Ruler Postulate The points on a line can be put into a one-to-one correspondence with the real numbers. (Ruler Post.; p. 13) Post. 1-2-2 Segment Addition Postulate If B is between A and C, then AB + BC = AC. (Seg. Add. Post.; p. 14) Post. 1-3-1 Protractor Postulate Given   AB and a point O on   AB , all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from 0 to 180. (Protractor Post.; p. 20) Post. 1-3-2 Angle Addition Postulate If S is in the interior of ∠PQR, then m∠PQS + m∠SQR = m∠PQR. (∠ Add. Post.; p. 22) Thm. 1-6-1 Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. (Pyth. Thm.; p. 45) Chapter 2 Thm. 2-6-1 Linear Pair Theorem If two angles form a linear pair, then they are supplementary. (Lin. Pair Thm.; p. 110) Thm. 2-6-2 Congruent Supplements Theorem If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent. (≅ Supps. Thm.; p. 111) Thm. 2-6-3 Right Angle Congruence Theorem All right angles are congruent. (Rt. ∠ ≅ Thm.; p. 112) Thm. 2-6-4 Congruent Complements Theorem If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent. (≅ Comps. Thm.; p. 112) S82 S82 Postulates, Theorems, and Corollaries Thm. 2-7-1 Common Segments Theorem Given collinear points A, B, C, and D arranged as shown, if Segs. Thm.; p. 118) ̶̶ BD . (Common ̶̶ CD , then ̶̶ AC ≅ ̶̶ AB ≅ Thm. 2-7-2 Vertical Angles Theorem Vertical angles are congruent. (Vert.  Thm.; p. 120) Thm. 2-7-3 If two congruent angles are supplementary, then each angle is a right angle. (≅  supp. → rt. ; p. 120) Chapter 3 Post. 3-2-1 Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. (Corr.  Post.; p. 155) Thm. 3-2-2 Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. (Alt. Int.  Thm.; p. 156) Thm. 3-2-3 Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles are congruent. (Alt. Ext.  Thm.; p. 156) Thm. 3-2-4 Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary. (Same-Side Int.  Thm.; p. 156) Post. 3-3-1 Converse of the Corresponding Angles Postulate If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. (Conv. of Corr.  Post.; p. 162) Post. 3-3-2 Parallel Postulate Through a point P not on line ℓ, there is exactly one line parallel to ℓ. (Parallel Post.; p. 163) Thm. 3-3-3 Converse of the Alternate Interior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel. (Conv. of Alt. Int.  Thm.; p. 163) Thm. 3-3-4 Converse of the Alternate Exterior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel. (Conv. of Alt. Ext.  Thm.; p. 163) �� Thm. 3-3-5 Converse of the Same-Side Interior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are parallel. (Conv. of Same-Side Int.  Thm.; p. 163) Thm. 3-4-1 If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. (2 intersecting lines form lin. pair of ≅  → lines ⊥; p. 173) Thm. 3-4-2 Perpendicular Transversal Theorem In a plane, if a transversal is perpendicular to one of two parallel lines, then
it is perpendicular to the other line. (⊥ Transv. Thm.; p. 173) Thm. 3-4-3 If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. (2 lines ⊥ to same line → 2 lines ǁ; p. 173) Thm. 3-5-1 Parallel Lines Theorem In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. (ǁ Lines Thm.; p. 184) Thm. 3-5-2 Perpendicular Lines Theorem In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular. (⊥ Lines Thm.; p. 184) Chapter 4 Thm. 4-2-1 Triangle Sum Theorem The sum of the angle measures of a triangle is 180°. (△ Sum Thm.; p. 223) Cor. 4-2-2 The acute angles of a right triangle are complementary. (Acute  of rt. △ are comp.; p. 224) Cor. 4-2-3 The measure of each angle of an equiangular triangle is 60°. (p. 224) Thm. 4-2-4 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. (Ext. ∠ Thm.; p. 225) Thm. 4-2-5 Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. (Third  Thm.; p. 226) Post. 4-4-1 Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. (SSS; p. 242) Post. 4-4-2 Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. (SAS; p. 243) Post. 4-5-1 Angle-Side-Angle (ASA) Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. (ASA; p. 252) Thm. 4-5-2 Angle-Angle-Side (AAS) Congruence Theorem If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent. (AAS; p. 254) Thm. 4-5-3 Hypotenuse-Leg (HL) Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. (HL; p. 255) Thm. 4-8-1 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the sides are congruent. (Isosc. △ Thm.; p. 273) Thm. 4-8-2 Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. (Conv. of Isosc. △ Thm.; p. 273) Cor. 4-8-3 If a triangle is equilateral, then it is equiangular. (equilateral △ → equiangular; p. 274) Cor. 4-8-4 If a triangle is equiangular, then it is equilateral. (equiangular △ → equilateral; p. 275) Chapter 5 Thm. 5-1-1 Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. (⊥ Bisector Thm.; p. 300) Thm. 5-1-2 Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. (Conv. of ⊥ Bisector Thm.; p. 300) Thm. 5-1-3 Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. (∠ Bisector Thm.; p. 301) Postulates, Theorems, and Corollaries S83 S83 Thm. 5-1-4 Converse of the Angle Bisector Thm. 5-7-2 Pythagorean Inequalities Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. (Conv. of ∠ Bisector Thm.; p. 301) Theorem In △ABC, c is the length of the longest side. If c 2 > a 2 + b 2 , then △ABC is an obtuse triangle. If c 2 < a 2 + b 2 , then △ABC is an acute triangle. (Pyth. Inequal. Thm.; p. 351) Thm. 5-2-1 Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle. (Circumcenter Thm.; p. 307) Thm. 5-8-1 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times √  2 . (45°-45°-90° △ Thm.; p. 356) Thm. 5-2-2 Incenter Theorem The incenter of a triangle is equidistant from the sides of the triangle. (Incenter Thm.; p. 309) Thm. 5-3-1 Centroid Theorem The centroid of a triangle is located 2 __ 3 of the distance from each vertex to the midpoint of the opposite side. (Centroid Thm.; p. 314) Thm. 5-4-1 Triangle Midsegment Theorem A midsegment of a triangle is parallel to a side of a triangle, and its length is half the length of that side. (△ Midsegment Thm.; p. 323) Thm. 5-5-1 If two sides of a triangle are not congruent, then the larger angle is opposite the longer side. (In △, larger ∠ is opp. longer side; p. 333) Thm. 5-5-2 If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. (In △, longer side is opp. larger ∠; p. 333) Thm. 5-5-3 Triangle Inequality Theorem The sum of any two side lengths of a triangle is greater than the third side length. (△ Inequal. Thm.; p. 334) Thm. 5-6-1 Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is across from the larger included angle. (Hinge Thm.; p. 340) Thm. 5-6-2 Converse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side. (Conv. of Hinge Thm.; p. 340) Thm. 5-7-1 Converse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. (Conv. of Pyth. Thm.; p. 350) Thm. 5-8-2 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times √  3 . (30°-60°-90° △ Thm.; p. 358) Chapter 6 Thm. 6-1-1 Polygon Angle Sum Theorem The sum of the interior angle measures of a convex polygon with n sides is (n - 2) 180°. (Polygon ∠ Sum Thm.; p. 383) Thm. 6-1-2 Polygon Exterior Angle Sum Theorem The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360°. (Polygon Ext. ∠ Sum Thm.; p. 384) Thm. 6-2-1 If a quadrilateral is a parallelogram, then its opposite sides are congruent. ( → opp. sides ≅; p. 391) Thm. 6-2-2 If a quadrilateral is a parallelogram, then its opposite angles are congruent. ( → opp.  ≅; p. 392) Thm. 6-2-3 If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. ( → cons.  supp.; p. 392) Thm. 6-2-4 If a quadrilateral is a parallelogram, then its diagonals bisect each other. ( → diags. bisect each other; p. 392) Thm. 6-3-1 If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. (quad. with pair of opp. sides ǁ and ≅ → ; p. 398) Thm. 6-3-2 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. (quad. with opp. sides ≅ → ; p. 398) Thm. 6-3-3 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. (quad. with opp.  ≅ → ; p. 398) S84 S84 Postulates, Theorems, and Corollaries Thm. 6-3-4 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. (quad. with ∠ supp. to cons.  → ; p. 399) Thm. 6-3-5 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. (quad. with diags. bisecting each other → ; p. 399) Thm. 6-4-1 If a quadrilateral is a rectangle, then it is a parallelogram. (rect. → ; p. 408) Thm. 6-4-2 If a parallelogram is a rectangle, then its diagonals are congruent. (rect. → diags. ≅; p. 408) Thm. 6-4-3 If a quadrilateral is a rhombus, then it is a parallelogram. (rhombus → ; p. 409) Thm. 6-4-4 If a parallelogram is a rhombus, then its diagonals are perpendicular. (rhombus → diags. ⊥; p. 409) Thm. 6-4-5 If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. (rhombus → each diag. bisects opp. ; p. 409) Thm. 6-5-1 If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. ( with one rt. ∠ → rect.; p. 418) Thm. 6-5-2 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. ( with diags. ≅ → rect.; p. 418) Thm. 6-5-3 If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. ( with one pair cons. sides ≅ → rhombus; p. 419) Thm. 6-5-4 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. ( with diags. ⊥ → rhombus; p. 419) Thm. 6-5-5 If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. ( with diags. bisecting opp.  → rhombus; p. 419) Thm. 6-6-1 If a quadrilateral is a kite, then its diagonals are perpendicular. (kite → diags. ⊥; p. 427) Thm. 6-6-2 If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. (kite → one pair opp.  ≅; p. 427) Thm. 6-6-3 If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent. (isosc. trap. → base  ≅; p. 429) Thm. 6-6-4 If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles. (trap. with pair base  ≅ → isosc. trap.; p. 429) Thm. 6-6-5 A trapezoid is isosceles if and only if its diagonals are congruent. (isosc. trap ↔ diags. ≅; p. 429) Thm. 6-6-6 Trapezoid Midsegment Theorem The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases. (Trap.
Midsegment Thm.; p. 431) Chapter 7 Post. 7-3-1 Angle-Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. (AA ∼ Post.; p. 470) Thm. 7-3-2 Side-Side-Side (SSS) Similarity Theorem If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar. (SSS ∼ Thm.; p. 470) Thm. 7-3-3 Side-Angle-Side (SAS) Similarity Theorem If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. (SAS ∼ Thm.; p. 471) Thm. 7-4-1 Triangle Proportionality Theorem If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally. (△ Proportionality Thm.; p. 481) Thm. 7-4-2 Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. (Conv. of △ Proportionality Thm.; p. 482) Cor. 7-4-3 Two-Transversal Proportionality Corollary If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. (2-Transv. Proportionality Cor.; p. 482) Thm. 7-4-4 Triangle Angle Bisector Theorem An angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides. (△ ∠ Bisector Thm.; p. 483) Thm. 7-5-1 Proportional Perimeters and Areas Theorem If the similarity ratio of two similar figures is a __ , then the ratio of their perimeters is b , and the ratio of their areas is a 2 ___ a __ b 2 b or ( a __ ) . (p. 490) b 2 Postulates, Theorems, and Corollaries S85 S85 Chapter 8 Thm. 8-1-1 The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. (p. 518) Cor. 8-1-2 Geometric Means Corollary The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse. (p. 519) Cor. 8-1-3 Geometric Means Corollary The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg. (p. 519) Thm. 8-5-1 The Law of Sines For any △ABC ____ ____ ____ a = sin B = sin C . c b with side lengths a, b, and c, sin A (p. 552) Thm. 8-5-2 The Law of Cosines For any △ABC with sides a, b, and c, a 2 = b 2 + c 2 - 2bc cos A, b 2 = a 2 + c 2 - 2ac cos B, and c 2 = a 2 + b 2 - 2ab cos C. (p. 553) Chapter 9 Post. 9-1-1 Area Addition Postulate The area of a region is equal to the sum of the areas of its nonoverlapping parts. (Area Add. Post.; p. 589) Chapter 11 Thm. 11-1-1 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. (line tangent to ⊙ → line ⊥ to radius; p. 748) Thm. 11-1-2 If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. (line ⊥ to radius → line tangent to ⊙; p. 748) Thm. 11-1-3 If two segments are tangent to a circle from the same external point, then the segments are congruent. (2 segs. tangent to ⊙ from same ext. pt. → segs. ≅; p. 749) Post. 11-2-1 Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. (Arc Add. Post.; p. 757) Thm. 11-2-2 In a circle or congruent circles: (1) congruent central angles have congruent chords, (2) congruent chords have congruent arcs, and (3) congruent arcs have congruent central angles. (≅ arcs have ≅ central  have ≅ chords; p. 757) S86 S86 Postulates, Theorems, and Corollaries Thm. 11-2-3 In a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc. (Diam. ⊥ chord → diam. bisects chord and arc; p. 759) Thm. 11-2-4 In a circle, the perpendicular bisector of a chord is a radius (or diameter). (⊥ bisector of chord is diam.; p. 759) Thm. 11-4-1 Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. (Inscribed ∠ Thm.; p. 772) Cor. 11-4-2 If inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc, then the angles are congruent. (p. 773) Thm. 11-4-3 An inscribed angle subtends a semicircle if and only if the angle is a right angle. (p. 774) Thm. 11-4-4 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. (Quad. inscribed in circle → opp.  supp.; p. 775) Thm. 11-5-1 If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. (p. 782) Thm. 11-5-2 If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs. (p. 783) Thm. 11-5-3 If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs. (p. 784) Thm. 11-6-1 Chord-Chord Product Theorem If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal. (p. 792) Thm. 11-6-2 Secant-Secant Product Theorem If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (whole • outside = whole • outside; p. 793) Thm. 11-6-3 Secant-Tangent Product Theorem If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. (whole • outside = tangent 2 ; p. 794) Thm. 11-7-1 Equation of a Circle The equation of a circle with center (h, k) and radius r is (x - h) 2 + (y - k) 2 = r 2 . (p. 799) Chapter 12 Thm. 12-4-1 A composition of two isometries is an isometry. (p. 848) Thm. 12-4-2 The composition of two reflections across two parallel lines is equivalent to a translation. The translation vector is perpendicular to the lines. The length of the translation vector is twice the distance between the lines. The composition of two reflections across two intersecting lines is equivalent to a rotation. The center of rotation is the intersection of the lines. The angle of rotation is twice the measure of the angle formed by the lines. (p. 849) Thm. 12-4-3 Any translation or rotation is equivalent to a composition of two reflections. (p. 850) Constructions Angle Bisector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 23 Parallel Lines . . . . . . . . . . . . . . . pp. 163, 170, 171, 179 Center of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . p. 774 Parallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 404 Centroid of a Triangle . . . . . . . . . . . . . . . . . . . . . p. 314 Perpendicular Bisector Circle Circumscribed of a Segment . . . . . . . . . . . . . . . . . . . . . . . . . p. 172 About a Triangle . . . . . . . . . . . . . . . . pp. 313, 778 Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . p. 179 Circle Inscribed in a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 313 Circle Through Three Noncollinear Points . . . . . . . . . . . . . . . . . . . p. 763 Circumcenter of a Triangle . . . . . . . . . . . . . . . . . . . . . . pp. 307, 313 Congruent Angles . . . . . . . . . . . . . . . . . . . . . . . . . .p. 22 Congruent Segmentsp. 14 Congruent Triangles Rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 424 Reflection. . . . . . . . . . . . . . . . . . . . . . . . . . pp. 824, 829 Regular Decagon . . . . . . . . . . . . . . . . . . . . . . . . . p. 381 Regular Dodecagon . . . . . . . . . . . . . . . . . . . . . . . p. 380 Regular Hexagon . . . . . . . . . . . . . . . . . . . . . . . . . p. 380 Regular Octagon. 380 Regular Pentagon. 381 Rhombus . . . . . . . . . . . . . . . . . . . . . . . . . . pp. 415, 424 Using ASA . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 253 Right Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 258 Congruent Triangles Using SAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 243 Congruent Triangles Using SSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 248 Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . pp. 872, 878 Equilateral Triangle . . . . . . . . . . . . . . . . . . . . . . . p. 220 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . pp. 839, 844 Segment Bisector . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 16 Square. . . . . . . . . . . . . . . . . . . . . . . . . . . . . pp. 380, 424 Tangent to a Circle at a Point. 748 Tangent to a Circle from Incenter of a Triangle . . . . . . . . . . . . . . . . . . . . . p. 313 an Exterior Point . . . . . . . . . . . . . . . . . . . . . . p. 779 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . p. 363 Translation . . . . . . . . . . . . . . . . . . . . . . . . pp. 831, 836 Kite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 435 Triangle Circumscribed Midpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 16 Midsegment of a Triangle. 327 Orthocenter of a Triangle . . . . . . . . . . . . . . . . . . p. 320 About a Circle . . . . . . . . . . . . . . . . . . . . . . . . p. 779 Trisecting a Segment . . . . . . . . . . . . . . . . . . . . . . p. 487 Constructions S87S87 Selected Answers Chapter 1 43. 14.02 m 45. 12 47. -23 49. -8x + 6 51. ̶̶ BD 53. ̶̶ AD ,  CB 1-1 1-3 Check It Out! 1. Possible answer: plane R and plane ABC 2. 3. Possible answer: plane GHF 4. Exercises 3. A, B, C, D, E 5. Possible answer: planes ABC and  7. 9. Possible answer:   AB 11.
13. B, E, A 15. Possible answer: ABC 17. 19. Possible answer: planes  and  21. 23. 25. U 27. U 29. If 2 lines intersect, then they intersect in exactly 1 pt. 31. A 33. A 35. Post. 1-1-3 37. Post. 1-1-2 39. C 41. D n (n - 1) _______ 43. 6 45. 2 twins are 11. 49. no 51. mean: 0.44; median: 0.442; mode: 0.44 47. Mother is 36; 1-2 Check It Out! 1a. 3 1 __ 2 1b. 4 1 __ 2 3a. 1 2 __ 3 3b. 24 4. 591.25 m 5. RS = 4; ST = 4; RT = 8 ̶̶̶ MY 3. 3.5 ̶̶̶ XM and Exercises 1. 7. 29 9. x = 4; KL = 7; JL = 14 11. 5 11 __ 12 15. 5 17. DE = EF =14; DF = 28 19a. C is the mdpt. of b. 16 21. 7.1 23. 4 25. S 27. Statement A 29. 6.5; -1.5 31. 3.375 33. 9 37. J 39. H ̶̶ AE . 41. S88 S88 Selected Answers Check It Out! 1. ∠RTQ, ∠T, ∠STR, ∠1, ∠2 2a. 40°; acute 2b. 125°; obtuse 2c. 105°; obtuse 3. 62° 4a. 34° 4b. 46° Exercises 1. ∠A, ∠R, ∠O 3. ∠AOB, ∠BOA, or ∠1; ∠BOC, ∠COB, or ∠2; ∠AOC or ∠COA 5. 105°; obtuse 7. 70° 9. 28° 11. ∠1 or ∠JMK; ∠2 or ∠LMK; ∠M or ∠JML 13. 93°; obtuse 15. 66.6° 17. 20° 19. acute 21. acute 27. 67.5°; 22.5° 29. 16 1 __ 3 31. 9 33a. 9 b. 12 c. 0 < x < 15.6 35. m∠COD = 72°; m∠BOC = 90° 37. No; an obtuse ∠ measures greater than 90°, so it cannot be ≅ to an acute ∠ (less than 90°). 41. D 43. C 45. The  are acute. An obtuse ∠ measures between 90° and 180°. Since 1 __ 2 of 180 is 90, the resulting  must measure less than 90°. 47. 36° or 4° 49. 8100 51. 22.4 53. 55. 57. 6 1-4 Check It Out! 1a. adj.; lin. pair 1b. not adj. 1c. not adj. 2a. (102 - 7x) ° 2b. 63 1 __ 2 ° 3. 68° 4. m∠1 = m∠2 = 62.4°; m∠4 = 27.6° 5. Possible answer: ∠EDG and ∠FDH; m∠EDG ≈ m∠FDH ≈ 45° Exercises 1. (90 - x) °; (180 - x) ° 3. adj.; lin. pair 5. not adj. 7. 98.8° 9. (185 - 6x) ° 11. 69° 13. ∠ABE, ∠CBD; ∠ABC, ∠EBD 15. adj.; lin. pair 17. not adj. 19. 33.6° 21. (94 - 2x) ° 23. m∠2 = 22.3°; m∠3 = m∠4 = 67.7° 25. 1 __ 3 27. 72°; 108° 29. 61°; 29° 31. 10°; 80° 33a. m∠JAH = 64°; m∠KAH = 26° b. m∠JAH = 131.5°; m∠KAH = 48.5° c. m∠JAH = m∠KAH = 7° 35. F 37. T 39. C 41. C 43. 12 45. 30° 47. x = 8 49. y = 3 51. 17 53. 32 55. 52° 1-5 Check It Out! 1. P =14 in.; A = 12.75 in 2 2. 65 i n 2 3. C ≈ 88.0 m; A ≈ 615.8 m 2 Exercises 1. Both terms refer to the dist. around a figure. 3. P = 30 mm; A = 44 m m 2 5. P = (x + 21) m; A = (2x + 6) m 2 7. C ≈ 13.2 m; A ≈ 13.9 m 2 9. C ≈ 50.3 cm; A ≈ 201.1 cm 2 11. P = 4x + 12; A = x 2 + 6x 13. 72 in 2 15. C ≈ 39.3 ft; A ≈ 122.7 ft 2 17. 82.81 yd 2 19. 6.1875 in 2 21. 17.1 cm 23. Statement A 25. 9 y 2 π 27. For a square, the length and width are both s, so P = 2l + 2w = 2s + 2s = 4s and A = lw = s (s) = s 2 . 29. b = 41 in.; h = 38 in. 31a. ac + ad + bc + bd b. (a + 1) (c + 1) ; ac + a + c + 1 c. (a + 1) 2 ; a 2 + 2a + 1 33. 28 ft 35. 26.46 ft 2 37. 25 2 __ 3 yd 2 or 231 ft 2 39. 10 ft 41. 14 __ π 43. 50 45. Measure any side as the base. Then measure the height of the △ at a rt. ∠ to the base. 47. B 49. A 51. 83.7 in 2 53. 5; 8; 9 55. width = 16 in.; length = 20 in. 57. D: {4, -2, 16}; R: {-2, 8, 0} 59. line or segment 61. 3 1-6 Check It Out! 1. ( 3 __ 2 , 0) 2. (4, 3) ̶̶ 3. EF = 5; GH = 5; EF ≅ 4a. 6.7 4b. 8.5 5. 60.5 ft ̶̶̶ GH Exercises 1. hypotenuse 3. (1 1 __ 2 , -4) 5. (0, -2) 7. √  29 ; 3 √  5 ; no 9. 15.0 11. 27.2 ft 13. (3 1 __ 2 , -4 1 __ 2 ) 15. (8, 4) 17. 2 √  5 ; √  29 ; no 19. 8.9 21. 18 in. 23. 4.47 25. Divide each coord. by 2. 27. 2.5 mi 31. 1 33. Let M be the ̶̶ AC ; AM = MC = 5.0 ft; mdpt. of MB = MD ≈ 6.4 ft. 35. G 37. J 39. ±2 41. AB = √  x 2 + y 2 43. yes 45. 90°; rt. 47. 135°; obtuse 49. 4 ft 2 �� 1-7 Check It Out! 1a. translation; MNOP → M′N′O′P′ 1b. rotation; △XYZ → △X′Y′Z′ 2. rotation; 90° 3. J′ (-1, -5) ; K′ (1, 5) ; L′ (1, 0) ; M′ (-1, 0) 4. (x, y) → (x - 4, y - 4) Exercises 1. Preimage is △XYZ ; image is △X′Y′Z′ 3. reflection; △ABC → △A′B′C′ 5. reflection across the y-axis 7. (x, y) → (x + 4, y + 4) 9. reflection; WXYZ → W′X′Y′Z′ 11. A′ (-1, -1) , B′ (4, -1) , C (4, -4) , D (-1, -4) 13. reflection 15. reflection 17. acute; ∠XYW: rt.; ∠ZYW: acute; ∠VYW: straight 17. 59° 18. 96° 19. only adj. 20. adj. and a lin. pair 21. not adj. 22. 15.4°; 105.4° 23. (94 - 2x) °; (184 - 2x) ° 24. 73° 25. 14x - 2; 12x 2 - 3x 26. 4x + 16; x 2 + 8x + 16 27. x + 15; 4x - 20 28. 10x + 54; 100x + 140 29. A ≈ 1385.4 m 2 ; C ≈ 131.9 m 30. A ≈ 153.9 ft 2 ; C ≈ 44.0 ft 31. 12 m 32. Y (1, 3) 33. B (- 9, 6) 34. A (0, 2) 35. 8.5 36. 7.3 37. 8.1 38. 90° rotation; DEFG → D′E′F′G′ 39. translation; PQRS → P′Q′R′S′ 40. X′ (-1, 1) ; Y′ (1, 4) ; Z′ (2, 3) 19. B 21. D 23. R′ (-1, -12) ; S′ (-3, -9) ; T ′ (-7, -7) 25. 29. A 31. A 33a. R″ (1, 0) ; S″ (0, 3) ; T′′ (4, 3) b. (x, y) → ( x + 3, y + 2) 35. 37. (-x, y) 39. x = -6 or x = 3 41. x = 1 or x = 2 43. 13.9° 45. 4.1 47. 6.3 SGR 1. angle bisector 2. complementary angles 3. hypotenuse 4. A, F, E, G or C, G, D, B 5. Possible answer:   GC 6. Possible answer: plane AEG 7. 8. 9. Chapter 2 2-1 Check It Out! 1. 0.0004 2. odd 3. Female whales are longer than male whales. 4a. Possible answer: x = 1 __ 2 4b. Possible answer: 4c. Jupiter or Saturn Exercises 3. 4 __ 6 5. even 7. The number of bacteria doubles every 20 minutes. 9. The 3 pts. are collinear. 11. 5 P.M. 13. ̶̶ 09 , 2 __ 11 = 0. 15. n - 1 17. Possible answer: y = -1 19. m∠1 = m∠2 = 90° 21. Possible answer: each term is the previous term multiplied by 1 __ 2 ; 1 __ 16 ; 1 __ 32 . 23. 2n + 1 25. F ̶̶ 18 , 3 __ 11 27. T 29. 1 __ 11 = 0. ̶̶ = 0. 27 ,…; the fraction pattern is multiples of 1 __ 11 , and the decimal pattern is repeating multiples of 0.09. 31. 34, 55, 89; each term is the sum of the 2 previous terms. 33. odd 37. C 39. D 41. 12 years 43. m∠CAB = m∠CBA; AC = CB 45. yes 47. no 49. 10x - 6 51. 6πx 53. (3, -2) , (4, 0) , (8, -1) 2-2 10. 3.5 11. 5 12. 7.6 13. 22 14. 13; 13; 26 15. 18; 18; 36 16. ∠VYX: rt.; ∠VYZ: obtuse; ∠XYZ: Check It Out! 1. Hypothesis: A number is divisible by 6. Conclusion: A number is divisible by 3. 2. If 2  are comp., then they are acute. 3. F; possible answer: 7 4. Converse: If an animal has 4 paws, then it is a cat; F. Inverse: If an animal is not a cat, then it does not have 4 paws; F. Contrapositive: If an animal does not have 4 paws, then it is not a cat; T. 2 b < a __ . 9. T 11. F b Exercises 1. converse 3. Hypothesis: A person is at least 16 years old. Conclusion: A person can drive a car. 5. Hypothesis: a - b < a. Conclusion: b is a positive number. 7. If 0 < a < b, then ( a __ ) 13. Hypothesis: An animal is a tabby. Conclusion: An animal is a cat. 15. Hypothesis: 8 ounces of cereal cost $2.99. Conclusion: 16 ounces of cereal cost $5.98. 17. If the batter makes 3 strikes, then the batter is out. 19. T 21. T 25. T 27. F 29. F 35. If a person is a Texan, then the person is an American. 37a. H: Only you can find it. C: Everything’s got a moral. b. If only you can find it, then everything’s got a moral. 43. If a mineral has a hardness less than 5, then it is not apatite; T. 45. If a mineral is not apatite, then it is calcite; F. 47. If a mineral is calcite, then it has a hardness less than 5; T. 51. H 53. J 55. Some students are adults. Some adults are students. 57. 3 59. y = 2x + 1 61. T 63. T 65. 2 __ 81 2-3 Check It Out! 1. deductive reasoning 2. valid 3. valid 4. Polygon P is not a quad. Exercises 3. deductive reasoning 5. valid 7. invalid 9. deductive reasoning 11. invalid 13. Dakota gets better grades in Social Studies. 15. valid 17. valid 19. yes; no; because the first conditional is false 23. D 25. 196 27a. If you live in San Diego, then you live in the United States. b. If you do not live in California, then you do not live in San Diego. If you do not live in the United States, then you do not live in California. c. If you do not Selected Answers S89 S89 �� live in the United States, then you do not live in San Diego. d. They are contrapositives of each other. 29. 2x + 10 31. -7c + 14 33. (-1.5, 3.5) 2-4 Check It Out! 1a. Conditional: If an ∠ is acute, then its measure is greater than 0° and less than 90°. Converse: If an ∠’s measure is greater than 0° and less than 90°, then the ∠ is acute. 1b. Conditional: If Cho is a member, then he has paid the $5 dues. Converse: If Cho has paid the $5 dues, then he is a member. 2a. Converse: If it is Independence Day, then the date is July 4th. Biconditional: It is July 4th if and only if it is Independence Day. 2b. Converse: If pts. are collinear, then they lie on the same line. Biconditional: Pts. lie on the same line if and only if they are collinear. 3a. T 3b. F; y = 5 4a. A figure is a quad. if and only if it is a 4-sided polygon. 4b. An ∠ is a straight ∠ if and only if its measure is 180°. Exercises 3. Conditional: If your medicine will be ready by 5 P.M. , then you dropped your prescription off by 8 A.M. Converse: If you drop your prescription off by 8 A.M. , then your medicine will be ready by 5 P.M. 5. Converse: If 2 segs. are ≅, then they have the same length. Biconditional: 2 segs. have the same length if and only if they are ≅. 7. F 9. An animal is a hummingbird if and only if it is a tiny, brightly colored bird with narrow wings, a slender bill, and a long tongue. 11. Conditional: If a  is a rect., then it has 4 rt. . Converse: If a  has 4 rt. , then it is a rect. 13. Converse: If it is the weekend, then today is Saturday or Sunday. Biconditional: Today is Saturday or Sunday if and only if it is the weekend. 15. Converse: If a △ is a rt. △, then it contains a rt. ∠. Biconditional: A △ contains a rt. ∠ if and only if it is a rt. △. 17. T 19. A player is a catcher if and only if the player is positioned behind home plate and catches throws from the pitcher. 21. yes 23. no 25. A square is a quad. with 4 ≅ sides and 4 rt. . 31. no 33. 5 37a. If I say it, then I mean it. If I mean it, then I say it. 39. G 43a. If an ∠ does not measure 105°, then the ∠ is not obtuse. b. If an ∠ is not obtuse, then it does not measure 105°. c. It is the contrapositive of the original. d. F; the inverse is false, and its converse is true. 47. The graph is reflected across the x-axis and shif
ted 1 unit down and is narrower than the graph of the parent function. 49. T 51. S 53. F 2-5 Check It Out! 1. 1 __ 2 t = -7 (Given); 2 ( 1 __ 2 t) = 2 (-7) (Mult. Prop. of =); t = -14 (Simplify.) 2. C = 5 __ 9 (F - 32) (Given); C = 5 __ 9 (86 - 32) (Subst.); C = 5 __ 9 (54) (Simplify.); C = 30 (Simplify.) 3. ∠ Add. Post.; Subst.; Simplify.; Subtr. Prop. of =; Mult. Prop. of = 4a. Sym. Prop. of = 4b. Reflex. Prop. of = 4c. Trans. Prop. of = 4d. Sym. Prop. of ≅ Exercises 3. t - 3.2 = -8.3 (Given); t = -5.1 (Add. Prop. of =) 5. x + 3 ____ -2 = 8 (Given); x + 3 = -16 (Mult. Prop. of =); x = -19 (Subtr. Prop. of =) 7. 0 = 2 (r - 3) + 4 (Given); 0 = 2r - 6 + 4 (Distrib. Prop.); 0 = 2r - 2 (Simplify.); 2 = 2r (Add. Prop. of =); 1 = r (Div. Prop. of =) 9. C = $5.75 + $0.89m (Given); $11.98 = $5.75 + $0.89m (Subst.); $6.23 = $0.89m (Subtr. Prop. of =); m = 7 (Div. Prop. of =) 11. Seg. Add. Post.; Subst.; Subtr. Prop. of =; Add. Prop. of =; Div. Prop. of = 13. Trans. Prop. of = 15. Trans. Prop. of ≅ 17. 1.6 = 3.2n (Given); 0.5 = n (Div. Prop. of =) 19. - (h + 3) = 72 (Given); -h - 3 = 72 (Distrib. Prop.); -h = 75 (Add. Prop. of =); h = -75 (Mult. Prop. of =) 21. 1 __ 2 (p - 16) = 13 (Given); 1 __ 2 p - 8 = 13 (Distrib. Prop.); 1 __ 2 p = 21 (Add. Prop. of =); p = 42 (Mult. Prop. of =) 23. ∠ Add. Post.; Subst.; Simplify.; Subtr. Prop. of =; Add. Prop. of =; Div. Prop. of = S90 S90 Selected Answers 25. Sym. Prop. of ≅ 27. Trans. Prop. of = 29. x = 16; 2 (3.1x - 0.87) = 94.36 (Given); 6.2x - 1.74 = 94.36 (Distrib. Prop.); 6.2x = 96.1 (Add. Prop. of =); x = 15.5 (Div. Prop. of =); possible answer: the exact solution rounds to the estimate. 31. ∠A ≅ ∠T 33. x + 1 ____ 2 x + 1 = 6 (Mult. Prop. of =); x = 5 = 3 (Mdpt. Formula;) 1 + y ____ 2 = 5 (Mdpt. (Subtr. Prop. of =); Formula); 1 + y = 10 (Mult. Prop. of =); y = 9 (Subtr. Prop. of =) 35a. 1733.65 = 92.50 + 79.96 + 983 + 10,820x (Given); 1733.65 = 1155.46 + 10,820x (Simplify.); 578.19 = 10,820x (Subtr. Prop. of =); 0.05 ≈ x (Div. Prop. of =) b. $1.71 37a. x + 15 ≤ 63 (Given); x ≤ 48 (Subtr. Prop. of Inequal.) b. -2x > 36 (Given); x < -18 (Div. Prop. of Inequal.) 39. B 41. D 43. PR = PA + RA (Seg. Add. Post.); PA = QB, QB = RA (Given); PA = RA (Trans. Prop. of =); PR = PA + PA (Subst.); PA = 18 (Given) ; PR = 18 + 18 (Subst.); PR = 36 in. (Simplify.) 45. 7 - 3x > 19 (Given); -3x > 12 (Subtr. Prop. of Inequal.); x < -4 (Div. Prop. of Inequal.) 49. deductive reasoning 2-6 Check It Out! 1. 1. Given 2. Def. of mdpt. 3. Given 4. Trans. Prop. of ≅ 2a. ∠1 and ∠2 are supp., and ∠2 and ∠3 are supp. 2b. m∠1 + m∠2 = m∠2 + m∠3 2c. Subtr. Prop. of = 2d. ∠1 ≅ ∠3 3. 1. ∠1 and ∠2 are comp. ∠2 and ∠3 are comp. (Given) 2. m∠1 + m∠2 = 90°, m∠2 + m∠3 = 90° (Def. of comp. ) 3. m∠1 + m∠2 = m∠2 + m∠3 (Subst.) 4. m∠2 = m∠2 (Reflex. Prop. of =) 5. m∠1 = m∠3 (Subtr. Prop. of =) 6. ∠1 ≅ ∠3 (Def. of ≅ ) Exercises 1. statements; reasons 3. 1. Given ∠3 ≅ ∠4. By the Trans. Prop. of ≅, ∠2 ≅ ∠4. Similarly, ∠2 ≅ ∠3. the right. The next 2 items are and . 2. Subst. 3. Simplify. 4. Add. Prop. of = 5. Simplify. 6. Def. of supp.  ̶̶ AY . Y is the 5. 1. X is the mdpt. of ̶̶ XB . (Given) ̶̶ ̶̶ ̶̶ YB (Def. of XY ≅ XY , ̶̶ YB (Trans. Prop. of ≅) mdpt. of ̶̶ 2. AX ≅ mdpt.) ̶̶ 3. AX ≅ 7a. m∠1 + m∠2 = 180°, m∠3 + m∠4 = 180° b. Subst. c. m∠1 = m∠4 d. Def. of ≅  ̶̶ ̶̶ AE (Given) CE , ̶̶ DE ≅ 2. BE = CE, DE = AE (Def. of ≅ segs.) 3. AE + BE = AB, CE + DE = CD (Seg. Add. Post.) 4. DE + CE = AB (Subst.) 5. AB = CD (Subst.) ̶̶ 6. CD (Def. of ≅ segs.) ̶̶ AB ≅ ̶̶ BE ≅ 9. 1. Exercises 1. flowchart 3. 1. ∠1 ≅ ∠2 (Given) 2. ∠1 and ∠2 are supp. (Lin. Pair Thm.) 3. ∠1 and ∠2 are rt. . (≅  supp. → rt. ) 5. 1. ∠2 ≅ ∠4 (Given) ̶̶ AB ≅ 2. ∠1 ≅ ∠2 , ∠3 ≅ ∠4 (Vert.  Thm.) 3. ∠1 ≅ ∠4 (Trans. Prop. of ≅) 4. ∠1 ≅ ∠3 (Trans. Prop. of ≅) ̶̶ AC . (Given) 7. 1. B is the mdpt. of ̶̶ 2. BC (Def. of mdpt.) 3. AB = BC (Def. of ≅ segs.) 4. AD + DB = AB, BE + EC = BC (Seg. Add. Post.) 5. AD + DB = BE + EC (Subst.) 6. AD = EC (Given) 7. DB = BE (Subtr. Prop. of =) 11. 132° 13. 59° 17. S 19. N 21. x = 16 25. C 27. D 29. a = 17; 37.5°, 52.5°, and 37.5° 31. 24% 35. Sym. Prop. of ≅ 2-7 Check It Out! 1. 1. RS = UV, ST = TU (Given) 2. RS + ST = TU + UV (Add. Prop. of =) 3. RS + ST = RT, TU + UV = TV (Seg. Add. Post.) 4. RT = TV (Subst.) ̶̶ TV (Def. of ≅ segs.) 5. ̶̶ RT ≅ 2. 3. 1. ∠WXY is a rt. ∠. (Given) 2. m∠WXY = 90° (Def. of rt. ∠) 3. m∠2 + m∠3 = m∠WXY (∠ Add. Post.) 4. m∠2 + m∠3 = 90° (Subst.) 5. ∠1 ≅ ∠3 (Given) 6. m∠1 = m∠3 (Def. of ≅ ) 7. m∠2 + m∠1 = 90° (Subst.) 8. ∠1 and ∠2 are comp. (Def. of comp. ) 4. It is given that ∠1 ≅ ∠4. By the Vert.  Thm., ∠1 ≅ ∠2 and 9. 1. ∠1 ≅ ∠4 (Given) 2. ∠1 ≅ ∠2 (Vert.  Thm.) 3. ∠4 ≅ ∠2 (Trans. Prop. of ≅) 4. m∠4 = m∠2 (Def. of ≅ ) 5. ∠3 and ∠4 are supp. (Lin. Pair Thm.) 6. m∠3 + m∠4 = 180° (Def. of supp. ) 7. m∠3 + m∠2 = 180° (Subst.) 8. ∠2 and ∠3 are supp. (Def. of supp. ) 11. 13 cm; conv. of the Common Segs. Thm. 13. 37°, Vert.  Thm. 15. y = 11 17. A 21. C 23. D 25. 1. ∠AOC ≅ ∠BOD (Given) 2. m∠AOC = m∠BOD (Def. of ≅ ) 3. m∠AOB + m∠BOC = m∠AOC, m∠BOC + m∠COD = m∠BOD (∠ Add. Post.) 4. m∠AOB + m∠BOC = m∠BOC + m∠COD (Subst.) 5. m∠BOC = m∠BOC (Reflex. Prop. of =) 6. m∠AOB = m∠COD (Subtr. Prop. of =) 7. ∠AOB ≅ ∠COD (Def. of ≅ ) 27. x = 31 and y = 11.5; 86°, 94°, 86°, and 94° 29. (-4, 5) SGR 1. theorem 2. deductive reasoning 3. counterexample 4. conjecture 5. The rightmost △ is duplicated, rotated 180°, and shifted to and 1. 7. The white section 6. Each item is 1 __ 6 greater than the previous one. The next 2 items are 5 _ 6 is halved. If the white section is a rect. but not a square, it is halved horiz. and the upper portion is colored yellow. If the white section is a square, it is halved vert. and the left portion is colored yellow. The next 2 items are and . 8. odd 9. positive 10. F; 0 11. T 12. T 13. F; during a leap year, there are 29 days in February. 14. Check students’ constructions. Possible answer: The 3 ∠ bisectors of a △ intersect in the int. of the △. 15. If it is Monday, then it is a weekday. 16. If something is a lichen, then it is a fungus. 17. T 18. F; possible answer: √  2 and √  2 19. Converse: If m∠X = 90°, then ∠X is a rt. ∠; T. Inverse: If ∠X is not a rt. ∠, then m∠X ≠ 90°; T. Contrapositive: If m∠X ≠ 90°, then ∠X is not a rt. ∠; T. 20. Converse: If x = 2, then x is a whole number; T. Inverse: If x is not a whole number, then x ≠ 2; T. Contrapositive: If x ≠ 2, then x is not a whole number; F. 21. F 22. T 23. F 24. Sara’s call lasted 7 min. 25. The cost of Paulo’s long-distance call is $2.78. 26. No conclusion; the number and length of calls are unknown. 27. yes 28. no; possible answer: x = 2 29. no; possible answer: a seg. with endpoints (3, 7) and (-5, -1) 30. yes 31. comp. 32. positive 33. greater than 50 mi/h 34. 4s 35. m ___ -5 + 3 = -4.5 (Given); m ___ -5 = -7.5 (Subtr. Prop. of =); m = 37.5 (Mult. Prop. of =) 36. -47 = 3x - 59 (Given); 12 = 3x (Add. Prop. of =); 4 = x (Div. Prop. of =) 37. Reflex. Prop. of = 38. Sym. Prop. of ≅ 39. Trans. Prop. of = 40. figure ABCD 41. m∠5 = ̶̶ ̶̶ m∠2 42. EF 43. I = Prt CD ≅ (Given) ; 4200 = P (0.06) (4) (Subst.); 4200 = P (0.24) (Simplify.); Selected Answers S91 S91 �� 17,500 = P (Div. Prop. of =) 44. 1. Given 2. Def. of comp.  3. Given 4. Def. of ≅  5. Subst. 6. Def. of comp.  45a. Given b. TU = UV c. SU + UV = SV d. Subst. 46. z = 22.5 47. x = 17 48. 49. It is given that ∠ADE and ∠DAE are comp. and ∠ADE and ∠BAC are comp. By the ≅ Comps. Thm., ∠DAE ≅ ∠BAC. By the Reflex. Prop. of ≅, ∠CAE ≅ ∠CAE. By the Common  Thm., ∠DAC ≅ ∠BAE. 50. w = 45; Vert.  Thm. 51. x = 45; ≅  supp. → rt.  Chapter 3 3-1 ̶̶ EJ ̶̶ BF and ̶̶ BF ⊥ ̶̶ BF ǁ ̶̶ DE are skew. Check It Out! 1–2. Possible answers given. 1a. 1b. 1c. BCD 2a. ∠1 and ∠3 2b. ∠2 and ∠7 2c. ∠1 and ∠8 2d. ∠2 and ∠3 3. transv. n; same-side int.  ̶̶ FJ 1d. plane FJH ǁ plane ̶̶ AB and ̶̶ AB and Exercises 1. alternate interior angles 3–9. Possible answers ̶̶̶ given. 3. DH are skew. 5. plane ABC ǁ plane EFG 7. ∠6 and ∠8 9. ∠2 and ∠3 11. transv. m; alt. ext.  13. transv. p, sameside int.  15–21. Possible answers given. 15. 17. plane ABC ǁ plane DEF 19. ∠1 and ∠8 21. ∠2 and ∠5 23. transv. q; alt. int.  25. transv. p; sameside int.  27. corr. 29. sameside int.  31. Possible answer: ̶̶ FG 33a. plane MNR ǁ plane and KLP; plane LMQ ǁ plane KNP; plane PQR ǁ plane KLM ̶̶ CF are skew. ̶̶ CD b. same-side int.  35–39. Possible answers given. 35. ∠5 and ∠8 37. ∠1 and ∠5 39. transv. n; alt. int.  41. The lines are skew. 45. G 47. F 49. transv. m: ∠1 and ∠3, ∠2 and ∠4, ∠5 and ∠7, ∠6 and ∠8; transv. n: ∠9 and ∠11, ∠10 and ∠12, ∠13 and ∠15, ∠14 and ∠16; transv. p: ∠1 and ∠9, ∠2 and ∠10, ∠5 and ∠13, ∠6 and ∠14; transv. q: ∠3 and ∠11, ∠4 and ∠12, ∠7 and ∠15, ∠8 and ∠16 51. transv. m: ∠1 and ∠8, ∠4 and ∠5; transv. n: ∠9 and ∠16, ∠12 and ∠13; transv. p: ∠1 and ∠14, ∠2 and ∠13; transv. q: ∠3 and ∠16, ∠4 and ∠15 53. corr.  55. -3; -7; -3; 9; 29 57. -8; -9; -8; -5; 0 59. C = 11.9 m; A = 11.3 m 2 61. Lin. Pair Thm. 3-2 Check It Out! 1. m∠QRS = 62° 2. m∠ABD = 60° 3. 55° and 60° Exercises 1. m∠JKL = 127° 3. m∠1 = 90° 5. x = 8; y = 9 7. m∠VYX = 100° 9. m∠EFG = 102° 11. m∠STU = 90° 13. 120°; Corr.  Post. 15. 60°; Same-Side Int.  Thm. 17. 60°; Lin. Pair Thm. 19. 120°; Vert.  Thm. 21. x = 4; Same-Side Int.  Thm.; m∠3 = 103°; m∠4 = 77° 23. x = 3; Corr.  Post.; m∠1 = m∠4 = 42° 25a. ∠1 ≅ ∠3 b. Corr.  Post. c. ∠1 ≅ ∠2 d. Trans. Prop. of ≅ 29a. same-side int.  b. By the Same-Side Int.  Thm., m∠QRT + m∠STR = 180°. m∠QRT = 25° + 90° = 115°, so m∠STR = 65°. 31. A 35. J 37. m∠1 = 75° 39. x = 4; y = 12 41. increase 43. m∠1 + m∠2 = 180° 45–47. Possible answers given. 45. ∠3 and ∠6 47. ∠3 and ∠5 3-3 Check It Out! 1a. ∠1 ≅ ∠3, so ℓ ǁ m by the Conv. of Corr.  Post. 1b. m∠7 = 77° and m∠5 = 77°, so ∠7 ≅ ∠5. ℓ ǁ m by the Conv. of Corr.  Post. 2a. ∠4 ≅ ∠8, so r ǁ s by the Conv. of Alt. Int.  Thm. 2b. m∠3 = 100° and m∠7 = 100°, so
∠3 ≅ ∠7. r ǁ s by the Conv. of Alt. Int.  Thm. 3. 1. ∠1 ≅ ∠4 (Given) 2. m∠1 = m∠4 (Def. ≅ ) 3. ∠3 and ∠4 are supp. (Given) 4. m∠3 + m∠4 = 180° (Def. supp. ) 5. m∠3 + m∠1 = 180° (Subst.) 6. m∠2 = m∠3 (Vert.  Thm.) 7. m∠2 + m∠1 = 180° (Subst.) 8. ℓ ǁ m (Conv. of Same-Side Int.  Thm.) 4. 4y - 2 = 4 (8) - 2 = 30°; 3y + 6 = 3 (8) + 6 = 30°; The  are ≅, so the oars are ǁ by the Conv. of Corr.  Post. Exercises 1. ∠4 ≅ ∠5, so p ǁ q by the Conv. of Corr.  Post. 3. m∠4 = 47°, and m∠5 = 47°, so ∠4 ≅ ∠5. p ǁ q by the Conv. of Corr.  Post. 5. ∠3 and ∠4 are supp., so r ǁ s by the Conv. of Same-Side Int.  Thm. 7. m∠4 = 61°, and m∠8 = 61°, so ∠4 ≅ ∠8. r ǁ s by the Conv. of Alt. Int.  Thm. 9. m∠2 = 132°, and m∠6 = 132°, so ∠2 ≅ ∠6. r ǁ s by the Conv. of Alt. Ext.  Thm. 11. m∠1 = 60°, and m∠2 = 60°, so ∠1 ≅ ∠2. By the Conv. of Alt. Int.  Thm., the landings are ǁ. 13. m∠4 = 54°, and m∠8 = 54°, so ∠4 ≅ ∠8. ℓ ǁ m by the Conv. of Corr.  Post. 15. m∠1 = 55°, and m∠5 = 55°, so ∠1 ≅ ∠5. ℓ ǁ m by the Conv. of Corr.  Post. 17. ∠2 ≅ ∠7, so n ǁ p by the Conv. of Alt. Ext.  Thm. 19. m∠1 = 105°, and m∠8 = 105°, so ∠1 ≅ ∠8. n ǁ p by the Conv. of Alt. Ext.  Thm. 21. m∠3 = 75°, and m∠5 = 105°. 75° + 105° = 180°, so ∠3 and ∠5 are supp. ℓ ǁ m by the Conv. of Same-Side Int.  Thm. 23. If x = 6, then m∠1 = 20° ̶̶ and m∠2 = 20°. So EK by the Conv. of Corr.  Post. 25. Conv. of Alt. Ext.  Thm. 27. Conv. of Corr.  Post. 29. Conv. of Same-Side Int.  Thm. 31. m ǁ n; Conv. of SameSide Int.  Thm. 33. m ǁ n; Conv. of Alt. Ext.  Thm. 35. ℓ ǁ n; Conv. of Same-Side Int.  Thm. 37a. ∠URT ; m∠URT = m∠URS + m∠SRT by the ∠ Add. Post. It is given that m∠SRT = 25° and m∠URS = 90°, so m∠URT = 25° + 90° = 115°. b. It is given that m∠SUR = 65°. From part a, m∠URT = 115°. 65° + 115° = 180°, so   SU ǁ   RT by the Conv. of Same- ̶̶ DJ ǁ S92 S92 Selected Answers �� Side Int.  Thm. 39. It is given that ∠1 and ∠2 are supp., so m∠1 + m∠2 = 180°. By the Lin. Pair Thm., m∠2 + m∠3 = 180°. By the Trans. Prop. of =, m∠1 + m∠2 = m∠2 + m∠3. By the Subtr. Prop. of =, m∠1 = m∠3. By the Conv. of Corr.  Post., ℓ ǁ m. 41. The Reflex. Prop. is not true for ǁ lines, because a line is not ǁ to itself. The Sym. Prop. is true, because if ℓ ǁ m, then ℓ and m are coplanar and do not intersect. So m ǁ ℓ . The Trans. Prop. is not true for ǁ lines, because if ℓ ǁ m and m ǁ n, then ℓ and n could be the same line. So they would not be ǁ. 43. C 45. 15 47. No lines can be proven ǁ. 49. q ǁ r by the Conv. of Alt. Int.  Thm. 51. s ǁ t by the Conv. of Alt. Ext.  Thm. 53. No lines can be proven ǁ. 55. By the Vert.  Thm., ∠6 ≅ ∠3, so m∠6 = m∠3. It is given that m∠2 + m∠3 = 180°. By subst., m∠2 + m∠6 = 180°. By the Conv. of Same-Side Int.  Thm., ℓ ǁ m. 57. a = b - c ̶̶ 59. y = - 3 __ 2 x + 3 63. BC ̶̶ 65. AD ̶̶ AB ⊥ ̶̶ AD ǁ 3-4 Check It Out! 1a. 2. 1. ∠EHF ≅ ∠HFG (Given) ̶̶ AB 1b. x < 17 2.   EH ǁ   FG (Conv. of Alt. Int.  Thm.) 3.   FG ⊥   GH (Given) 4.   EH ⊥   GH (⊥ Transv. Thm.) 3. The shoreline and the path of the swimmer should both be ⊥ to the current, so they should be ǁ to each other. ̶̶ Exercises 1. AB and   CD are ̶̶ ̶̶ BC are ≅. 3. x >-5 ⊥. AC and 5. The service lines are coplanar lines that are ⊥ to the same line (the center line), so they must be ǁ to each other. 7. x < 11 9. Both the frets are lines that are ⊥ to the same line (the string), so the frets must be ǁ to each other. 11. x > 8 __ 3 13. x = 6, y = 15 15. x = 60, y = 60 17. no 19. no 21. yes 23a. It is ̶̶ ̶̶ RS , PQ and given that ̶̶ ̶̶ RS by the ⊥ Transv. Thm. QR ⊥ so ̶̶ ̶̶ QR QR . Since It is given that ̶̶ ̶̶ RS by the ⊥ Transv. ⊥ PS ⊥ ̶̶ Thm. b. It is given that QR ̶̶ QR ⊥ ̶̶ PQ ǁ ̶̶ PS ǁ ̶̶ PS ǁ ̶̶ RS , ̶̶ PQ ⊥ ̶̶ PQ . So ̶̶ PS by the ̶̶ QR ⊥ and ⊥Transv. Thm. 25. Possible answer: 1.6 cm 31. C 33. D 35a. n ⊥ p b. AB; AB; the shortest distance from a point to a line is measured along a perpendicular segment. c. The distance between two parallel lines is the length of a segment that is perpendicular to both lines and has one endpoint on each line. 39. 30 games 41. 25° 43. Conv. of Alt. Ext. ∠ Thm. 45. Conv. of Same-Side Int. ∠ Thm. 3-5 Check It Out! 1. m = 2 2. 390 m 3a. ⊥ 3b. neither 3c. ǁ Exercises 1. rise; run 3. m = - 5 __ 9 5. m = 5 __ 2 7. ǁ 9. neither 11. m = 0 13. m = - 7 __ 3 15. ǁ 17. ⊥ 19. m = 1 __ 10 21. m = 1 __ 2 23. m <-1 25a. 66 ft/s b. 45 mi/h 27. F 29.   JK is a vert. line. 33. Possible answer: x = 1, y = -6 35. x-int.: 0.25; y-int.: 1 37. 1. ∠1 is supp. to ∠3. (Given) 2. ∠1 and ∠2 are supp. (Lin. Pair Thm.) 3. ∠2 ≅ ∠3 (≅ Supps. Thm.) 39. T: Corr.  Post. 3-6 Check It Out! 1a. y = 6 1b. y - 2 = 0 2a. 2b. 2c. 3. parallel 4. The lines would be ǁ. Exercises 1. The slope-intercept form of an equation is solved for y. The x term is first, and the constant term is second. 3. y - 2 = 3 __ 4 (x + 4) 7. 5. 9. intersect 11. ǁ 13. y + 2 = 2x 15. y + 4 = 2 __ 3 (x - 6) 17. 19. intersect 21. coincide 23. $1000 per week 33. no 35. yes 37. ǁ line: y = 3x - 3; ⊥ line: y = - 1 __ 3 x + 11 __ 3 39. ǁ line: y = - 4 __ 3 x + 10 __ 3 ; ⊥ line: y = 3 __ 4 x - 5 41. yes; ∠B 43. no 45. For 4 toppings, both pizzas will cost $14. 47. y = - 1 __ 2 x + 17 __ 2 49. y = 2x + 7 __ 2 51. y = 2x - 1; (2, 3) ; √  5 units 53a–b. b. the time when the car has traveled 300 ft c. Possible answer: 3.5 s 59. J 61. J 63. Possible answer: y = - 8 __ 15 x + 8 65. no 67. 6 69. (1, 0) 71. m = 2 __ 5 73. m = - 4 __ 3 SGR ̶̶ BC are ̶̶ DE ̶̶ DE and ̶̶ AD ⊥ 1. alternate interior angles 2. skew lines 3. transversal 4. point-slope form 5. rise; run 6. Possible answer: ̶̶ ̶̶ skew. 7. DE 8. AB ǁ 9. plane ABC ǁ plane DEF 10. ℓ; alt. int.  11. n; corr.  12. ℓ; sameside int.  13. m; alt. ext.  14. m∠WYZ = 90° 15. m∠KLM = 100° 16. m∠DEF = 79° 17. m∠QRS = 76° 18. ∠4 ≅ ∠6, so c ǁ d by the Conv. of Alt. Int.  Thm. 19. m∠1 = 107° and m∠5 = 107°, so ∠1 ≅ ∠5. c ǁ d by the Conv. of Corr.  Post. 20. m∠6 = 66°, m∠3 =114°, and 66° + 114° ≠ 180°, so ∠6 and ∠3 are supp. c ǁ d by the Conv. of Same-Side Int.  Thm. 21. m∠1 ≠ 99°, and m∠7 = 99°, so ∠1 ≅ ∠7. c ǁ d by the Conv. of Alt. Ext.  Thm. 22. ̶̶̶ KM 23. x < 13 ̶̶ AD ⊥ ̶̶ BC , ̶̶ BC 24. 1. ̶̶ AB , ̶̶ DC ⊥ ̶̶ BC (⊥ Transv. Thm.); ̶̶ CD (2 lines ⊥ to same ̶̶ AD ǁ (Given); ̶̶ 2. AB ⊥ ̶̶ 3. AB ǁ line → 2 lines ǁ) 25. m = - 1 __ 7 26. m = 5 __ 3 27. neither 28. ǁ 29. ⊥ 30. y = - 4 __ 9 x + 11 __ 3 31. y = 2 __ 3 x - 2 Selected Answers S93 S93 �� 32. y - 0 = 2 (x - 1) 33. ǁ 34. intersect 35. coincide 55. scalene 57.△ACD is equil. 4-3 Chapter 4 4-1 Check It Out! 1. equiangular 2. scalene 3. 17; 17; 17 4a. 4 4b. 3 Exercises 1. An equilateral △ has 3 ≅ sides. 3. rt. 5. obtuse 7. scalene 9. 36; 36; 36 11. 6 13. obtuse 15. equil. 17. scalene 19. 8.6; 8.6 21. 18 ft; 18 ft; 24 ft 23. 25. 27. not possible 29. 35 in. 31. isosc. rt. 33a. 173 ft; 87 ft b. scalene 35. S 37. A 41. D 43. D 45. It is an isosc. △ since 2 sides of the △ have length a. It is a rt. △ since 2 sides of the △ lie on the coord. axes and form a rt. ∠. 47. y = -3 49. y = x 2 51. y = x 2 53. T 55. ǁ 57. coincides 4-2 Check It Out! 1. 32° 2a. 26.3° 2b. (90 - x) ° 2c. 41 3 __ 5 ° 3. 141° 4. 32°; 32° Exercises 3. auxiliary lines 5. 36°; 80°; 64° 7. (90 - y) ° 9. 28° 11. 52°; 63° 13. 89°; 89° 15. 84° 17. (90 - 2x) ° 19. 162° 21. 48°; 48° 23. 15°; 60°; 105° 29. 36° 31. 48° 33. 120°; 360° 35. 18° 37. The ext.  at the same vertex of a △ are vert. . Since vert.  are ≅, the 2 ext.  have the same measure. 41. C 43. D 45. y = 7 or y = -7 47. Since an ext. ∠ is = to a sum of 2 remote int. , it must be greater than either ∠. Therefore it cannot be ≅ to a remote int. ∠. 49. 38° 51. 53. 6 in.; Seg. Add. Post. S94 S94 Selected Answers Check It Out! 1. ∠L ≅ ∠E, ∠M ≅ ̶̶ ∠F, ∠N ≅ ∠G, ∠P ≅ ∠H, EF , ̶̶ ̶̶̶ MN ≅ NP ≅ 2a. 4 2b. 37° 3. 1. ∠A ≅ ∠D (Given) ̶̶̶ LM ≅ ̶̶ EH ̶̶ LP ≅ ̶̶̶ GH , ̶̶ FG , ̶̶ DE (Given) ̶̶ AB ≅ ̶̶ AD bisects ̶̶ AD . (Given) ̶̶ ̶̶ AC ≅ EC , 2. ∠BCA ≅ ∠ECD (Vert.  are ≅.) 3. ∠ABC ≅ ∠DEC (Third  Thm.) 4. 5. bisects ̶̶ 6. BC ≅ bisector) 7. △ABC ≅ △DEC (Def. of ≅ ) ̶̶ DC (Def. of ̶̶ BE , and ̶̶ BE 4. 1. ̶̶ JK ǁ ̶̶̶ ML (Given) ̶̶̶ ML (Given) 2. ∠KJN ≅ ∠MLN, ∠JKN ≅ ∠LMN (Alt. Int.  Thm.) 3. ∠JNK ≅ ∠LNM (Vert.  Thm.) 4. 5. bisects ̶̶ 6. JN ≅ bisector) 7. △JKN ≅ △MLN (Def. of ≅ ) ̶̶ JK ≅ ̶̶̶ MK bisects ̶̶̶ MK . (Given) ̶̶̶ ̶̶ MN ≅ LN , ̶̶ KN (Def. of ̶̶ JL , and ̶̶ JL ̶̶ BE ; ̶̶ CE , ̶̶ PN ≅ Exercises 1. You find the  and sides that are in the same, or ̶̶̶ matching, places in the 2 . 3. LM 5. ∠M 7. ∠R 9. KL = 9 11a. Given b. Alt. Int.  Thm. c. Given ̶̶ ̶̶ d. Given e. DE ≅ AE ≅ f. Vert.  Thm. g. Def. of ≅  ̶̶̶ 13. LM 15. ∠N 17. m∠C = 31° 19a. Given b. Given c. ∠NMP ≅ ∠RMP d. ∠NPM ≅ ∠RPM e. Given ̶̶ f. PR g. Given h. Reflex. Prop. of ≅ 21. △GSR ≅ △KPH; △SGR ≅ △PHK; △RGS ≅ △HKP 23. x = 30; AB = 50 25. x = 2; BC = 17 29. solution A 31. B 33. D 35. x = 5.5; yes; UV = WV= 41.5, and UT = WT = 33. TV = TV by the Reflex. Prop. of =. It is given that ∠VWT ≅ ∠VUT and ∠WTV ≅ ∠UTV. ∠WVT ≅ ∠ UVT by the Third  Thm. Thus △TUV ≅ △TWV by the def. of ≅ . 39. 1 __ 9 41. rt. 43. 72° 45. 146° 4-4 ̶̶ AB ̶̶ BC ≅ ̶̶ AC ≅ Check It Out! 1. It is given that ̶̶ ̶̶ DA . By the Reflex. ≅ CD and ̶̶ Prop. of ≅, AC . So △ABC ≅ △CDA by SSS. 2. It is given that ̶̶ BD and ∠ABC ≅ ∠DBC. By the ≅ ̶̶ BA ̶̶ BC ≅ ̶̶ BC . So ̶̶ DB ≅ ̶̶ DB by ̶̶ DC by def. of ≅. Reflex. Prop. of ≅, △ABC ≅ △DBC by SAS. 3. DA = ̶̶ DA = DC = 13, so m∠ADB = m∠CDB = 32°, so ∠ADB ≅ ∠CDB by def. of ≅. the Reflex. Prop. of ≅. Therefore △ADB ≅ △CDB by SAS. ̶̶ 4. 1. QS (Given) 2. 3. ∠RQP ≅ ∠SQP (Def. of bisector) ̶̶ ̶̶ 4. QP (Reflex. Prop. of ≅) QP ≅ 5. △RQP ≅ △SQP (SAS Steps 1, 3, 4) ̶̶ QR ≅  QP bisects ∠RQS. (Given) ̶̶ GL . ̶̶ KJ ≅ ̶̶ NP ≅ ̶̶ GK ≅ ̶̶ QP ≅ ̶̶̶ NQ ≅ ̶̶ CB c. ̶̶ LJ and ̶̶̶ MN ≅ ̶̶̶ MP by the Reflex. Prop. of ≅. ̶̶ DC d. Def. of ⊥ e.
Rt. ∠ ≅ Thm. ̶̶ ̶̶ AB g. SAS Steps 2, 5, 6 AB ≅ Exercises 1. ∠T 3. It is given ̶̶ ̶̶̶ QP . MQ and that ̶̶̶ MP ≅ Thus △MNP ≅ △MQP by SSS. 5. When x = 4, HI = GH = 3, and ̶̶ ̶̶ IJ = GJ = 5. HJ by the Reflex. HJ ≅ Prop. of ≅. Therefore △GHJ ≅ △IHJ by SSS. 7a. Given b. ∠JKL ≅ ∠MLK c. Reflex. Prop. of ≅ d. SAS Steps 1, 2, 3 9. It is given ̶̶ ̶̶ GJ GJ ≅ that by the Reflex. Prop. of ≅. So △GJK ≅ △GJL by SSS. 11. When y = 3, NQ = NM = 3, and QP = MP = ̶̶̶ 4. So by the def. of ≅, NM ̶̶̶ MP . m∠M = m∠Q = 90°, and so ∠M ≅ ∠Q by the def. of ≅. Thus △MNP ≅ △QNP by SAS. ̶̶ ̶̶ 13a. Given b. AB DB ≅ ⊥ f. 15. SAS 17. neither 19. QS = TV = √  5 . SR = VU = 4. QR = TU = √  13 . The  are ≅ by SSS. 21a. Given b. Def. of ≅ c. m∠WVY = m∠ZYV d. Def. of ≅ e. Given ̶̶ f. VY g. SAS Steps 6, 5, 7 25. Measure the lengths of the logs. If the lengths of the logs in 1 wing deflector match the lengths of the logs in the other wing deflector, the  will be ≅ by SAS or SSS. 27. Yes; if each side is ≅ to the corr. side of the second △, they can be in any order. 29. G 31. J 35. x = 27; FK = FH = 171, so of ≅. ∠KFJ ≅ ∠HFJ by the def. of ∠ bisector. of ≅. So △FJK ≅ △FJH by SAS. 37. a < 4 ̶̶ FH by the def ̶̶ FJ by the Reflex. Prop. ̶̶ VY ≅ ̶̶ FK ≅ ̶̶ FJ = 43. 34° �� 4-5 ̶̶ LN ≅ Check It Out! 1. Yes; the △ is uniquely determined by AAS. 2. By the Alt. Int.  Thm., ∠KLN ̶̶ ≅ ∠MNL. LN by the Reflex. Prop. of ≅. No other congruence relationships can be determined, so ASA cannot be applied. 3. Given: ≅ ∠M. Prove: △JKL ≅ △JML ̶̶ JL bisects ∠KLM, and ∠K ̶̶ DF , ̶̶ BC ≅ ̶̶ ≅ EF , and ∠C and ∠F are rt. . ∠C ≅ ∠F by the Rt. ∠ ≅ Thm. Thus △ABC ≅ △DEF by SAS. 27. J 29. G 31. Yes; the sum of the ∠ measures in each △ must be 180°, which makes it possible to solve for x and y. The value of x is 15, and the value of y is 12. Each △ has  measuring 82°, 68°, and ̶̶ VU by the Reflex. Prop. 30°. of ≅. So △VSU ≅ △VTU by ASA or AAS. 35. 2; -6 37. 1; 5 39. 36.9° ̶̶ VU ≅ 4-6 Check It Out! 1. 41 ft 2. ̶̶ CB ̶̶ AC ≅ ̶̶ 4. Yes; it is given that DB . ̶̶ ≅ CB by the Reflex. Prop. of ≅. Since ∠ABC and ∠DCB are rt. , △ABC and △DCB are rt. , △ABC ≅ △DCB by HL. ̶̶ BC Exercises 1. The included side is enclosed between ∠ABC and ∠ACB. 3. Yes, the △ is determined by AAS. 5. No; you need to know that a pair of corr. sides are ≅. 7. Yes; it is given that ∠D and ∠B ̶̶ ̶̶ BC . △ABC and are rt.  and AD ≅ ̶̶ ̶̶ CA by AC ≅ △CDA are rt.  by def. the Reflex. Prop. of ≅. So △ABC ≅ △CDA by HL. 9. ̶̶ BE ̶̶ AE ≅ ̶̶ CE , and 11. No; you need to know that ∠MKJ ≅ ∠MKL. 13a. ∠A ≅ ∠D b. Given c. ∠C ≅ ∠F d. AAS 15. Yes; E is a mdpt. So by def., ̶̶ DE . ∠A and ∠D ≅ are ≅ by the Rt. ∠ ≅ Thm. By def. △ABE and △DCE are rt. . So △ABE ≅ △DCE by HL. 17. △FEG ≅ △QSR; rotation 19a. No; there is not enough information given to use any of the congruence theorems. b. HL 21. It is given that △ABC and △DEF are rt.. ̶̶ AC 3. 1. J is the mdpt. of ̶̶̶ KM and ̶̶ NJ ≅ ̶̶ LJ (Def. of ̶̶ NL . (Given) ̶̶ ̶̶ 2. MJ , KJ ≅ mdpt.) 3. ∠KJL ≅ ∠MJN (Vert.  Thm.) 4. △KJL ≅ △MJN (SAS Steps 2, 3) 5. ∠LKJ ≅ ∠NMJ (CPCTC) ̶̶̶ ̶̶ 6. MN (Conv. of Alt. KL ǁ Int.  Thm.) 4. RJ = JL = √  5 , RS = JK = √  10 , and ST = KL = √  17 . So △JKL ≅ △RST by SSS. ∠JKL ≅ ∠RST by CPCTC. Exercises 1. corr.  and corr. sides. 3a. Def. of ⊥ b. Rt. ∠ ≅ Thm. c. Reflex. Prop. of ≅ d. Def. of mdpt. e. △RXS ≅ △RXT f. CPCTC 5. EF = JK = 2 and EG = FG = JL = KL = √  10 . So △EFG ≅ △JKL by SSS. ∠EFG ≅ ∠JKL by CPCTC. 7. 420 ft ̶̶̶ 9. 1. WX ≅ ̶̶ 2. ZX ≅ 3. △WXZ ≅ △YZX (SSS) 4. ∠W ≅ ∠Y (CPCTC) ̶̶ ̶̶̶ XY ≅ ZW (Given) ̶̶ ZX (Reflex. Prop. of ≅) ̶̶ YZ ≅ 11. 1. ̶̶̶ LM bisects ∠JLK. (Given) 3, 2, 4) ̶̶̶ ̶̶ 6. KM (CPCTC) JM ≅ 7. M is the mdpt. of of mdpt.) ̶̶ JK . (Def. 13. AB = DE = √  13 , BC ≠ EF ± 5, and AC = DF = √  18 ≠ 3 √  2 . So △ABC ≅ △DEF by SSS. ∠BAC ≅ ∠EDF by CPCTC. 15. 1. E is the mdpt. of ̶̶ AC and ̶̶ BD . ̶̶ CE ; ̶̶ BE ≅ ̶̶ DE (Def. (Given) ̶̶ 2. AE ≅ of mdpt.) 3. ∠AEB ≅ ∠CED (Vert.  Thm.) 4. △AEB ≅ △CED (SAS Steps 2, 3) 5. ∠A ≅ ∠C (CPCTC) ̶̶ ̶̶ 6. CD (Conv. of Alt. Int. AB ǁ  Thm.) 17. 14 25. G 27. G 29. Any diag. on any face of the cube is the hyp. of a rt. △ whose legs are edges of the cube. Any 2 of these  are ≅ by SAS. Therefore any 2 diags. are ≅ by CPCTC. 33. 94 35. reflection across the x-axis 37. Yes; it is given that ∠B ≅ ∠D and the Vert. ∠ Thm., ∠BCA ≅ ∠DCE. Therefore △ABC ≅ △EDC by ASA. ̶̶ DC . By ̶̶ BC ≅ 4-7 Check It Out! 1. Possible answer: , 6 + 0 ____ 2 2. △ABC is a rt. △ with height AB and base BC. The area of △ABC is 1 __ 2 (4) (6) = 12 square units. By the Mdpt. Formula, the coords. of D ̶̶ are ( 0 + 4 ) = (2, 3) . With ____ 2 AB as the base of △ADB, the x-coord. of D gives the height of △ADB. The area of △ADB = 1 __ 2 bh = 1 __ 2 (6) (2) = 6 square units. Since 6 = 1 __ 2 (12) , the area of △ADB is the area of △ABC. 3. Possible answer: 2. ∠JLM ≅ ∠KLM (Def. of ∠ bisector) ̶̶ ̶̶ 3. JL ≅ KL (Given) ̶̶̶ 4. LM ≅ 5. △JLM ≅ △KLM (SAS Steps ̶̶̶ LM (Reflex. Prop. of ≅) Selected Answers S95 S95 �� 4. △ABC is a rt. △ with height 2j and base 2n. The area of △ABC = 1 __ 2 bh = 1 __ 2 (2n) (2j) = 2nj square units. By the Mdpt. Formula, the coords. of D are (n, j) . The base of △ABD is 2j units and the height is n units. So the area of △ADB = 1 __ 2 bh = 1 __ 2 (2j) (n) = nj square units. Since nj = 1 __ 2 (2nj) , the area of △ADB is 1 __ 2 the area of △ABC. 2 2 - y 1 + y 2 _____ 2 y 1 + y 2 _____ 2  _____ 2  2 = √ + ( ___ 2 ) ( ___ _____ - 2 = √  1 __ __ __ 2 √  ( . So AM = 1 __ 2 AB. 27. B 29. D 31. (a + c, b) 35. x = -2.5 or x = 0.25 37. x = 2 or x = -1.67 39. 22 Exercises 7. 4-8 Check It Out! 1. 4.2 × 10 13 ; since it is 6 months between September and March, the ∠ measures will be the same between Earth and the star. By the Conv. of the Isosc. △ Thm., the  created are isosc. and the dist. is the same. 2a. 66° 2b. 48° 3. 10 4. By the Mdpt. Formula, the coords. of X are (-a, b) , the coords. of Y are (a, b) , and the coords. of Z are (0, 0) . By the Dist. Formula, XZ = YZ = ̶̶ √  XZ ≅ isosc. ̶̶ YZ and △XYZ is a 2 + b 2 . So ̶̶ KJ and ̶̶ KL ; ̶̶ BC , and ̶̶ AB ≅ ̶̶ AC . By the Mdpt. Exercises 1. legs: ̶̶ JL ; base : ∠J and ∠L 3. 118° base: 5. 27° 7. y = 5 9. 20 11. It is given that △ABC is rt. isosc., X is the mdpt. of Formula, the coords. of X are(a, a). By the Dist. Formula, AX = BX = a √  2 . So △AXB is isosc. by def. of an isosc. △. 13. 69° 15. 130° or 172° 17. z = 92 19. 26 21. It is given that △ABC is isosc., P is the mdpt. of mdpt. of the coords. of P are (a, b) and the coords. of Q are (3a, b) . By the Dist. Formula, PC = QB = √  9 a 2 + b 2 , ̶̶ QB by the def of ≅ segs. so 23. S 25. N 27a. 38° b. m∠PQR = m∠ PRQ = 53° 29. m∠1 = 127°; m∠2 = 26.5°; m∠3 = 53° 33. 20 39. 1. △ABC ≅ △CBA (Given) ̶̶ AB ≅ ̶̶ AB , and Q is the ̶̶ AC . By the Mdpt. Formula, ̶̶ PC ≅ ̶̶ AC , ̶̶ AB ≅ ̶̶ 2. CB (CPCTC) 3. △ABC is isosceles (Def. of Isosc) 43. H 47. (2a, 0) , (0, 2b) , or any pt. ̶̶ AB 49. x = 3 on the ⊥ bisector of or x = 1 51. m = -3 53. m = 2 __ 3 By the Mdpt. Formula, the coords. of A are (0, a) and the coords. of B are (b, 0) . By the Dist. Formula, PQ = √  (0 - 2b) 2 + (2a) 2 = √  (-2b) 2 + (2a) 2 = √  4b 2 + 4a 2 = 2 √  b 2 + a 2 units. AB = √  (0 - b) 2 + (a - 0) 2 = √  (-b) 2 + a 2 = √  So AB = 1 __ 2 PQ. 13. b 2 + a 2 units. By the Mdpt. Formula, the coords. of E are (0, a) and the coords. of F are (2c, a) . By the Dist. Formula, AD = √  (2c - 0) 2 + (2a - 2a) 2 = √  (2c) 2 = 2c units. Similarly, EF = √  (2c - 0) 2 + (a - a) 2 = √  (2c) 2 = 2c units. So EF = AD. 15a. b. 8.5 mi 17. 2s + 2t units; st square units 19. (p, 0) 21. AB ≈ 128 nautical miles; AP = BP ≈ 64 nautical miles; so P is the mdpt. ̶̶ AB . 23. By the Dist. Formula, AB ( and AM of = √  S96 S96 Selected Answers SGR ̶̶ DB ≅ ̶̶ XZ 9. ∠Q 10. 25 11. 7 ̶̶ ̶̶ AE (Given) DE , ̶̶ DA (Reflex. Prop. of ≅) 1. isosceles triangle 2. corresponding angles 3. included side 4. equiangular; equil. 5. obtuse; scalene 6. 60° 7. 66.5° 8. ̶̶ 12. 1. AB ≅ ̶̶ 2. DA ≅ 3. △ADB ≅ △DAE (SSS Steps 1, 2) ̶̶ GJ bisects ̶̶ GJ . (Given) bisects ̶̶ ̶̶ ̶̶ 2. FK ≅ GK ≅ JK , seg. Bisect) 3. ∠GKF ≅ ∠JKH (Vert.  Thm.) 4. △FGK ≅ △HJK (SAS Steps 2, 3) ̶̶ HK (Def. of ̶̶ FH , and 13. 1. ̶̶ FH ̶̶ BC ≅ ̶̶ YZ ; ∠C ≅ ̶̶ XZ ; so △ABC ≅ △XYZ 14. BC = (-6) 2 + 36 = 72; YZ = 2 (-6) 2 = 72; ̶̶ AC ≅ ∠Z; by SAS. 15. PQ = 25 - 1 = 24; QR = 25; PR = 25 2 - (25 - 1) 2 - 42 = ̶̶ ̶̶ ̶̶̶ PR ; so QR ; MN ≅ 7; △LMN ≅ △PQR by SSS. 16. 1. C is the mdpt. of ̶̶ AG . (Given) ̶̶̶ LM ≅ ̶̶ LN ≅ ̶̶ PQ ; ̶̶ GC ≅ ̶̶ HA ǁ ̶̶ AC (Def. of mdpt.) 2. ̶̶ 3. GB (Given) 4. ∠HAC ≅ ∠BGC (Alt. Int.  Thm.) 5. ∠HCA ≅ ∠BCG (Vert.  Thm.) 6. △HAC ≅ △BGC (ASA Steps 4, 2, 5) ̶̶ ̶̶̶ XZ , WX ⊥ ̶̶ ̶̶ ZX (Given) YZ ⊥ 2. ∠WXZ and ∠YZX are rt. . (Def. of ⊥) 3. △WZX and △YXZ are rt. . (Def. of rt. △) 4. 5. 6. △WZX ≅ △YXZ (HL Steps 5, 4) ̶̶ XZ (Reflex. Prop. of ≅) ̶̶ YX (Given) ̶̶ XZ ≅ ̶̶̶ WZ ≅ 17. 1. ̶̶ RT ≅ 18. 1. ∠S and ∠V are rt. . (Given) 2. ∠S ≅ ∠V (Rt. ∠ ≅ Thm.) 3. RT = UW (Given) ̶̶̶ 4. UW (Def. of ≅) 5. m∠T = m∠W (Given) 6. ∠T ≅ ∠W (Def. of ≅) 7. △RST ≅ △UVW (AAS Steps 2, 6, 4) 19. 1. M is the mdpt. of ̶̶ BD . (Given) ̶̶̶ MB ≅ ̶̶ BC ≅ ̶̶̶ CM ≅ ̶̶̶ DM (Def. of mdpt.) ̶̶ DC (Given) ̶̶̶ CM (Reflex. Prop. of ≅) 2. 3. 4. 5. △CBM ≅ △CDM (SSS Steps 2, 3, 4) 6. ∠1 ≅ ∠2 (CPCTC) �� ̶̶ PQ ≅ ̶̶ PS ≅ ̶̶ QS ≅ ̶̶ RQ (Given) ̶̶ RS (Given) ̶̶ QS (Reflex. Prop. of ≅) 20. 1. 2. 3. 4. △PQS ≅ △RQS (SSS Steps 1, 2, 3) 5. ∠PQS ≅ ∠RQS (CPCTC) 6. bisect) ̶̶ QS bisects ∠PQR. (Def. of ̶̶ GJ , and L is ̶̶ KL (Def. of ≅) 21. 1. H is mdpt. of line ̶̶̶ MK . (Given) mdpt. of 2. GH = JH, ML = KL (Def. of
mdpt.) ̶̶̶ ̶̶ ̶̶̶ 3. ML ≅ GH ≅ JH , ̶̶ ̶̶̶ KM (Given) 4. GJ ≅ ̶̶̶ ̶̶ KL (Div. Prop. of ≅) 5. GH ≅ ̶̶̶ ̶̶ 6. KJ , ∠G ≅ ∠K (Given) GM ≅ 7. △GMH ≅ △KJL (SAS Steps 5, 6) 8. ∠GMH ≅ ∠KJL (CPCTC) 22. (0, 0) , (r, 0) , (0, s) 23. (0, 0) , (2p, 0) , (2p, p) , (0, p) 24. (0, 0) , (8m, 0) , (8m, 8m) , (0, 8m) 25. Use coords. A (0, 0) , B (2a, 0) , C (2a, 2b) , and D (0, 2b) . Then, by the Mdpt. Formula, E (a, 0) , F (2a, b) , G (a, 2b) , and H (0, b) . By the Dist. Formula, EF = √  (2a - a) 2 + (b - 0) 2 = √  a 2 + b 2 , and GH = √  (0 - a) 2 + (b - 2b) 2 = √  a 2 + b 2 . ̶̶̶ ̶̶ GH by the def. of ≅. EF ≅ So 26. Use coords. P (0, 2b) , Q (0, 0) , and R (2a, 0) . Then, by the Mdpt. Formula, M (a, b) . By the Dist. Formula, QM = √  PM = √  = √  a 2 + b 2 , and RM = √  So QM = PM = RM. By def. M is equidistant from the vertices of △PQR 27. To be a rt. △, the side lengths must have lengths such that a 2 + b 2 = c 2 . √  (3 - 3) 2 + (5 - 2) 2 = 3, √  (3 - 2) 2 + (2 - 5) 2 = √  10 , and √  3 2 + 1 2 = ( √  triangle is a rt. △. 28. x = -5 29. RS = 13.5 30. 70 units (a - 0) 2 + (b - 0) 2 = √  a 2 + b 2 , (a - 0) 2 + (b - 2b) 2 (2 - 3) 2 + (5 - 5) 2 = 1. Since 10 2 ) , or 9 + 1 = 10, the (2a - a) 2 + (0 - b) 2 = √  a 2 + b 2 . Chapter 5 5-1  PQ , ̶̶ SQ ⊥  PS bisects ∠QPR.  PR (Given) Check It Out! 1a. 14.6 1b. 10.4 2a. 3.05 2b. 126° 3.  QS bisects ∠PQR. 4. y + 1 = - 2 __ 3 (x - 3) Exercises 1. perpendicular bisector 3. 25.9 5. 21.9 7. 38° 9. y - 1 = x + 2 11. y - 2 = 4 __ 3 (x + 3) 13. 26.5 15. 1.3 17. 54° 19. y + 3 = - 1 __ 2 (x + 2) 21. y + 3 = 5 __ 2 (x - 2) 23. 38 25. 38 27. 24 29. Possible answer: C (3, 2) 31. 1. ̶̶ SR ⊥ 2. ∠QPS ≅ ∠RPS (Def. of ∠ bisector) 3. ∠SQP and ∠SRP are rt. . (Def. of ⊥) 4. ∠SQP ≅ ∠SRP (Rt. ∠ ≅ Thm.) 5. 6. △PQS ≅ △PRS (AAS) ̶̶ 7. SR (CPCTC) 8. SQ = SR (Def. of ≅ segs.) 33a. y = - 3 __ 4 x + 2 b. 2 c. 6.4 mi 35. D 39. the lines y = x and y = -x 43. parallel 45. perpendicular 47. y = - 1 __ 2 x - 10 ̶̶ PS (Reflex. Prop. of ≅) ̶̶ SQ ≅ ̶̶ PS ≅ 5-2 Check It Out! 1a. 14.5 1b. 18.6 1c. 19.9 2. (4, -4.5) 3a. 19.2 3b. 52° 4. By the Incenter Thm., the incenter of a △ is equidistant from the sides of the △. Draw the △ formed by the streets and draw the ∠ bisectors to find the incenter, point M. The city should place the monument at point M. Exercises 1. They do not intersect at a single point. 3. 5.64 5. 3.95 7. (2, 6) 9. 42.1 11. The largest possible 〇 in the int. of the △ is its inscribed 〇, and the center of the inscribed 〇 is the incenter. Draw the △ and its ∠ bisectors. Center the 〇 at E, the pt. of concurrency of the ∠ bisectors. 13. 63.9 15. 63.9 17. (-1.5, 9.5) 19. 55° 23. perpendicular bisector 25. angle bisector 27. neither 29. S 31. N 33. (4, 3) 35a. ∠ Bisector Thm. b. the bisector of ∠B c. PX = PZ 37a. (4, - 7 __ 6 ) b. outside c. 4.2 mi 41. F 45. t = 4 47. y = 120 49. 35° 51. yes 53. no 5-3 Check It Out! 1a. 21 1b. 5.4 2. 3; 4; possible answer: the x-coordinate of the centroid is the average of the x-coordinates of the vertices of the △, and the y-coordinate of the centroid is the average of the y-coordinates of the vertices of the △. 3. Possible answer: An equation of the altitude ̶̶ JK is y = - 1 __ 2 x + 3. It is true that to 4 = - 1 __ 2 (-2) + 3, so (-2, 4) is a solution of this equation. Therefore this altitude passes through the orthocenter. Exercises 1. centroid 3. 136 5. 156 7. (4, 2) 9. (2, -3) 11. (-1, 2) 13. 7.2 15. 5.8 17. (0, -2) 19. (-2, 9) 21. 12 23. 5 25. 36 units 27. (10, -2) 29. 54 31. 48 33. Possible answer: ⊥ bisector of the base; bisector of the vertex ∠; median to the base; altitude to the base 35. A 37. A 41. D 43. D ̶̶ RS = c __ 45a. slope of ; slope of b ̶̶ ̶̶ ST = c ____ RT = 0 ; slope of b - a ̶̶ RS , slope of ℓ = - b __ c . b. Since ℓ ⊥ ̶̶ ____ ST , slope of m = - b - a Since m ⊥ c ̶̶ ____ = a - b RT , n is a vertical c line, and its slope is undefined. c. An equation of ℓ is y - 0 = - b __ c (x - a) , or y = - b __ c x + ab __ c . An equation of m is y - 0 ____ ____ = a - b (x - 0) , or y = a - b x. c c An equation of n is x = b. d. (b, ab - b 2 ______ c of line n is x = b and the xcoordinate of P is b, P lies on n. f. Lines ℓ, m, and n are concurrent at P. 47. F 49. 14.0 51. 108° ) e. Since the equation . Since n ⊥ 5-4 Check It Out! 1. M (1, 1) ; N (3, 4) ; ̶̶ ̶̶̶ RS = 3 __ 2 ; MN = 3 __ 2 ; slope of slope of since the slopes are the same, ̶̶̶ MN ǁ = 2 √  13 ; the length of the length of ̶̶ RS . MN = √  13 ; RS = √  52 ̶̶̶ MN is half ̶̶ RS . 2a. 72 2b. 48.5 Selected Answers S97 S97 � 2c. 102° 3. 775 m Exercises 1. midpoints 3. 5.1 5. 5.6 7. 29° 9. less than 5 yd 11. 38 13. 19 15. 55° 17. yes 19. 17 21. n = 36 23. n = 8 25. n = 4 27. B 29. Possible answer: about 18 parking spaces 31. 11 33. 57° 35. 123° 37a. 2.25 mi b. 28.5 mi 39. D 41. D 43. equilateral and equiangular 45. 7 47a. 32; 16; 8; 4 b. 1 __ 4 c. 64 ( 1 __ 2 ) 51. (4, -2) , (8, -1) , (5, -4) 53. 6 55. 9 n = 2 6 - n 49. 2.25% 5-5 Check It Out! 1. Possible answer: Given: △RST Prove: △RST cannot have 2 rt. . Proof: Assume that △RST has 2 rt. . Let ∠R and ∠S be the rt. . By the def. of rt. ∠, m∠R = 90° and m∠S = 90°. By the △ Sum Thm., m∠R + m∠S + m∠T = 180°. But then 90° + 90° + m∠T = 180° by subst., so m∠T = 0°. However, a △ cannot have an ∠ with a measure of 0°. So there is no △RST, which contradicts the given information. This means the assumption is false, and △RST cannot have 2 rt. . ̶̶ DF , 2a. ∠B, ∠A, ∠C 2b. 3a. No; 8 + 13 = 21, which is not greater than the third side length. 3b. Yes; the sum of each pair of 2 lengths is greater than the third length. 3c. Yes; when t = 4, the value of t - 2 is 2, the value of 4t is 16, and the value of t 2 + 1 is 17. The sum of each pair of 2 lengths is greater than the third length. 4. greater than 5 in. and less than 39 in. 5. 28 mi < d < 72 mi ̶̶ EF , ̶̶ DE Exercises 3. Possible answer: Given: △PQR is an isosc. △ with base ̶̶ PR . Prove: △PQR cannot have a base ∠ that is a rt. ∠. Proof: Assume that △PQR has a base ∠ that is a rt. ∠. Let ∠P be the rt ∠. By the Isosc. △ Thm., ∠R ≅ ∠P, so ∠R is also a rt. ∠. By the def. of rt. ∠, m∠P = 90° S98 S98 Selected Answers and m∠R = 90°. By the △ Sum Thm., m∠P + m∠Q + m∠R = 180°. By subst., 90° + m∠Q + 90° = 180°, so m∠Q = 0°. However, a △ cannot have an ∠ with a measure of 0°. So there is no △PQR, which contradicts the given information. This means the assumption is false, and therefore △PQR cannot have a base ∠ that is rt. ̶̶ ̶̶ XY 7. no 9. no 11. yes YZ , ̶̶ XZ , ̶̶ ST , ̶̶ EF , ̶̶ RS , ̶̶ DE , ̶̶ AB ⊥ 5. 13. greater than 0 ft and less than 32 ft 15a. the path from the refrigerator to the stove b. no ̶̶ 19. RT 21. no 23. yes 25. no 27. greater than 5 km and less than 51 km 29. greater than 1.18 m and less than 4.96 m 31. greater than 2 2 __ 3 ft and less than 10 1 __ 3 ft 33. a > 7.5, where a is the ̶̶ length of a leg. 35. DF 37. m∠Y < 90°, and ∠Y is an obtuse ̶̶ ̶̶ angle. 39. AB ǁ BC , and 41. x is a multiple of 4, and x is prime. 43. < 45. = 47. > 49. > 51. < 53. = 55. ∠L, ∠K, ∠J 57. ∠J, ∠L, ∠K 59a. 0.4 h < t < 2 h b. no 61. 1 < n < 6 63. n > 0 65. n > 0.5 67a. def. of ≅ segs. b. Isosc. △ Thm. c. def. of ≅  d. m∠1 + m∠3 e. subst. f. m∠S g. Trans. Prop. of Inequal. 71. H 73. 3 __ 10 , or 30% 77. -2x + y = 6 79. BC = 10, EF = 11, and m∠ABC = 102°, so △ABC ≅ △EFD by SAS. 81. (0, 0) ̶̶ BC . 5-6 Check It Out! 1a. m∠EGF > m∠EGH 1b. BC > AB 2. The ∠ of the swing at full speed is greater than the ∠ at low speed. 3a. 1. C is the mdpt. of ̶̶ BD . m∠1 = ̶̶ BC ≅ m∠2, m∠3 > m∠4 (Given) ̶̶ 2. DC (Def. of mdpt.) 3. ∠1 ≅ ∠2 (Def. of ≅ ) 4. Thm.) 5. AB > ED (Hinge Thm.) 3b. 1. ∠SRT ≅ ∠STR, TU > RU ̶̶ EC (Conv. of Isosc. △ ̶̶ AC ≅ ̶̶ SR (Conv. of Isosc. △ (Given) ̶̶ 2. ST ≅ Thm.) ̶̶ ̶̶ 3. SU (Reflex. Prop. of ≅ ) SU ≅ 4. m∠TSU > m∠RSU (Conv. of Hinge Thm.) Exercises 1. AC < XZ 3. KL > KN 5. 1.2 < x < 3 7. the second position 9. m∠DCA > m∠BCA 11. TU > SV 13. -3.5 < z < 32.5 15. the second position 17. BC = YZ 19. > 21. = 23. < 25. m∠RSV < m∠TSV 27. m∠YMX > m∠ZMX 31. D 33. Group A is closer to the camp. 37. 14; none 39. m∠2 = m∠6 = 36°; m ǁ n by the Conv. of the Corr.  Post. 41. 2.5 43. 85° 5-7 Check It Out! 1a. x = 4 √  5 1b. x = 16 2. 29 ft 1 in. 3a. 2 √  41 ; no; 2 √  41 is not a whole number. 3b. 10; yes; the 3 side lengths are nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2 . 3c. 2.6; no; 2.4 and 2.6 are not whole numbers. 3d. 34; yes; the 3 side lengths are nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2 . 4a. yes; obtuse 4b. no 4c. yes, acute Exercises 1. no 3. x = 6 √  2 5. width: 14.8 in.; height: 11.9 in. 7. 16; yes 9. triangle; acute 11. triangle; right 13. triangle; acute 15. x = 10 17. x = 24 19. 6; no 21. 3 √  5 ; no 23. not a triangle 25. triangle; right 27. triangle; acute 29. B 31. x = 8 + √  13 33. x = 4 √  6 35. x = 6 √  13 39. perimeter: 16 + 4 √  7 units; area: 12 √  7 square units 41. perimeter: 14 + 2 √  13 units; area: 18 square units 43. perimeter: 22 units; area: 26 square units 47a. King City b. m∠SRM > 90° 49. B 51a. PA = √  2 ; PB = √  3 ; PC = √  4 ; PD = √  5 ; PE = √  6 ; PF = √  7 55a. no b. yes. c. no d. no 57. x = -5 61. - 1 __ 3 < x < 2 5-8 Check It Out! 1a. x = 20 1b. x = 8 √  2 2. 43 cm 3a. x = 9 √  3 ; y = 27 3b. x = 5 √  3 ; y = 10 3c. x = 12; y = 12 √  3 3d. 34.6 cm Exercises 1. x = 14 √  2 3. x = 9 5. x = 3; y = 3 √  3 7. x = 21; y = 14 √  3 9. x = 15 √  2 _____ 11. x = 18 2 13. x = 48; y = 24 √  3 15. x = 2 √  3 ____ 3 ; y = 4 √  3 ____ 3 17. perimeter: (12 + 12 √  2 ) in.; area: 36 in 2 19. perimeter: 36 √  2 m; area: 162 m 2 21. perimeter: 60 √  3 yd; area: 300 √  3 yd 2 23. no 25. (10, 3) 27. (5, 10 - 12 √  3 ) 29a. 640 m b. 453 m c. 234 m 31. F 33. 443.4 35. x = 32 __ 9 39. y = (x - 5) 2 - 27; x = 5 41. obtuse 43. right 45. 132° SGR ̶̶ AB and ̶̶ AB , ̶̶ CP , P is on the ̶̶ AP ⊥ ̶̶ AP ⊥ 1. equidistant 2. midsegment 3. incenter 4. lo
cus 5. 7.4 6. 13.4 7. 5.8 8. 52° 9. y = x - 1 10. y - 6 = -0.25 (x - 4) 11. No; to apply the Conv. of the ∠ Bisector Thm., you need to know that ̶̶ ̶̶ CB . 12. Yes; because CP ⊥ ̶̶ ̶̶ ̶̶ CP ⊥ AP ≅ CB , and bisector of ∠ABC by the Conv. of the ∠ Bisector Thm. 13. 42.2 14. 46 15. 57.6 16. 46 17. 18 18. 37° 19. (4, 3) 20. (-6, -3.5) 21. 16.4 22. 8.2 23. 5.8 24. 17.4 25. (-6, 0) 26. (1, 2) 27. (7, 4) 28. (3, 0) 29. (3, 4) 30. 35.1 31. 64.8 32. 32.4 33. 42° 34. 138° 35. 42° 36. V (-1, -1) ; W (6, 1) ; ̶̶ ̶̶̶ GJ = 2 __ 7 ; VW = 2 __ 7 ; slope of slope of since the slopes are the same, ̶̶ ̶̶̶ GJ . VW = √  53 ; GJ = 2 √  53 ; VW ǁ since √  53 = 1 __ 2 (2 √  53 ) , VW = 1 __ 2 GJ. 37. 39. greater than 9 cm and less than 18 cm 40. Yes; possible answer: the sum of each pair of 2 lengths is greater than the third length. 41. No; possible answer: when z = 5, the value of 3z is 15. So the 3 lengths are 5, 5, and 15. The sum of 5 and 5 is 10, which is not greater than 15. By the △ Inequality Thm., a △ cannot have these side lengths. 43. PS < RS 44. m∠BCA < m∠DCA 45. -1.4 < n < 3 46. 2.75 < n < 12.5 47. x = 2 √  10 48. x = 2 √  33 49. 6; the lengths do not form a Pythagorean triple because 4.5 and 7.5 are not whole numbers. 50. 40; the lengths do form a Pythagorean triple because they are nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2 . 51. triangle; ̶̶ AB 38. ∠F, ∠H, ∠G ̶̶ BC , ̶̶ AC , obtuse 52. not a triangle 53. triangle; right 54. triangle; acute 55. x = 26 √  2 56. x = 6 √  2 57. x = 32 58. x = 24; y = 24 √  3 59. x = 6 √  3 ; y = 12 60. x = 14 √  3 _____ ; 3 y = 28 √  3 _____ 3 61. 21 ft 3 in. 62. 15 ft 7 in. Chapter 6 6-1 Check It Out! 1a. not a polygon 1b. polygon, nonagon 1c. not a polygon 2a. regular, convex 2b. irregular, concave 3a. 2340° 3b. 144° 4a. 30° 4b. r = 15 5. 45° Exercises 3. not a polygon 5. not a polygon 7. irregular, concave 9. m∠A = m∠D = 81°; m∠B = 108°; m∠C = m∠E = 135° 11. 3240° 13. 72° 15. m∠Q = m∠S = 135° 17. not a polygon 19. irregular, concave 21. irregular, convex 23. 160° 25. 40° 27. 120° 29. 61.5 31. 72 33. 10 35. pentagon 37. dodecagon 39. 3; 60° 41. 10; 144° 43. A 45a. heptagon b. 900° c. 140° 53. A 55. D 57. x = 36; y = 36; z = 72 59. Yes, if you allow for ∠ measures greater than 180° 61. x = -3 or x = 4 63. 0 < x < 8 65. 4 < x < 10 67. 5 √  3 6-2 Check It Out! 1a. 28 in. 1b. 74° 1c. 13 in. 2a. 12 2b. 18 3. (7,6) 4. 1. GHJN and JKLM are . (Given) 2. ∠N and ∠HJN are supp. ∠K and ∠MJK are supp. ( → cons.  supp.) 3. ∠HJN ≅ ∠MJK (Vert.  Thm.) 4. ∠N ≅ ∠K (≅ Supps. Thm.) Exercises 3. 36 5. 18 7. 70° 9. 24.5 11. 51° 13. (- 6, - 1) 15. 82.9 17. 82.9 19. 130° 21. 10 23. 28 25. (-1, 3) 27. PQ = QR = RS = SP = 21 29. PQ = RS = 17.5; QR = SP = 24.5 31a. ∠3 ≅ ∠1 (Corr.  Post.); ∠6 ≅ ∠1 ( → opp.  ≅); ∠8 ≅ ∠1 ( → opp.  ≅) b. ∠2 is supp. to ∠1 ( → cons.  supp.); ∠4 is supp. to ∠1 ( → cons.  supp.); ∠5 is supp. to ∠1 ( → cons.  supp.); ∠7 is supp. to ∠1 (Subst.) 33. ∠KMP ( → ̶̶̶ KM ( → opp. sides ̶̶ RP (Def. of ) 39. ∠RTP opp.  ≅) 35. ≅) 37. (Vert.  Thm.) 41. x = 119; y = 61; z = 119 43. x = 24; y = 50; z = 50 47. x = 5; y = 8 49a. no b. no 51. A 53. 26.4 55. (2, 4) , (4, -6) , (-6, -2) 59. no correlation 61. alt. ext.  63. corr.  65. 8 sides; 45° 6-3 ̶̶ PQ ǁ ̶̶ PQ ≅ Check It Out! 1. PQ = RS = 16.8, ̶̶ RS . m∠Q = 74°, and m∠R so = 106°, so ∠Q and ∠R are supp., ̶̶ RS . So 1 which means that pair of opp. sides of PQRS are \|\| and ≅. By Thm. 6-3-1, PQRS is a . 2a. Yes; possible answer: the diag. of the quad. forms 2 . 2  of 1 △ are ≅ to 2  of the other, so the third pair of  are ≅ by the Third  Thm. So both pairs of opp.  of the quad. are ≅. By Thm. 6-3-3, the quad. is a . 2b. No; 2 pairs of cons. sides are ≅. None of the sets of conditions for a  are met. ̶̶ 3. Possible answer: slope of KL = ̶̶̶ ̶̶̶ MN = - 7 __ 2 ; slope of LM = slope of ̶̶ NK = - 1 __ 4 ; both pairs of slope of opp. sides have the same slope, ̶̶ ̶̶̶ NK ; by def., MN and so KLMN is a . 4. Possible answer: Since ABRS is a , it is always ̶̶ RS . Since true that ̶̶ RS also remains vert. no vert., matter how the frame is adjusted. Therefore the viewing ∠ never changes. ̶̶ AB stays ̶̶̶ LM ǁ ̶̶ AB ǁ ̶̶ KL ǁ ̶̶ UR = 0; ̶̶ FH . EJ ̶̶ FH ̶̶ ST and ̶̶ ST ǁ Exercises 1. FJ = HJ = 10, so ̶̶ ̶̶ ̶̶ EG bisects FJ ≅ HJ . Thus ̶̶ ̶̶ = GJ = 18, so GJ . Thus EJ ≅ ̶̶ EG . So the diags. of EFGH bisects bisect each other. By Thm. 6-3-5, EFGH is a . 3. yes 5. yes ̶̶ 7. Possible answer: slope of ST = ̶̶ UR have slope of ̶̶ UR ; ST = the same slope, so UR = 6; 1 pair of opp. sides are \|\| and ≅; by Thm. 6-3-1, RSTU is a . 9. BC = GH = 16.6, so CG = HB = 28, so both pairs of opp. sides of BCGH are ≅, BCGH is a  by Thm. 6-3-2. 11. yes 13. no 15. Possible answer: slope of and ̶̶ RS = 5 __ 3 ; ̶̶ RS have the same slope, so ̶̶ ̶̶̶ GH . BC ≅ ̶̶ HB . Since ̶̶ PQ = slope of ̶̶ PQ ̶̶ PQ ̶̶ CG ≅ Selected Answers S99 S99 ̶̶ RS ; PQ = RS = √  34 ; 1 pair of opp. ǁ sides are \|\| and ≅; by Thm. 6-3-1, PQRS is a . 17. no 19. yes 21. a = 16.5; b = 23.2 23. a = 8.4; b = 20 27a. ∠Q b. ∠S. c. 35 B 37. no 39. (3, 1) ; (-6, -3.5) 41. ̶̶ RS e.  ̶̶ SP d. x -5 -2 0 0.5 y -38 -17 -3 0.5 43. x y -5 -2 77 14 0 2 0.5 2.75 47. 12 49. 12 6-4 Check It Out! 1a. 48 in. 1b. 61.6 in. 2a. 42.5 2b. 17° 3. SV ̶̶̶ = TW = √  TW . Slope ̶̶̶ TW = -11, of ̶̶ SV ≅ 122 , so ̶̶ SV = 1 __ 11 , and slope of ̶̶ SV ⊥ so mdpt. of ̶̶ SV and so So the diags. of STVW are ≅ ⊥ bisectors of each other. 4. Possible answer: ̶̶̶ TW . The coordinates of the ̶̶ ̶̶̶ TW are ( 1 __ 2 , - 7 __ 2 ) , SV and ̶̶̶ TW bisect each other. 1. PQTS is a rhombus. (Given) ̶̶ 2. PT bisects∠QPS. (Rhombus → each diag. bisects opp. ) 3. ∠QPR ≅ ∠SPR (Def. of ∠ bisector) ̶̶ 4. PQ ≅ ̶̶ 5. PR ≅ 6. △QPR ≅ △SPR (SAS) 7. ̶̶ PS (Def. of rhombus) ̶̶ PR (Reflex. Prop. of ≅) ̶̶ RS (CPCTC) ̶̶ RQ ≅ ̶̶ TY ̶̶ RX ≅ ̶̶ XY (Reflex. Prop. of ≅) Exercises 1. rhombus; rectangle; square 3. 160 ft 5. 380 ft 7. 122° 9. Possible answer: 1. RECT is a rect. (Given) ̶̶ 2. XY ≅ 3. RX = TY , XY = XY (Def. of ≅ segs.) 4. RX + XY = TY + XY (Add. Prop. of =) 5. RX + XY = RY, TY + XY = TX (Seg. Add. Post.) 6. RY = TX (Subst.) ̶̶ 7. TX (Def. of ≅ segs.) 8. ∠R and ∠T are rt. . (Def. of rect.) 9. ∠R ≅ ∠T (Rt. ∠ ≅ Thm.) 10. RECT is a . (Rect. → ) 11. 12. △REY ≅ △TCX (SAS) ̶̶ TC ( → opp. sides ≅) ̶̶ RE ≅ ̶̶ RY ≅ S100 S100 Selected Answers 11. 25 13. 14 1 __ 2 15. m∠VWX = 132°; m∠WYX = 66° 17. Possible answer: ̶̶ HB is a ̶̶̶ MH ≅ ̶̶ HB bisects ∠RHM. 1. RHMB is a rhombus. diag. of RHMB. (Given) ̶̶ 2. RH (Def. of rhombus) 3. (Rhombus → each diag. bisects opp. ) 4. ∠MHX ≅ ∠RHX (Def. of ∠ bisector) ̶̶ 5. HX ≅ 6. △MHX ≅ △RHX (SAS) 7. ∠HMX ≅ ∠HRX (CPCTC) ̶̶ HX (Reflex. Prop. of ≅) 19. m∠1 = 54°; m∠2 = 36°; m∠3 = 54°; m∠4 = 108°; m∠5 = 72° 21. m∠1 = 126°; m∠2 = 27°; m∠3 = 27°; m∠4 = 126°; m∠5 = 27° 23. m∠1 = 64°; m∠2 = 64°; m∠3 = 26°; m∠4 = 90°; m∠5 = 64° 25. S 27. S 29. A 31. S 35a. Rect. →  ̶̶̶ b. HG c. Reflex. Prop. of ≅ d. Def. of rect. e. ∠GHE f. SAS g. CPCTC 41. 28 √  2 in. ≈ 39.60 in.; 98 in 2 45. D 47. H 51. 45 53. T 55. no 6-5 Check It Out! 1. Both pairs of opp. sides of WXYZ are ≅, so WXYZ is a . The contractor can use the carpenter’s square to see if 1 ∠ of WXYZ is a rt. ∠. If 1 ∠ is a rt. ∠, then by Thm. 6-5-1 the frame is a rect. 2. Not valid; by Thm. 6-5-1, if 1 ∠ of a  is a rt. ∠, then the  is a rect. To apply this thm., you need to know that ABCD is a . 3a. rect., rhombus, square 3b. rhombus ̶̶ QS ̶̶ PR ≅ Exercises 3. valid 5. rhombus 7. valid 9. square, rect., rhombus 11. , rect. 13. , rect., rhombus, square 15. , rect., rhombus, square 17. B 19. 21. (2, 6) 23. (-2, -2) 25. x = 3 ̶̶ 27. rhombus 29a. slope of AB = ̶̶ ̶̶ CD = - 1 __ 3 ; slope of AD = slope of ̶̶ ̶̶ CB = -3 b. Slope of AC = slope of ̶̶ BD = 1; the slopes are -1; slope of negative reciprocals of each other, so since it is a  and its diags. are ⊥ (Thm. 6-5-4.) 33b.  c. square 39. A 41a. 15x = 13x + 12; x = 6 b. yes c. not necessarily d. yes 43b. no c. no 45. linear 47. linear 49. 31 + √  61 ≈ 38.8 51. y = 4 ̶̶ BD . c. ABCD is a rhombus, ̶̶ AC ⊥ 6-6 Check It Out! 1. about 191.2 in.; 3 packages 2a. 51° 2b. 110° 2c. 62° 3a. 131° 3b. 25.4 4. x = 4 or x = -4 5. 8 ̶̶ QT ̶̶ RS and ̶̶ PV ; ̶̶ VS ; midsegment: Exercises 1. bases: ̶̶ PR and legs: 3. about 20.1 in.; 3 sun catchers 5. 63° 7. 106° 9. z = 2 or z = -2 11. 14 13. about 56.6 in.; about 418.3 in. 15. 122° 17. 62° 19. ±4 √  5 21. 3.6 23. S 25. N 27. m∠1 = 82°; m∠2 = 128° 29. m∠1 = 51°; m∠2 = 16° 31. m∠1 = 120° 33a. EF = FG = √  17 , and GH = HE ̶̶ ̶̶ = √  29 , so HE . FG , and Thus EFGH is a kite, since it has exactly 2 pairs of ≅ cons. sides. b. m∠E = m∠G = 126° 35. 13 37. m∠PAQ = 108°; m∠OAQ = 130°; m∠OBP = 22° 41. kite 43. isosc. trap. 47. B 49. 18 51. AD = 7.08 in.; AB = CD = 5.08 in.; BC = 10.16 in. 53. 2x < x + 6; x < 6 55. rect., rhombus, square ̶̶̶ GH ≅ ̶̶ EF ≅ SGR 1. vertex of a polygon 2. convex 3. rhombus 4. base of a trapezoid 5. not a polygon 6. polygon; △ 7. polygon; dodecagon 8. irregular; concave 9. irregular; convex 10. reg.; convex 11. 1800° 12. 162° 13. 90° 14. m∠A = m∠D = 144°; m∠B = m∠E = 126°; m∠C = m∠F = 90° 15. 37.5 16. 62.4 17. 37.5 18. 79° 19. 101° 20. 101° 21. 9.5 22. 9.5 23. 54° 24. 126° 25. 54° 26. 126° 27. T (6, - 5) 28. 1. GHLM is a . ∠L ≅ ∠JMG ̶̶ GJ ≅ (Given) 2. ∠G ≅ ∠L ( → opp.  ≅) 3. ∠G ≅ ∠JMG (Trans. Prop. of ≅) 4. Thm.) 5. △GJM is isosc. (Def. of isosc. △) ̶̶ MJ (Conv. of Isosc. △ 29. m∠A = m∠E = 63°; m∠G = 117°; since 117° + 63° = 180°, ∠G is supp. to ∠A and to ∠E. So 1 ∠ of ACEG is supp. to both of its cons. . By Thm. 6-3-4, ACEG is a . ̶̶ 30. RS = QT = 25, so QT . m∠R = 76°, m∠Q = 104°, and m∠R + m∠Q = 180°, so ∠R is supp. to ̶̶ RS ≅ ̶̶ RS ǁ ̶̶ BD ǁ ̶̶ BH ǁ ̶̶ FH and ̶̶ SU . ̶̶ SU ; slope of 5 ̶̶ DF ; by def., ∠Q. Since ∠R and ∠Q are a pair of same-side int. , and they are ̶̶ QT . So 1 pair of opp. supp., sides of QRST are \|\| and ≅. By Thm. 6-3-1, QRST is a . 31. Yes; the diags. of the quad. bisect each other. By Thm. 6-3-5, the
quad. is a . 32. No; a pair of alt. int.  are ≅, so 1 pair of opp. sides are \|\|. A different pair of opp. sides are ≅. None of the conditions ̶̶ for a  are met. 33. slope of BD FH = 1 __ ̶̶ ̶̶ = slope of BH = ̶̶ DF = -6; both pairs of slope of opp. sides have the same slope, so BDFH is a . 34. 18 35. 39.6 36. 39.6 37. 19.8 38. 25.5 39. 10.5 40. 25.5 41. 21 42. 41° 43. 49° 44. 82° 45. 98° 46. m∠1 = 57°; m∠2 = 66°; m∠3 = 33°; m∠4 = 114°; m∠5 = 57° 47. m∠1 = 37°; m∠2 = 53°; m∠3 = 90°; m∠4 = 37°; m∠5 = 53° ̶̶ 48. RT = SU = 2 √  10 , so RT ≅ ̶̶ RT = -3, and slope of Slope of ̶̶ = 1 __ 3 , so RT ⊥ of the mdpt. of ̶̶ (-4, -3) , so RT and other. So the diags. of RSTU are ≅ ⊥ bisectors of each other. ̶̶ 49. EG = FH = 3 √  2 , so EG ≅ Slope of = 1, so ̶̶ EG and of the mdpt. of ̶̶ ( 7 __ 2 , - 1 __ 2 ) , so FH bisect each other. So the diags. of EFGH are ≅ ⊥ bisectors of each other. 50. Not valid; by Thm. 6-5-2, if the diags. of a  are ≅, then the  is a rect. By Thm. 6-5-4, if the diags. of a  are ⊥, then the  is a rhombus. If a  is both a rect. and a rhombus, then the  is a square. To apply this chain of reasoning, you must first know that EFRS is a . 51. valid 52. valid 53. rhombus 54. rect. 55. rect., rhombus, square 56. 64° 57. 25° 58. 65° 59. 123° 60. m∠R = 126°; m∠S = 54° 61. 51.6 62. 48.5 63. 3.5 64. n = 3 or n = -3 65. kite 66. trap. 67. isosc. trap. ̶̶ EG = -1, and slope of ̶̶ EG ⊥ ̶̶ ̶̶ RT and SU are ̶̶ SU bisect each ̶̶ SU . The coordinates ̶̶ FH . The coordinates ̶̶ FH . ̶̶ FH ̶̶ EG and ̶̶ FH are Chapter 7 7-1 Check It Out! 1. 1 __ 4 2. 9°; 54°; 117° 3a. x = 21 3b. y = ± 3 3c. d = 9 3d. x = 3 or -9 4. 4 : 5 5. 1527.2 m b. 800 in., Exercises 1. means: 3 and 2; extremes: 1 and 6 3. 1 __ 2 5. - 2 __ 3 7. 95° 9. y = 9 11. y = ± 9 13. x = 0 or x = 3 15. 2 : 9 17. 3 __ 1 19. 3 __ 2 21. 72°; 108°; 72°; 108° 23. x = 20 25. m = 2 or m = -4 27. x = ± 8 29. 3 : 5 31. 5b 33. b __ 7 35. 1 __ 3 37. -3 ______ ______ = x in. 39a. 1.25 in. 9600 in. 15 in. or 66 ft 8 in. 41. 4 __ 9 45. H 47. First, cross multiply: 36x = 15 (72) , or 36x = 1080. Then divide both sides by ____ 36: 36x ___ 36 = 1080 36 . Finally, simplify: x = 30. You must assume that x ≠ 0. 49. Given a __ = c __ , add 1 to both sides d b + d __ = c __ + b __ of the eqn. as shown: a __ . d d b b Adding the fractions on both sides of the eqn. gives a + b = c + d ____ ____ . b d 51. x + 3 ____ x - 6 , where x ≠ ±6 53. 1 55. 96° 57. acute 59. right 7-2 = AC ___ JH = BC ___ JK = DA ___ LH = BC ___ GH Check It Out! 1. ∠A ≅ ∠J; ∠B ≅ ∠G; ∠C ≅ ∠H; AB ___ = 2 JG 2. yes; 5 __ 2 ; △LMJ ∼ △PNS 3. 5 in. Exercises 3. ∠A ≅ ∠H; ∠B ≅ ∠J; ∠C ≅ ∠K; ∠D ≅ ∠L; AB ___ HJ = 2 __ 3 5. yes; 2 __ 3 ; △RMP ∼ = CD ___ KL △XWU 7. ∠J ≅ ∠S; ∠K ≅ ∠T; ∠L ≅ ∠U; ∠M ≅ ∠V; JK ___ = MJ ___ = LM ___ UV VS ST = 5 __ 6 9. yes; 7 __ 8 ; △RSQ ∼ △UZX 11. 14 ft 13. S 15. N 17. S 19. 5 23. ∠O; ∠Q 27. C 29. The ratios of the sides are not the same; 12 ___ 3.5 = 24 __ 7 ; 10 ___ 2.5 = 4; 6 ___ 1.5 = 4. 33a. rect. ABCD ∼ rect. BCFE b. ℓ __ 1 = 1 ____ ℓ - 1 c. ℓ = 1 + √  5 ______ d. ℓ ≈ 1.6 35. 90° 2 37. 70° 39. 4 __ x = KL ___ TU 7-3 Check It Out! 1. By the △ Sum Thm., m∠C = 47°, so∠C ≅ ∠F. ∠B ≅ ∠E by the Rt. ∠ ≅ Thm. Therefore △ABC ∼ △DEF by AA ∼. 2. ∠TXU ≅ ∠VXW by the Vert.  Thm. TX ___ = VX = 15 __ 20 = 3 __ 4 . Therefore 12 __ 16 = 3 __ 4 , and XU ___ XW △TXU ∼ △VXW by SAS ∼. 3. It is given that ∠RSV ≅ ∠T. By the Reflex. Prop. of ≅, ∠R ≅ ∠R. Therefore △RSV ∼ △RTU by AA ∼. RT = 15. 4. 1. M is the mdpt. of ̶̶ JK , N is the ̶̶ KL , and P is the mdpt. mdpt. of ̶̶ JL . (Given) of 2. MP = 1 __ 2 KL, MN = 1 __ 2 JL, NP = 1 __ 2 KJ (△Midsegs. Thm.) 3. MP ___ = NP ___ = MN ___ KJ JL KL of =) 4. △JKL ∼ △NPM (SSS ∼ Step 3) 5. 5 = 1 __ 2 (Div. Prop. = EF ___ KL Exercises 1. By the △ Sum Thm., m∠A = 47°. So by the def. of ≅ , ∠A ≅ ∠F, and ∠C ≅ ∠H. Therefore △ABC ∼ △FGH by AA ∼. 3. DF ___ JL = DE ___ JK SSS ∼. 5. It is given that ∠AED ≅ ∠ACB. ∠A ≅ ∠A by the Reflex. Prop. of ≅. Therefore △AED ∼ △ACB by AA ∼. AB = 10 ̶̶̶ 7. 1. MN ǁ = 1 __ 2 , so △DEF ∼ △JKL by ̶̶ KL (Given) 2. ∠JMN ≅ ∠JKL, ∠JNM ≅ ∠JLK (Corr.  Post.) 3. △JMN ∼ △JKL (AA ∼ Step 2) 9. SAS or SSS ∼ Thm. 11. It is given that ∠GLH ≅ ∠K. ∠G ≅ ∠G by the Reflex. Prop. of ≅. Therefore △HLG ∼ △JKG by AA ∼. 13. ∠K ≅ ∠K by = 3 __ 2 . the Reflex. Prop. of ≅. KL ___ KN Therefore △KLM ∼ △KNL by SAS ∼. 15. It is given that ∠ABD ≅ ∠C. ∠A ≅ ∠A by the Reflex. Prop. of ≅. Therefore △ABD ∼ △ACB by AA ∼. AB = 8 = KM ___ KL 17. 1. CD = 3AC , CE = 3BC (Given) = 3, CE ___ BC = 3 (Div. Prop. 2. CD ___ AC of =) 3. ∠ACB ≅ ∠DCE (Vert.  Thm.) 4. △ABC ≅ △DEC (SAS ∼ Steps 2, 3) 19. 1.5 in. 21. yes; SSS ∼ 23. x = 3 25a. Pyramids A and C are ∼ because the ratios of their corr. side lengths are =. b. 5 __ 4 27. 2 ft; 4 ft 31a. The  are ∼ by AA ∼ if you assume that the camera is ǁ to the hurricane (that is, b. △YWZ ∼ △BCZ, and △XWZ ∼ △ACZ, also by AA ∼. c. 105 mi 35. J 37. 30 41. 94 43. Possible answer: (0, k) , (2k, k) , (2k, 0) , (0, 0) 45. y = ±15 ̶̶ YX ǁ ̶̶ AB ). Selected Answers S101 S101 7-4 AD = 1, and ED ___ DB Check It Out! 1. 7.5 2. AD = 16, and BE = 12 , so DC ___ = 20 __ 16 = 5 __ 4 , and AD EC ___ = EC ___ = 15 __ 12 = 5 __ 4 . Since DC ___ , BE BE ̶̶ ̶̶ DE ǁ AB by the Conv. of the △ Proportionality Thm. 3. LM ≈ 1.5 cm; MN ≈ 2.4 cm 4. AC = 16; DC = 9 Exercises 1. 30 3. EC ___ AC ̶̶ = ED ___ = 1. Since EC ___ AB ǁ , DB AC the Conv. of the △ Proportionality Thm. 5. 286 ft 7. CD = 4; AD = 6 9. 20 11. PM ___ MQ PN ___ NR MN ǁ QR by the Conv. of the △ Proportionality Thm. 13. BC = 6; CD = 5 15. CE 17. BD 19. DF 21. 15 in. or 26 2 __ 3 in. = AF ___ 23. 1. AE ___ EB FC = 6.3 ___ 2.7 = 2 1 __ 3 , and = PN ___ , NR = 7 __ 3 = 2 1 __ 3 . Since PM ___ ̶̶ CD by (Given) MQ 2. ∠A ≅ ∠A (Reflex. Prop. of □) 3. △AEF ∼ △ABC (SAS ∼ Steps 1, 2) 4. ∠AEF ≅ ∠ABC (Def. of ∼ △) 5.   EF ǁ Thm.) ̶̶ BC (Conv. of Corr.  = QS ___ SU 25a. PR = 6; RT = 8; QS = 3; SU = 4 b. PR ___ , or 6 __ 8 = 3 __ 4 27. 15 33. J RT 39. Check students’ work. 41. 3n 43. (5, -18) 45. ∠KLJ ≅ ∠NLM by the Vert.  Thm. By the △ Sum Thm., m∠MNL = 68°. So ∠JKL ≅ ∠MNL. Therefore △JKL ∼ △MLN by AA ∼. 7-5 Check It Out! 1. 15 ft 7 in. 2. 900 m, or 0.9 km 3. Check students’ work. The drawing should be 3.7 in. by 3 in. 4. P = 14 mm; A = 10 2 __ 3 mm 2 Exercises 1. indirect measurement 3. 12 ft 5. 60 ft 11. 27 cm 2 13. ≈ 61 km 19. 864 m 2 21. 175 ft 23. 375 ft 25. 4 __ 5 27. 0.3 ft by 1.2 ft 29. 20 in.; 12 in. 31a. 1 __ 24 b. 1 ___ 576 c. 24 ft 2 33. 1 __ 9 cm 35. 1 cm : 5 m; since each centimeter will equal 5 m, this drawing will be 1 __ 5 the size of the drawing with a scale of 1 cm : 1 m. 39. D 41. C 43a. 150 m b. 1.28 cm 47. x = -4 or x = 10 49. x ≈ 0.65 or x ≈ -4.65 51. The slopes of ̶̶ KL and of ̶̶ JK and ̶̶ JM = -1. Since both ̶̶̶ LM = 1. The slopes S102 S102 Selected Answers pairs of opp. sides have the same ̶̶ JK ǁ slope, def., JKLM is a . ̶̶̶ LM , and ̶̶ JM . By ̶̶ KL ǁ 7-6 = RT ___ RV = 1 __ 3 . ∠R ≅ ∠R by Check It Out! 1. The photo should have vertices A′ (0, 0) , B′ (0, 2) , C′ (1.5, 2) , and D′ (1.5, 0) . 2. N (0, -20) ; 2 __ 3 3. RS = √  2 , RU = 3 √  2 , RT = √  5 , and RV = 3 √  5 , so RS ___ RU the Reflex. Prop. of ≅. So △RST ∼ △RUV by SAS ∼. 4. Check students’ work. The image of △MNP has vertices M′ (-6, 3) , N′ (6, 6) , and P′ (-3, -3) . MP = √  5 , MN = √  17 , and PN = 3 √  2 . M′P′ = 3 √  5 , M′N′ = 3 √  17 , and P′N′ = ____ ____ = M′N′ 9 √  2 . M′P′ = 3. So MN MP △M′N′P′ ∼ △MNP by SSS ∼. = P′N′ ___ PN = JL __ JN = S′T ′ ___ ST = R′T ′ ___ RT = 3 __ 2 . So △RST ∼ Exercises 1. dilation 5. S (0, -8) ; 5 __ 2 7. JK = 2 √  5 , JM = 3 √  5 , JL = 2 √  5 , and JN = 3 √  5 , so JK ___ = 2 __ 3 . ∠J ≅ ∠J by the JM Reflex. Prop. of ≅. So △JKL ∼ △JMN by SAS ∼. 9. The image of △RST has vertices R′ (-3, 3) , S′ (3, 6) , and T ′ (0, -3) . RS = 2 √  5 , RT = 2 √  5 , and ST = 2 √  10 . R′S′ = 3 √  5 , R′T ′ = 3 √  5 , and S′T ′ = 3 √  10 . R′S′ ___ RS △R′S′T ′ by SSS ∼. 11. X (-24, 0) ; 8 __ 3 13. DE = 2 √  5 , DG = 3 √  5 , DF = = DF ___ 4 √  2 , and DH = 6 √  2 , so DE ___ DH DG = 2 __ 3 . ∠D ≅ ∠D by the Reflex. Prop. of ≅. So △DEF ∼ △DGH by SAS ∼. 15. The image of △JKL has vertices J′ (-6, 0) , K′ (-3, -3) , and L′ (-9, -6) . JK = √  2 , JL = √  5 , and LK = √  5 . J′K′ = 3 √  2 , J′L′ = 3 √  5 , and L′K′ = 3 √  5 . J′K′ ___ JK So △JKL ∼ △J′K′L′ by SSS ∼. 17. It is not a dilation; because it changes the shape of the figure. 21. A 23. A 25. 12 31. 5 33. 12 35. 6 = L′K′ ___ LK = J′L′ ___ JL = 3. 19. 1. JM 2. JL __ = 1 __ 3 , JK ___ = 1 __ 3 JN (Div. Prop. of =) = JK ___ 3. JL __ (Trans. Prop. of =) JM JN 4. ∠J ≅ ∠J (Reflex. Prop. of ≅) 5. △JKL ∼ △JMN (SAS ∼ Steps 3, 4) ̶̶ QR ǁ ̶̶ ST (Given) 2. ∠RQP ≅ ∠STP (Alt. Int. ∠ Thm.) 3. ∠RPQ ≅ ∠SPT (Vert.  Thm.) 4. △PQR ∼ △PTS (AA ∼ Steps 2, 3) ̶̶ BC ǁ ̶̶ CE (Given) 20. 1. EA = BD ___ CE (Def. of ∼ polygons) 2. ∠ABD ≅ ∠C (Corr. ∠ Post.) 3. ∠ADB ≅ ∠E (Corr. ∠ Post.) 4. △ABD ∼ △ACE (AA ∼ Steps 2, 3) 5. AB ___ AC 6. AB (CE) = AC (BD) (Cross Products Prop.) = JL __ 21. 10 22. 3 1 __ 3 23. JK ___ = 1 __ 2 . JN JM ̶̶̶ ̶̶ Since JK ___ = JL __ MN by the KL ǁ , JN JM Conv. of the △ Proportionality Thm. 24. EC ___ = 3 __ 7 . Since EC ___ = ED ___ = EA EB ̶̶ ̶̶ ED ___ CD by the Conv. of the △ AB ǁ , EB Proportionality Thm. 25. SU = 4; SV = 6 26. 18 27. 4x + 8 28. 25 ft 4 in. 29. 3 ft 30. By the Dist. Formula, RS = 2 √  2 , RU = 4 √  2 , RT = √  10 , and RV = 2 √  10 . = 1 __ 2 . ∠R ≅ ∠R by the RS ___ RU Reflex. Prop. of ≅. So △RST ∼ △RUV by SAS ∼. 31. By the Dist. Formula, JK = √  5 , JM = 4 √  5 , JL = 2, and JN = 8. JK ___ = 1 __ 4 . ∠J ≅ ∠J JM by the Reflex. Prop. of ≅. So △JKL ∼ △JMN by SAS ∼. 32. (0, -6) ; 2 __ 3 33. The image of △KLM has vertices K′ (0, 9) , L′ (0, 0) , and M′ (12, 0) . By the Dist. Formula, KL = 3, LM = 4, KM = 5, K′L′ = 9, L′M′ = 12, and K′M′ = 15. K′L′ ___ ____ = K′M′ KM KL △KLM ∼ △K′L′M′ by SSS ∼. = 3 __ 1 . Therefore ____ = L′M′ LM = RT ___ RV = JL __ JN SGR 1. proportion 2. dilation 3. means 4. ratio 5. 1 __ 5 6. -
1 __ 2 7. 3 __ 2 8. 54 9. 17.5; 30; 17.5; 30 10. y = 21 11. s = 10 12. x = ±6 13. z = 13 or z = -11 14. x = ±8 15. y = 3 or y = -5 16. yes; 5 __ 3 ; JKLM ∼ PQRS 17. yes; 2; △TUV ∼ △WXY 18. 1. JL = 1 __ 3 JN , JK = 1 __ 3 JM (Given) Chapter 8 8-1 Check It Out! 1. △LJK ∼ △JMK ∼ △LMJ 2a. 4 2b. 10 √  3 2c. 6 √  2 3. 27; 3 √  10 ; 9 √  10 4. 148 ft Exercises 1. 8 is the geometric mean of 2 and 32. 3. △BED ∼ △ECD ∼ △BCE 5. 10 7. 2 9. 20 41. B 43. By 11. 2 √  15 ; 2 √  6 ; 2 √  10 13. 12; 4 √  13 ; 8 15. △MPN ∼ △PQN ∼ △MQP 17. △RSU ∼ △RTS ∼ △STU 19. 3 √  5 21. 2 √  5 23. 3 √  5 ____ 10 25. 20 √  3 ; 10 √  21 ; 20 √  7 27. 1670 ft 29. 10 __ 3 , or 3 1 __ 3 31. x + y 33. z 35. x 37. 4 √  5 39. √  10 ____ 2 Corollary 8-1-3, a 2 = x (x + y) , and b 2 = y (x + y) . So a 2 + b 2 = x (x + y) + y (x + y) . By the Distrib. Prop., this expression simplifies to (x + y) (x + y) = (x + y) 2 = c 2 . So a 2 + b 2 = c 2 . 47. D 49. A 51. 7; √  35 ; 2 √  15 53. AC ≈ 15.26 cm; AB ≈ 8.53 cm 55. -4; 2 57. 6 59. 4 61. 39° 8-2 Check It Out! 1a. 24 __ 25 = 0.96 1b. 24 __ 7 ≈ 3.43 1c. 24 __ 25 = 0.96 2. s _ s = 1 3a. 0.19 3b. 0.88 3c. 0.87 4a. 21.87 m 4b. 7.06 in. 4c. 36.93 ft 4d. 6.17 cm 5. 14.34 ft 2 3. 4 __ 5 = 0.8 5. 4 __ 5 = Exercises 1. LK ___ JL 0.8 7. 4 __ 3 ≈ 1.33 9. 1 __ 2 11. √  2 ___ 2 13. 0.39 15. 0.03 17. 0.16 19. 9.65 m 21. 7 ft 6 in. 23. 15 __ 8 ≈ 1.88 25. 15 __ 17 ≈ 0.88 27. 15 __ 17 ≈ 0.88 29. 1 __ 2 31. 1.23 33. 0.22 35. 0.82 37. 3.58 cm 39. 19.67 ft 41. 5.27 ft 43. 6.10 m 45. sine; cosine 47. 60° 49. 1.2 ft 2 + ( √  3 ___ 2 ) 53. 0.6 55. 753 ft 59. ( 1 __ 2 ) = 1 __ 4 + 3 __ 4 = 1 61a. sin A = a __ c ; cos A = b __ c b. (sin A) 2 + (cos A) 2 = ( a __ __ = c 2 __ + b 2 ___ ______ c 2 c 2 c 2 c 2 63. 18.64 cm; 16.00 cm 2 65. 22.60 in.; 14.69 in 2 69. H 71. x ≈ 5; AB ≈ 20; BC ≈ 18; AC ≈ 27 75. 1.25 77. 0.75 79. Possible answers: (-2, 11) ; (0, 10) ; (2, 9) 81. Trans. Prop. of ≅ 83. Sym. Prop. of ≅ 85. 12 + ( b __ c ) = 1 2 2 8-3 Check It Out! 1a. ∠2 1b. ∠1 2a. 37° 2b. 87° 2c. 42° 3. DF ≈ 16.51; EF ≈ 8.75; m∠D = 32° 4. RS = ST = 7; RT ≈ 9.90; m∠S = 90°; m∠R = m∠T = 45° 5. 21° Exercises 1. ∠1 3. ∠1 5. ∠2 7. 65° 9. 34° 11. 38° 13. RP ≈ 9.42; m∠P ≈ 19°; m∠R ≈ 71° 15. YZ ≈ 13.96; m∠Y ≈ 38°; m∠Z ≈ 52° 17. RS = 5; ST = 6; RT ≈ 7.81; m∠S = 90°; m∠R ≈ 50°; m∠T ≈ 40° 19. AB = 2; BC = 4; AC ≈ 4.47; m∠B = 90°; m∠A ≈ 27°; m∠C ≈ 63° 21. ∠2 23. ∠1 25. ∠2 27. 18° 29. 37° 31. 57° 33. JK ≈ 2.88; LK ≈ 1.40; m∠L = 64° 35. QR ≈ 4.90; m∠P ≈ 36°; m∠R ≈ 54° 37. MN = NP = 4; MP ≈ 5.66; m∠N = 90°; m∠M = m∠P = 45° 39. 74° 41. cos 43. 0.93 47a. 5° b. 85° c. 31 ft 1 in. 49. 23°; 67° 51. The acute ∠ measure changes from about 58° to about 73°, an increase by a factor of 1.26. 53a. AB = √  50 ; BC = 2 √  10 ; AC = √  10 b. AC 2 + BC 2 = AB 2 , so △ABC is a rt. △, and ∠C is the rt. ∠. c. m∠A = 63°; m∠B = 27° 55. 35° 57. 62° 59. 72° 61. 39° 65. D 67. A 69. 58° 71. 34° 73. x 77. F 79. F 81. -1 83. 0.89 85. 2.05 8-4 Check It Out! 1a. angle of depression 1b. angle of elevation 2. 6314 ft 3. 1717 ft 4. 32,300 ft Exercises 1. elevation 3. angle of elevation 5. angle of elevation 7. 18 ft 9. 64.6 m 11. angle of elevation 13. angle of depression 15. 1962 ft 17. T 19. F 21. ∠1 and ∠3 25a. 424 ft b. 276 ft 27a. 2080 ft b. 14 s 29. J 31. 98 m 33. 1318 ft 35. 6 min 37. rhombus and square 39. rectangle, rhombus, and square 41. 4 43. 16 __ 3 8-5 Check It Out! 1a. -0.09 1b. -0.03 1c. 0.34 2a. 34.9 2b. 29° 2c. 26° 2d. 17.7 3a. 6.5 3b. 30° 3c. 7.0 3d. 65° 4. 68.6 m; 54° Exercises 1. 0.98 3. -28.64 5. -0.68 7. 0.54 9. -0.91 11. 43° 13. 44° 15. 17.3 17. -0.09 19. -1.88 21. 0.99 23. -0.87 25. 0.79 27. 20.6 29. 10.4 31. 65° 33. 33° 35. 8.4 37. 21° 39. 8.2 cm 41. 50° 43. no 45. 63° 47. 41.2 ft 49. 42°; 138° 51. Law of Sines 53. Law of Sines 55. BC ≈ 10.73; AB ≈ 10.34; m∠ABC ≈ 50° 57a. y 2 + h 2 b. b 2 c. a 2 = c 2 - 2cx + x 2 + h 2 d. a 2 = c 2 + b 2 - 2cx e. b cos A f. Subst. 59. A 61. C 63. 31° 65. 3n 67. 2n + 2 69. Alt. Int.  Thm. 71. Alt. Ext.  Thm. 73. ∠1 8-6 Check It Out! 1a. 〈-3, -4〉 1b. 〈7, 1〉 2. 3.2 3. 23°  PQ = 4a.  XY ǁ  MN 5. 4.4 mi/h; 58°, or N 32° E  RS 4b.  PQ ǁ  RS ;  XY  UV  RS =  RS =  CD =  LM 29. Exercises 1. equal 3. magnitude 5. 〈8, -8〉 7. 4.1 9. 5.8 11. 11° 13.  EF 15. 17. 4.6 mi; 20°, or N 70° E 19. 〈-3.5, 5.5〉 21. 2.0 23. 4.3 25. 36° 27.  DE = 31. 190.1 km/h; 54°, or N 36° E 33. 〈2, 2〉 35. 〈6, 2〉 37a. 98° b. 68.9 mi/h c. 36° d. N 81° E 39. 〈7.1, 1.1〉 41. 〈2.2, 5.4〉 43a. 1 __ 12 b. 1 __ 6 45. 4; 0° 47. 3.6; 56° 49. 〈0, 10〉, 〈10, 0〉; 〈10, 10〉; the magnitude of the resultant is 10 √  2 , and the direction of the resultant is tan -1 ( 10 __ 10 ) = 45°. 53. 〈3.5, 1〉; 3.6; 16° 55. 〈4, 4〉; 5.7; 45° 57a. 〈1, 3〉; 〈2, 6〉 b. √  10 ; 2 √  10 ; the magnitude  v is twice the magnitude of of 2 c. 72°; 72°; the direction of 2 same as the direction of d. Multiply each component by k. e. -  v = -1〈x, y〉 = 〈-x, -y〉 61. G 63. 8.2 65. 180° 67. 6.4 mi/h at a bearing of N 58° E 69. (2, 7) 71. (6, -1) 73. 54 cm 2 75. 73°  v .  v is the  v = -1  v . SGR 1. component form 2. equal vectors 3. geometric mean 4. angle of elevation 5. trigonometric ratio 6. △PRQ ∼ △RSQ ∼ △PSR 7. 5 8. √  51 9. x = √  35 ; y = 2 √  15 ; z = 2 √  21 10. x = 3 11. x = 5; y = √  5 ; z = √  30 12. 11.17 m 13. 6.30 m 14. 10.32 cm 15. 1.31 cm 16. m∠C = 68°; AB ≈ 4.82; AC ≈ 1.95 17. m∠H ≈ 53°; m∠G ≈ 37°; HG ≈ 5.86 18. m∠S = 40°; RS ≈ 42.43; RT ≈ 27.27 19. m∠Q ≈ 41°; m∠N ≈ 49°; QN ≈ 13.11 Selected Answers S103 S103 �� 20. angle of depression 21. angle of elevation 22. 36 ft 23. 458 m 24. 22° 25. 31.4 26. 20.1 27. 56° 28. 〈-7, 2〉 29. 〈1, -6〉 30. 〈-2, -5〉 31. 32. 5.8 33. 2 34. 5.7 35. 51° 16° 36. 641.6 mi/h; 32°, or N 58° E Chapter 9 formula for y and substitute the expression into the perimeter formula. Graph, and find the minimum value. 63. -2 ≤ y ≤ 2 65. P = 2x + 8; A = 7x __ 2 67. 〈6, 8〉 9-2 Check It Out! 1. A = (4 x 2 - 12x + 9) π m 2 2. C ≈ 31.4 in.; C ≈ 37.7 in.; C ≈ 44.0 in. 3. A ≈ 77.3 cm 2 Exercises 1. Draw a segment perpendicular to a side with one endpoint at the center. The apothem is 1 __ 2 s. 3. A = 9 x 2 π in 2 5. A ≈ 50.3 in 2 ; A ≈ 78.5 in 2 ; A ≈ 113.1 in 2 7. A ≈ 32.7 cm 2 9. A ≈ 279.9 m 2 11. C = 5π 13. A ≈ 962.1 ft 2 ; A ≈ 1963.5 f t 2 ; A ≈ 3421.2 ft 2 15. A ≈ 13.3 ft 2 17. A ≈ 14.5 ft 2 19. 90° 21. 60° 23. 45° 25. 36° 27. A ≈ 84.3 cm 2 29. A ≈ 46.8 m 2 31. A ≈ 90.8 ft 2 35. 20 √  π ___ π ; 10 √  π ___ π ; 20 √  π 37. 36; 18; 324π 39a. A ≈ 745.6 in 2 b. A ≈ 1073.6 in 2 c. 44% 43. B 45. B 47. A = C 2 ___ 4π 49. y = 3x - 13 51. m∠B = 124° 53. d 2 = 1.4 cm 9-1 9-3 Check It Out! 1. b = 0.5 yd 2. A = 96 m 2 3. d 2 = 8y m 4. P = (4 + 4 √  2 ) cm; A = 4 cm 2 Check It Out! 1. A = 1781.3 m 2 2. A ≈ 10.3 in 2 3. 23,296.5 gal 4. A ≈ 12 ft 2 Exercises 1. A ≈ 40.5 units 2 3. isosceles triangle; P = (6 + 6 √  2 ) units; A = 9 units 2 5. rectangle; P = 28 units; A = 40 units 2 7. A = 20 units 2 9. A = 6 units 2 ; P = 12 units; A = 5 units 2 ; P = 12 units 11. A ≈ 43.5 units 2 13. rhombus; P = 4 √  29 units; A = 20 units 2 15. isosceles trapezoid; P = (8 + 2 √  29 ) units; A = 20 units 2 17. A = 53 units 2 19. P = (6 + 3 √  2 ) units; A = 4.5 units 2 21a. A = 20 mi 2 b. A ≈ 150 mi 2 . The area represents the distance the boat traveled in 5 h. 23a. A = 6 units 2 b. Possible answer: C (2, 1) and H (8, 2) 25. J 27. A ≈ 10.5 units 2 29. A ≈ 17.5 units 2 31. P = 8 √  A = 2 √  2 units 2 33. -2 < a < 3 37. d = 22 ft 2 - √  2 units; 9-5 Check It Out! 1. The area is tripled. 2. The perimeter is tripled, and the area is multiplied by 9. 3. The side length is multiplied by 1 ___ . 4. Possible answer: √  2 29a. h = 31.2 in. Exercises 1. A = 120 cm 2 3. P = 52 cm 5. b = 13 in. 7. A = 336 in 2 9. d 2 = 8x y 2 cm 11. h = 1.25 m 13. A = (21 x 2 + 32x - 5) ft 2 15. h = 20 cm 17. A = 196 √  3 in 2 19. A = (12 x 2 + 34x + 20) ft 21. A = 4.5 in 2 23. A = 30 √  3 cm 2 25. A = 300 in 2 27. A = x 2 √  3 ____ 2 b. A = 561.6 in 2 c. 734.4 in 2 31. 8; 50 33. 9; 24 35. h = 5 cm 37. 9 39. 100 41. A = 108 ft 2 43a. A = 1 __ 2 (a + b) 2 b. 1 __ 2 ab; 1 __ 2 ab; 1 __ 2 c 2 c. 1 __ 2 (a + b) 2 = 1 __ 2 ab + 1 __ 2 ab + 1 __ 47a. Possible answers: A: 4.2 cm 2 ; B: 3.8 cm 2 ; C: 4.3 cm 2 b. C has the greatest area. 49. 23 cases 53. H 55. H 57. h = 4 in. 59. b = (7x + 5) cm; h = (6x + 3) cm 61a. A = x (12 - x) b. D: 0 < x < 12; R: 0 < y < 36 c. 6 ft by 6 ft d. Solve the area S104 S104 Selected Answers Exercises 3. A ≈ 16.3 ft 2 5. A = 17.5 m 2 7. A ≈ 4.5 in 2 9. A = 49.5 mm 2 11. A ≈ 2.3 m 2 13. 7 qt 15. A ≈ 9 m 2 17. A = 540 in 2 19. A = (25 √  3 + 75π ___ 2 ) in 2 21. Possible answer: 35,000 mi 2 23a. A = 675 in 2 b. c. 675 in 2 25. A = (26 + 2π) in 2 27. A = 2 29. Possible answer: A ≈ 10 c m 2 31. A 33. C 37. 15.96 39. 1.4 41. A ≈ 3.9 c m 2 9-4 Check It Out! 1. A ≈ 38 units 2 2. parallelogram; P ≈ 20.8 units 2 ; A = 25 units 2 3. A = 48 units 2 Exercises 1. The area is doubled. 3. The perimeter is tripled. The area is multiplied by 9. 5. The side length is multiplied by √  2 . 7. $147.00 9. The area is multiplied by 2 __ 3 . 11. The circumference is multiplied by 3 __ 5 . The area is multiplied by 9 __ 25 . 13. The side length is multiplied by √  3 . 15. The area is multiplied by 64. 17. The area is multiplied by �� 28. 19. The area is divided by 16. 21. The area is multiplied by 4. 23. 800,000 acres 25a. The area is multiplied by 3. b. The area is multiplied by 3 c. The area is multiplied by 9. 27a. The area is multiplied by 3. b. The area is multiplied by 3. c. The area is multiplied by 9. 29a. 8 √  2 in. b. 4 √  2 in. 31. G 33. 36 35. A = (9π x 2 + 54πx + 81π) in 2 37. t __ 2 = 36 39. 128.6°; 51.4° 41. 154.3°; 25.7° 43. A = 32 units 2 9-6 Check It Out! 1. 2 __ 3 2. 1 __ 2 3. 1 __ 2 4. 0.71 Exercises 3. 1 __ 2 5. 7 __ 10 7. 9 times 9. 3 __ 8
11. 5 __ 12 13. 0.08 15. 0.79 17. 0.78 19. 0.46 21. 0.62 23. 1 __ 2 25. 3 __ 4 27. 0.5 29. 0.11 31. A 33. 0.84 35. 0.13 37. 0.77 39–41. Possible answers given. 39. The point lies on AC. 41. The point lies in the blue triangle or the green triangle. 43. 1 __ 2 ; it does not matter which regions are shaded because they all have the same area. ____ 4 45. A 47. D 49. 4 - π ≈ 0.21 53. 4 m 10 55. By the Distance Formula, AB = 2 √  5 , AC = 2 √  5 , BC = 4, AD = 4 √  5 , AE = 4 √  5 , and DE = 8. AB ___ = 1 __ 2 , so △ABC ∼ = BC ___ DE AD △ADE by SSS. 57. A ≈ 10.6 in 2 = AC ___ AE SGR 1. apothem 2. center of a circle 3. geometric probability 4. A = 81 in 2 5. P = 22 cm 6. h = 3 x 2 in. 7. h = 8 ft 8. A = 252 yd 2 9. d 2 = 42 xy 4 in. 10. A = 288 m 2 11. C = 2 ft 12. A ≈ 153.9 yd 2 13. d = 16x m 14. A ≈ 172.0 ft 2 15. A ≈ 6.9 in 2 16. A ≈ 309.0 cm 2 17. A = 72 m 2 18. A ≈ 200.9 ft 2 19. A = 192 cm 2 20. A ≈ 21.4 mm 2 21. A ≈ 49.5 units 2 22. A ≈ 44 units 2 23. square; P = 12 √  2 units; A = 18 units 2 24. right triangle; P = (12 + √  74 ) units; A = 17.5 units 2 25. isosceles trapezoid; P = (12 + 4 √  5 ) units; A = 24 units 2 26. parallelogram; P = (8 + 2 √  13 ) units; A = 12 units 2 27. A = 30.5 units 2 28. A = 17.5 units 2 29. A = 12 units 2 30. A = 16 units 2 31. The perimeter is multiplied by 3. The area is multiplied by 9. 32. The perimeter is doubled. The area is multiplied by 4. 33. The circumference is multiplied by 1 __ 2 . The area is multiplied by 1 __ 4 . 34. The perimeter is multiplied by 4. The area is multiplied by 16. 35. 7 __ 13 36. 8 __ 13 37. 12 __ 13 38. 6 __ 13 39. 0.17 40. 0.05 41. 0.17 42. 0.66 33. 35. Chapter 10 10-1 37 a. pentagonal prism b. 2 pentagons and 5 rectangles c. ̶̶ VY , ̶̶ TV , ̶̶ UV , Check It Out! 1a. cone; vertex: N; edges: none; base: ⊙M 1b. triangular prism; vertices: T, U, ̶̶ ̶̶̶ TU , TW , V, W, X, Y; edges: ̶̶ ̶̶̶ ̶̶̶ ̶̶ UX , XY ; bases: △TUV, WY , WX , △WXY 2a. triangular pyramid 2b. cylinder 3a. hexagon 3b. triangle 4. Cut through the midpoints of 3 edges that meet at 1 vertex. ̶̶ JD , ̶̶ GF , ̶̶ HE , ̶̶ ST , ̶̶ XY , ̶̶ GK , ̶̶ DE , ̶̶̶ GH , ̶̶ CD , ̶̶̶ ̶̶ ̶̶ UV , SW , VS , ̶̶̶ ZW ; bases: Exercises 1. cylinder 3. rectangular prism; vertices: C, D, ̶̶ HJ , E, F, G, H, J, K; edges: ̶̶ ̶̶ ̶̶ ̶̶ EF ; FC , KC , JK , bases: GHJK, CDEF 5. rectangular prism 7. cube 9. pentagon 11. Cut parallel to the bases. 13. cube; vertices: S, T, U, V, W, X, ̶̶ ̶̶ TU , TX , Y, Z; edges: ̶̶ ̶̶̶ ̶̶ ̶̶ UY , YZ , WX , VZ , STUV, WXYZ 15. cylinder; vertices: none; edges: none; bases: ⊙R, ⊙Q 17. triangular pyramid 19. square 21. rectangle 23. Cut perpendicular to the ground. 25. rectangular prism 27. hexagonal prism 29. The figure is a cylinder whose bases each have a radius of 12 ft. The height of the cylinder is 9 ft. 31. 39. 41. D 43. B 45. 47. 49. 51a. A and B, C and F, D and G, E and H b. one 53. y = x 2 + 6 55. largest: ∠B; smallest: ∠C 57. largest: ∠I; smallest: ∠H 59. yes; 10:17 Selected Answers S105 S105 �� 10-2 Check It Out! 1. 11. no 13. no 15. c. 17. 19. 21. 23. yes 25. no 27. 29c. 9 31. 35. B 39. 41a. b. 2. 3a. 3b. 4. no Exercises 1. perspective 3. 5. 7. 9. S106 S106 Selected Answers 43. The first number is 20, and the second number is 10. 45. The first number is 0, and the second number is 5. 47. 2 49. 2 pentagons and 5 parallelograms 51. 4 triangles 10-3 Check It Out! 1a. V = 6; E = 12; F = 8; 6 - 12 + 8 = 2 1b. V = 7; E = 12; F = 7; 7 - 12 + 7 = 2 2. 5 √  3 ≈ 8.7 m 3. 4a. d ≈ 12.9 units; M (3, 4.5, 8.5) 4b. d ≈ 11.4 units; M (8.5, 12, 18) 5. 36.2 ft Exercises 1. because the bases are circles, which are not polygons 3. V = 6; E = 10; F = 6; 6 - 10 + 6 = 2 5. 15.0 ft 7. 17 in. 9. 11. d ≈ 14.4 units; M (2.5, 4.5, 5) 13. d ≈ 9.3 units; M (6.5, 9, 12.5) 15. V = 8; E = 12; F = 6; 8 - 12 + 6 = 2 17. V = 11; E = 20; F = 11; 11 - 20 + 11 = 2 19. h ≈ 5.3 m �� 21. 23. 45. d ≈ 2.8 units; M (2, 2, 2) 47. 25. d ≈ 10.3 units; M (5.5, 6.5, 8.5) 27. 6557 ft 29. 6 31. 12 33. V = n + 1; E = 2n; F = n + 1; (n + 1) - 2n + (n + 1) = 2 35. 6 √  3 37. 6 √  3 39. 41. 43. d ≈ 6.8 units; M (3.5, 4, 6.5) 49. d ≈ 4.6 units; M (4, 1.5, 7) 51. Possible answer: z = 9 53. Possible answer: 1.8 in. 55. AB = 11, AC = 11, and BC = 11 √  2 , so △ABC is an isosc. rt. △. 57. C 59. B 61. AB = BC = 2 √  6 , and AC = 4 √  6 , so AB + BC = AC. The points are collinear. 65. 0–9 yr old 67. A = 1 __ 2 h ( 1 __ 2 b 1 + b 2 ) 69. cone 71. ⊙C 10-4 Check It Out! 1. L = 256 cm 2 ; S = 384 cm 2 2. L = 196π in 2 ; S = 294π in 2 3. 239.7 cm 2 4. The surface area is multiplied by 1 __ 4 . 5. It will melt at about the same rate as the half cylinder. Exercises 1. 5 3. L = 24 cm 2 ; S = 36 cm 2 5. L = 24π ft 2 ; S = 42π ft 2 7. L = 80π m 2 ; S = 208π m 2 9. S ≈ 2855.0 ft 2 11. The surface area is multiplied by 4 __ 9 . 13. L = 200 cm 2 ; S = 250 cm 2 15. L = 336 ft 2 ; S ≈ 391.4 ft 2 17. L = 184π cm 2 ; S = 216π cm 2 19. S ≈ 352.0 cm 2 21. The surface area is multiplied by 9. 23. the cell that measures 35 µm by 7 µm by 10 µm 25. h = 3.5 m 27. S ≈ 121.5 units 2 29. 836.58 31. 1057.86 33. Multiply the radius and height by 1 __ 2 . 35. < 4.86 cm 2 37a. AB = 7 in.; BC = 4 √  2 in. ≈ 5.7 in. b. 4.1 in. c. 97.6 in 2 39. F 41. h = 18 cm 43. 198 cm 2 45. 70 ≤ s ≤ 110 47. 77° 49. 10-5 Check It Out! 1. L = 90 ft 2 ; S ≈ 105.6 ft 2 2. L = 80π cm 2 ; S ≈ 144π cm 2 3. The surface area is multiplied by 4 __ 9 . 4. S ≈ 28.9 yd 2 5. 9 in. Exercises 1. the vertex and the center of the base 3. L = 544 ft 2 ; S = 800 ft 2 5. L = 175π in 2 ; S = 224π in 2 7. L = 48π m 2 ; S = 84π m 2 9. The surface area is multiplied by 9. 11. S = 1056π m 2 13. L = 60 ft 2 ; S = 96 ft 2 15. L = 315 ft 2 ; S ≈ 442.3 ft 2 17. L = 444π in 2 ; S = 588π in 2 19. The surface area is divided by 9. 21. S = 287π in 2 23. 6 in. 25. 4 √  3 m 2 27. 3π ft 2 29. r = 8 m 31. P = 24 cm 33. S = 330 cm 2 35. Possible answer: 526,000 ft 2 39. F 41a. S = 500π cm 2 b. L = 100π cm 2 c. B = 25π cm 2 d. S = 500π - 100π + 25π = 425π cm 2 43a. c = 2πr b. C = 2πℓ c. c __ C larger circle is A = π ℓ 2 . The lateral surface area is c __ = r _ times the area ℓ C of the circle, so L = π ℓ 2 ( r _ ) = πrℓ 45. yes 47. 0.25 49. 0.21 51. S = 700 cm 2 = r _ d. The area of the ℓ = 2πr ___ 2πℓ ℓ 10-6 Check It Out! 1. V = 157.5 yd 3 2. 859,702 gal; 7,161,318 lb 3. V = 1088π in 3 ≈ 3418.1 in 3 4. The volume is multiplied by 8. 5. V ≈ 51.4 cm 3 Selected Answers S107 S107 �� Exercises 1. the same length as 3. V ≈ 748.2 m 3 5. 2552 gal; 12,071 lb 7. V = 45π m 3 ≈ 141.4 m 3 9. The volume is multiplied by 1 __ 64 . 11. V ≈ 1209.1 ft 3 13. V = 810 yd 3 15. V = 245 ft 3 17. V = 1764π cm 3 ≈ 5541.8 cm 3 19. V = 384π cm 3 ≈ 1206.4 cm 3 21. The volume is multiplied by 27 ___ 25a. 235.6 in 2 25b. 0.04 27. h = 11 ft 29. V = 392π m 3 31. 576 in 3 , or 1 __ 3 ft 3 33. 2,468,729 gal 37. A 39. B 41. V = x 3 + x 2 - 2x 43 __________ 4 49. 8 51. S ≈ 181.9 cm 2 125 . 23. V ≈ 242.3 ft 3 45. 139 47. 50° 10-7 Check It Out! 1. V = 36 cm 3 2. 107,800 yd 3 or 2,910,600 ft 3 3. V = 216π m 3 ≈ 678.6 m 3 4. The volume is multiplied by 8. 5. V = 3000 ft 3 Exercises 1. perpendicular 3. V = 96 cm 3 5. V ≈ 65 mm 3 7. V = 1440π in 3 ≈ 4523.9 in 3 9. The volume is multiplied by 27. 11. V = 2592 cm 3 13. V = 160 ft 3 15. V = 384 ft 3 17. V = 1107π m 3 ≈ 3477.7 m 3 19. V = 144π ft 3 ≈ 452.4 ft 3 21. The volume is multiplied by 216. 23. V = 150 ft 3 ____ 6 m 3 27. V = 240π cm 3 25. V = 25π 29. 1350 m 3 31. 166.3 cm 3 33. C = 10π √  3 cm 35. V = 1280 in 3 37. V = 17.5 units 3 39. 3 : 2 41a. 33.5 in 3 b. 134.0 in 3 c. $5; the large size holds 4 times as much. 43. H 45. 9 47. V = 2π ___ 3 ft 3 49. V = 1000 √  2 ______ c m 3 51. 38 and 14 3 53. 79 and 118 55. AA; PQ = 6 57. 10.0; (0.5, 0, -2) 59. 7; (-2, 2, 3.5) 10-8 Check It Out! 1. r = 12 ft 2. about 72.3 times as great 3. S = 2500π cm 2 4. The surface area is divided by 9. 5. S = 57π ft 2 ; V = 27π ft 3 Exercises 1. One endpoint is the center of the sphere, and the other is a point on the sphere. 3. V = 4π ___ 3 m 3 5. about 8 times as great 7. S = 196π cm 2 9. The surface area is multiplied by 4. S108 S108 Selected Answers m 3 units 3 11. S = 36π ft 2 ; V = 92π ___ 3 ft 3 13. V = 972π cm 3 15. d = 36 in. 17. S = 1764π in 2 19. V = 15,625π ______ 6 21. The volume is multiplied by 216. 23. S ≈ 1332.0 mm 2 ; V ≈ 1440.9 mm 3 25. C = 2π √  15 in. _____ 27. S = 196π units 2 ; V = 1372π 3 29. 5.28 in.; 8.87 in 2 ; 2.48 in 3 31. 7.85 in.; 19.63 in 2 ; 8.18 in 3 33. Possible answer: 14,293 in 3 35. about 1408 times as great 37. The surface area of Saturn is greater. 39. The cross section of the hemisphere is a circle with radius √  π ( r 2 - x 2 ) . The cross section of the cylinder with the cone removed has an outer radius of r and an inner radius of x, so the area is ) . 41a. 33.5 in 3 b. 44.6 in 3 43. H 45. 1 in. 47. The volume of the cylinder is 1.5 times the volume of the sphere. 49. y = x 2 + 1 51. 4.6 in 2 53. The volume is multiplied by 27 __ 64 . r 2 - x 2 , so its area is A = SGR 2. cross section 3. cone; vertex: M; edges: none; base: ⊙L 4. rectangular pyramid; vertices: N, P, Q, R, S; ̶̶ ̶̶ QR , NR , edges: ̶̶ ̶̶ SP ; base: PQRS 5. cylinder RS , 6. square pyramid 7. ̶̶̶ NQ , ̶̶ PQ , ̶̶ NP , ̶̶ NS , 8. 9. 10. 11. yes 12. no 13. V = 9; E = 16; F = 9; 9 - 16 + 9 = 2 14. V = 8; E = 12; F = 6; 8 - 12 + 6 = 2 15. d ≈ 7.7; M (4.5, 3.5, 2.5) 16. d ≈ 10.2; M (2.5, 5, 4) 17. d ≈ 2.4; M (8, 1.5, 5.5) 18. d ≈ 7.5; M (4
, 4.5, 6) 19. L ≈ 628.3 yd 2 ; S ≈ 785.4 yd 2 20. L = 100 ft 2 ; S = 150 ft 2 21. L = 126 m 2 ; S ≈ 157.2 m 2 22. L = 160 cm 2 ; S ≈ 215.1 cm 2 23. L = 630 ft 2 ; S = 855 ft 2 24. L = 175π m 2 ; S = 224π m 2 25. L = 150π in 2 ; S = 250π in 2 26. S = 800 ft 2 27. S = 448π m 2 28. V = 1080 ft 3 29. V ≈ 1651.7 cm 3 30. V = 900π in 3 31. V = 45π m 3 32. V = 112 m 3 33. V ≈ 10.4 cm 3 34. V = 120π cm 3 35. V = 48π ft 3 36. V = 512π ft 3 ____ 3 37. V ≈ 1533.3 cm 3 38. V = 500π 39. S = 144π in 2 40. d = 16 ft 41. S ≈ 338.3 cm 2 ; V ≈ 293.5 cm 3 42. S ≈ 245.0 ft 2 ; V ≈ 84.8 ft 3 m 3 Chapter 11 11-1 ̶̶ PQ , ̶̶ QR , ̶̶ Check It Out! 1. chords: ST ; secant:   ST ; tangent:   UV ; diam.: ̶̶ PS 2. radius of ⊙C: radii: 1; radius of ⊙D: 3; pt. of tangency: (2, -1) ; eqn. of tangent line: y = -1 3. 171 mi 4a. 2.1 4b. 7 ̶̶ PT , ̶̶ ST ; ̶̶ PV , ̶̶ PQ , ̶̶̶ VW ; radii: ̶̶̶ PW 13. radius Exercises 1. secant 3. congruent ̶̶ 5. chord: QS ; secant:   QS ; tangent: ̶̶ ̶̶ ̶̶   ST ; diam.: PS PR , QS ; radii: 7. radius of ⊙R: 2; radius of ⊙S: 2; pt. of tangency: (1, 2) ; eqn. of tangent line: x = 1 9. 19 11. chords: ̶̶ ̶̶̶ RS , VW ; secant:   VW ; tangent: ℓ; diam.: of ⊙C: 2; radius of ⊙D: 4; pt. of tangency: (-4, 0) ; eqn. of tangent line: x = -4 15. 413 km 17. 7 ̶̶ AC 27. 45° 19. N 21. A 23. 31. 8 33. 22 35a. rect.; ∠BCD and ∠EDC are rt.  because a line tangent to a ⊙ is ⊥ to a radius. It is given that ∠DEB is a rt. ∠. ∠CBE must also be a rt. ∠ because the sum of the  of a quad. is 360°. Thus BCDE has 4 rt.  and is a rect. b. 17 in.; 2 in. c. 17.1 in. 39. G 43. 18.6 in. 45. 3 __ 4 47. 13 __ 20 ̶̶ AC 25. �� 11-2 Check It Out! 1a. 108° 1b. 270° 1c. 36° 2a. 140° 2b. 295° 3a. 12 3b. 100° 4. 34.6 Exercises 1. semicircle 3. major arc 5. 162° 7. 61.2° 9. 39.6° 11. 129° 13. 108° 15. 24 17. 24.0 19. 136.3° 21. 136.3° 23. 223.7° 25. 152° 27. 155° 29. 147° 31. 6.6 33. F 35. T 37. 45°; 60°; 75° 39. 108° 41. 1. ⁀ BC ≅ ⁀ DE (Given) 2. m ⁀ BC = m ⁀ DE (Def. of ≅ arcs) 3. m∠BAC = m∠DAE (Def. of arc measures) 4. ∠BAC ≅ ∠DAE (Def. of ≅ ) ̶̶ JK is the ⊥ bisector of ̶̶̶ GH . 43. 1. (Given) 2. A is equidistant from G and H. (Def. of center of ⊙) 3. A lies on the ⊥ bisector of (⊥ Bisector Thm.) 4. (Def. of diam.) ̶̶ JK is a diam. of ⊙A. ̶̶̶ GH . 45. Solution A 47a. 13.5 in.; 6.5 in. b. 11.8 in. c. 23.7 in. 49. F 51. 48.2° 53a. 90°; 60°; 45° b. 3 __ 4 π; 3 __ 2 π 55. b 3 __ 16 57. 31 59. 28 61. 9 11-3 Check It Out! 1a. π __ 4 m 2 ; 0.79 m 2 1b. 25.6π in 2 ; 80.42 in 2 2. 203,575 ft 2 3. 4.57 m 2 4a. 4 __ 3 π m; 4.19 m 4b. 3π cm; 9.42 cm Exercises 1. seg. 3. 24π cm 2 ; 75.40 cm 2 5. 12 mi 2 7. 36.23 m 2 9. 4π ft; 12.57 ft 11. 2 __ 3 π in; 2.09 in. 13. 45 __ 2 π in 2 ; 70.69 in 2 15. 628 in 2 17. 15.35 in 2 19. 25 __ 18 π mm; 4.36 mm 21. 1 __ 10 π ft; 0.31 ft 23. N 25. A 27. 12 29a. 3.9 ft b. 103° 33. G 35. 7 __ 3 π 37a. 1 __ 8 b. 3 __ 8 c. 1 __ 2 39. neither 41. 36π cm 3 43. 122° 45. 302° 11-4 Check It Out! 1a. 270° 1b. 38° 2. 43°; 120° 3a. 12 3b. 39° 4. 51°; 129°; 72°; 108° Exercises 1. inscribed 3. 58° 5. 26° 7. 112.5 9. 46° 11. 70°; 110°; 115°; 65° 13. 47.5° 15. 47.6° 17. ±6 19. 100° 21. 100°; 39°, 80°; 141° 23. A 25. S 27. 115° 29a. 30° b. 120° c. Rt.; ∠FBC is inscribed in a semicircle, so it must be a rt. ∠; therefore △FBC is a rt. △. 33. 72°; 99°; 108°; 81° 35a. AB 2 + AC 2 = BC 2 , so by the Conv. of the Pyth. Thm., △ABC is a rt. △ with rt. ∠A. Since ∠A is an inscribed rt. ∠, it intercepts a semicircle. This means ̶̶ BC is a diam. b. 120° 39. D that 41. C 45. 133° 49. 13 __ 7 51. 5 __ 2 53. 3 m 2 11-5 Check It Out! 1a. 83° 1b. 142° 2a. 51° 2b. 22° 3. 33 4. 45° 5. 72° ̶̶ AB is ̶̶ AB is a diam. Exercises 1. 70° 3. 122° 5. 67° 7. 94° 9. 58 11. 142° 13. 96° 15. 116° 17. 124° 19. 260° 21. 107.5° 23. 57.5 25. 18 27. 45° 29. 90° 31. 2x° 33. (360 - 2x) ° 39. 150°; 30°; 35° 41a. 60° b. 120° c. obtuse isosceles 43. J 45. Case 1: Assume of the circle. Then m ⁀ AB = 180°, and ∠ABC is a rt. ∠. Thus m∠ABC = 1 __ 2 m ⁀ AB . Case 2: Assume not a diam. of the ⊙. Let X be the ̶̶ center of the ⊙ and draw radii XA ̶̶ ̶̶ XB . Since they are radii, XA and ̶̶ ≅ XB , so △AXB is isosceles. Thus ∠XAB ≅ ∠XBA and 2m∠XBA + m∠AXB = 180. This means that m∠XBA = 90 - 1 __ 2 m∠AXB. By Thm. 11-1-1, ∠XBC is a rt. ∠, so m∠XBA + m∠ABC = 90 or m∠ABC = 90 - m∠XBA. By subst., m∠ABC = 90 - (90 - 1 __ 2 m∠AXB) . Simplifying gives m∠ABC = 1 __ 2 m∠AXB. m∠AXB = m ⁀ AB because ∠AXB is a central ∠. Thus m∠ABC = 1 __ 2 ⁀ AB . 47. 95° 49. yes 51. no 53. 96π cm 3 ≈ 301.6 cm 3 55. 37° 57. 53° 11-6 Check It Out! 1. 3.75; AB = 11; CD = 11.75 2. 3 2 __ 3 in. 3. z = 14; JG = 27; LG = 39 4. 7 2 __ 7 Exercises 1. tangent seg. 3. x = 9; AB = 13; CD = 12 5. 51 1 __ 4 ft 7. y = 10.6; PR = 15.6; PT = 13 9. 4 11. √  33 13. x = 4.2; JL = 14.2; MN = 13 15. ≈ 1770 ft 17. y = 14.3; HL = 24.3; NL = 27 19. 2 √  21 21. 4 √  10 23a. 6 in. b. 12 in. 25. x = 8; y = 6 √  3 27. Solution B 33. B 35. CE = ED = 6 and by the Chord-Chord Product Thm., 6 · 6 = 3 · EF. So EF = 12 , FB = 15, and the radius AB must be 7.5. 37. 3.2 in. 39. 7.44 41. 72% 43. 45. 22π ft 2 ; 69.12 ft 2 47. 45°  CD 11-7 Check It Out! 1a. x 2 + (y + 3) 2 2 = 64 1b. (x - 2) 2a. + (y + 1) 2 = 16 2b. 3. (2, - 1) Exercises 1. (x - 3) 2 + (y + 5) 2 = 144 3. (x - 4) 2 + y 2 = 4 5. 7. 9a. (-2, 3) b. 10 ft 11. (x - 1.5) 2 + (y + 2.5) 2 = 3 13. (x - 1) 2 + (y + 2) 2 = 45 15. 17. 19. (x - 1) 2 + (y + 2) 2 = 4 21a. 80 ft b. x 2 + y 2 = 1600 23. T 25. T 29a. E (-3, -1) ; G (-6, 2) b. 6 c. (x + 3) 2 + (y - 2) 2 = 9 31. (0, -15) ; 5 33. A = 9π; C = 6π 35. A = 25π; C = 10π 37. (-200, -100) 39. (x - 1) 2 + (y + 2) 2 = 16 43. H 45a. (x - 2) 2 + (y + 4) 2 + (z - 3) 2 = 69 b. 15; if 2 segs. are tangent to a ⊙ or sphere from the same ext. pt., then the segs. are ≅. 47. (x - 3) 2 + (y - 4) 2 = 5 49. 9a + 2 51. 8 53. 196° Selected Answers S109 S109 �� 9. 11. 13. no 15. yes 17. 19. 21. 23. 27. 29. 31. (5, 2) → (5, -2) 33. (0, 12) → (0, -12) 35. (0, -5) → (-5, 0) 37a. no b. (7, 4) c. (6, 3.5) 39. y = x 41. 47. J 49. (4, 4) 51. (-1, 5) 53. Use the fact that the reflection of a seg. is ≅ to the preimage and the def. of ≅ segs. 55. Use the fact that the reflection of a seg. is ≅ to the preimage to prove △ABC ≅ △A′B′C′ by SSS. 59. 25 __ 36 61. 11 cm 63. 13.2 cm 65. 41° 12-2 Check It Out! 1a. Yes 1b. no 2. 3. 4. (16, -24) Exercises 1. no 3. yes 7. 9. 11. yes 13. no SGR ̶̶ UV ; tangent: ̶̶ QS , ̶̶ PS ; secant:   UV ; 1. segment of a circle 2. central angle 3. major arc 4. concentric circles 5. chords: ̶̶ PQ , ℓ; radii: ̶̶ ̶̶̶ ̶̶ QS 6. chords: MN ; KH , diam.: ̶̶ ̶̶ ̶̶ ̶̶ tangent:   KL ; radii: JM , JK , JH , JN ; ̶̶ ̶̶̶ KH 7. 25 secant:   MN ; diams.: MN , 8. 12 9. 7 10. 1.8 11. 81° 12. 210° 13. 99° 14. 279° 15. 17.0 16. 8.7 17. 12π in 2 ; 37.70 in 2 18. π __ 4 m 2 ; 0.79 m 2 19. 16π cm; 50.27 cm 20. 3π ft; 9.42 ft 21. 164° 22. 32° 23. 26 24. 39° 25. 82° 26. 79° 27. 67° 28. 90° 29. 11 2 __ 3 ; DE = 12; BC = 14 2 __ 3 30. 12; RQ = 22; ST = 23 31. 7; JG = 10; JL = 12 32. 8 1 __ 2 ; AC = 12 1 __ 2 ; AE = 10 33. (x + 4) 2 + (y + 3) 2 = 9 34. (x + 2) 2 + y 2 = 4 35. (x - 1) 2 + (y + 1) 2 = 16 36. Chapter 12 12-1 Check It Out! 1a. no 1b. yes 2. ̶̶ AX and ̶̶ BX would be ≅. 3. 4. Exercises 1. They are ≅. 3. no 5. no 7. S110 S110 Selected Answers �� Exercises 1. yes 3. no 5. 7. 9. 11. (8.7, 5) 13. yes 15. no around the center of rotation by the same ∠, pts. that are farther from the center of rotation move a greater distance than pts. that are closer to the center of rotation. 39. A′ (-2, 3) , B′ (-3, 0) , C′ (0, -3) , D′ (3, 0) , E′ (2, 3) , 43. H 45. 160° 47. Use the fact that the rotation of a seg. is ≅ to the preimage and the def. of ≅ segs. 49. Use the fact that the rotation of a seg. is ≅ to the preimage to prove △ABC ≅ △A′B′C′ by SSS. 51. If A, B, and C are collinear, then one pt. is between the other two. Case 1: If C is between A and B, then AC + BC = AB. Use the fact that the rotation of a seg. is ≅ to the preimage to prove A′C′ + B′C′ = A′B′. Then C′ is between A′ and B′, so A′, B′, and C′ are collinear. Prove the other two cases similarly. 53. - 3 __ 2 , 4 55. 94° 57. 〈4, -9〉 59. 〈3, 2〉 12-4 Check It Out! 1. 21. 17. 19. 21. 23a. 1 __ 4 b. 1 __ 2 . c. 0 27. No; there are no fixed pts. because, by def. of a translation, every pt. must move by the same distance. 29. 〈4, 0〉, (-3, 2) → (1, 2) 31. (-3, -2) , (3, -1) , (0, -3) 33. (-3, 1) , (3, -1) → (0, 0) 39. A 41. C 43a. the vector b. 3.46 cm 45. Use the fact that the reflection of a seg. is ≅ to the preimage and the def. of ≅ segs. 47. Use the fact that the translation of a seg. is ≅ to the preimage to prove △ABC ≅ △A′B′C′ by SSS. 51. (4, 5) 53. x = 15, y = 5 55. M′ (-2, 0) , N′ (-3, -2) , P′ (0, -4) 57. M′ (0, -2) , N′ (2, -3) , P′ (4, 0)  PQ 12-3 Check It Out! 1a. no 1b. yes 2. 17. 19. 23. 25. 2. a translation in direction ⊥ to n and p, by distance of 6 in. 3. Exercises 1. Draw a figure and translate it along a vector. Then reflect the image across a line. 3. 3. ̶̶ ST 31a. 72° 27. T 29. 31b. (4.9, 5.9) 33. 4. (20.9, 64.2) 35a. 90° 35b. 6 hours 37. No; although all pts. are rotated 5. a rotation of 100° about the pt. of intersection of the lines Selected Answers S111 S111 �� 7. 9. 11a. The move is a horiz. or vert. translation by 2 spaces followed by a vert. or horiz. translation by 1 space. 11b. 1b. yes; 1 line of symmetry 1c. yes; 1 line of symmetry 35. line symmetry; x = 2 37a. no b. yes; 180°; 2. c. Yes; if color is not taken into account the ∠ of rotational symmetry is 90. 39. parallelogram 41. square 43. 15° 45. It has rotational symmetry of order
3, with an ∠ of rotational symmetry of 120°. 47. 2a. yes; 120°; order: 3 2b. yes; 180°; order: 2° 2c. no 3a. line symmetry and rotational symmetry; 72°; order: 49. 51. A 53. C 55. 72 57. x = -4 59. x = 0 61. 7 63. $246.40 65. 5 cm 67. P′ (6, -5) 69. P′ (0,-4) 3b. line symmetry; 51.4°; order: 7 4a. both 4b. neither 12-6 11c. 13. Exercises 1. The line of symmetry is the ⊥ bisector of the base. 3. yes; 2 lines of symmetry 5. no 7. no 9. 72°; order: 5 11. both 13. yes; 1 line of symmetry 15. no 17. yes; 72°; order: 5 19. 90; order: 4 21. neither 23. isosc. 25. scalene 27. 0 29. line symmetry 17. never 19. always 23. A 25. C 29. yes 31. 6.4 33. 8 35. N′ (1, 3) 12-5 31. line symmetry Check It Out! 1a. yes; 2 lines of symmetry 33. rotational symmetry of order 4 S112 S112 Selected Answers Check It Out! 1a. translation symmetry 1b. translation symmetry and glide reflection symmetry 2. 3a. regular 3b. neither. 3c. semiregular 4a. yes 4b. no Exercises 3. translation symmetry and glide reflection symmetry 5. translation symmetry and glide reflection symmetry 9. regular 11. semiregular 13. yes; possible answer 15. translation symmetry 17. translation symmetry 19. 21. neither 23. neither 25. no 27. translation symmetry and glide reflection symmetry �� 29. translation, reflection, rotation 31. always 33. always 35. never 41. The tessellation has translation symmetry, reflection symmetry, and order 3 rotation symmetry. 43. 21. 23. 11. 12. 47. H 51. yes 53. 7.5% 55. (x + 2) 2 + (y - 3) 2 = 5 57. - (x - 5) 2 + (y + 3) 2 = 20 59. angle of rotational symmetry: 72°; order: 5 12-7 Check It Out! 1a. no 1b. yes 2. 3. 1600 in 2 4. Exercises 1. The center is the origin; the scale factor is 3. 3. yes 5. yes 7. 9. 11. 13. yes 15. no 19. 108 25. ABCDE ∼ MNPQR 27. 13. no 14. yes 15. no 16. no 17. 29. 31. B 35. -4.5 × 10 -12 37. k = -2; A′ (4, -4) , B′ (-2, -6) 39. k = 1 and k = -1 47. H 49. no 51. y = -x + 6 53. P = 24 units; A = -28 uni ts 2 55. yes SGR 1. reg. tessellation 2. frieze pattern 3. isometry 4. composition of transformations 5. yes 6. no 7. no 8. yes 9. 18. 19. 20. 10. 21. yes 22. yes 23. no 24. no 25. Selected Answers S113 S113 �� 26. 27. 28. 29. 30. 40. 41. neither 42. semiregular 43. yes 44. yes 45. 31. yes 32. yes 46. 33. yes; 120°; 3 34. no 35. yes; 120°; 3 36. yes; 180°; 2 37. 38. 39. S114 S114 Selected Answers �� Glossary/Glosario AA A ENGLISH acute angle (p. 21) An angle that measures greater than 0° and less than 90°. SPANISH ángulo agudo Ángulo que mide más de 0° y menos de 90°. EXAMPLES acute triangle (p. 216) A triangle with three acute angles. triángulo acutángulo Triángulo con tres ángulos agudos. adjacent angles (p. 28) Two angles in the same plane with a common vertex and a common side, but no common interior points. ángulos adyacentes Dos ángulos en el mismo plano que tienen un vértice y un lado común pero no comparten puntos internos. adjacent arcs (p. 757) Two arcs of the same circle that intersect at exactly one point. arcos adyacentes Dos arcos del mismo círculo que se cruzan en un punto exacto. alternate exterior angles (p. 147) For two lines intersected by a transversal, a pair of angles that lie on opposite sides of the transversal and outside the other two lines. ángulos alternos externos Dadas dos rectas cortadas por una transversal, par de ángulos no adyacentes ubicados en los lados opuestos de la transversal y fuera de las otras dos rectas. alternate interior angles (p. 147) For two lines intersected by a transversal, a pair of nonadjacent angles that lie on opposite sides of the transversal and between the other two lines. ángulos alternos internos Dadas dos rectas cortadas por una transversal, par de ángulos no adyacentes ubicados en los lados opuestos de la transversal y entre las otras dos rectas. altitude of a cone (p. 690) A segment from the vertex to the plane of the base that is perpendicular to the plane of the base. altura de un cono Segmento que se extiende desde el vértice hasta el plano de la base y es perpendicular al plano de la base. altitude of a cylinder (p. 680) A segment with its endpoints on the planes of the bases that is perpendicular to the planes of the bases. altura de un cilindro Segmento con sus extremos en los planos de las bases que es perpendicular a los planos de las bases. ∠1 and ∠2 are adjacent angles. ⁀ RS and ⁀ ST are adjacent arcs. ∠4 and ∠5 are alternate exterior angles. ∠3 and ∠6 are alternate interior angles. Glossary/Glosario S115 S115 �� EXAMPLES ENGLISH altitude of a prism (p. 680) A segment with its endpoints on the planes of the bases that is perpendicular to the planes of the bases. altitude of a pyramid (p. 689) A segment from the vertex to the plane of the base that is perpendicular to the plane of the base. SPANISH altura de un prisma Segmento con sus extremos en los planos de las bases que es perpendicular a los planos de las bases. altura de una pirámide Segmento que se extiende desde el vértice hasta el plano de la base y es perpendicular al plano de la base. altitude of a triangle (p. 316) A perpendicular segment from a vertex to the line containing the opposite side. altura de un triángulo Segmento perpendicular que se extiende desde un vértice hasta la recta que forma el lado opuesto. ambiguous case of the Law of Sines (p. 556) If two sides and a nonincluded angle of a triangle are given in order to solve the triangle using the Law of Sines, it is possible to have two different answers. angle (p. 20) A figure formed by two rays with a common endpoint. caso ambiguo de la Ley de los senos Si se conocen dos lados y un ángulo no incluido de un triángulo y se quiere resolver el triángulo aplicando la Ley de los senos, es posible obtener dos respuestas diferentes. ángulo Figura formada por dos rayos con un extremo común. angle bisector (p. 23) A ray that divides an angle into two congruent angles. bisectriz de un ángulo Rayo que divide un ángulo en dos ángulos congruentes. JK is an angle bisector of ∠LJM.  angle of depression (p. 544) The angle formed by a horizontal line and a line of sight to a point below. ángulo de depresión Ángulo formado por una recta horizontal y una línea visual a un punto inferior. angle of elevation (p. 544) The angle formed by a horizontal line and a line of sight to a point above. ángulo de elevación Ángulo formado por una recta horizontal y una línea visual a un punto superior. angle of rotation (p. 840) An angle formed by a rotating ray, called the terminal side, and a stationary reference ray, called the initial side. ángulo de rotación Ángulo formado por un rayo rotativo, denominado lado terminal, y un rayo de referencia estático, denominado lado inicial. angle of rotational symmetry (p. 857) The smallest angle through which a figure with rotational symmetry can be rotated to coincide with itself. ángulo de simetría de rotación El ángulo más pequeño alrededor del cual se puede rotar una figura con simetría de rotación para que coincida consigo misma. The angle of rotation is 135°. S116 S116 Glossary/Glosario �� ENGLISH annulus (p. 612) The region between two concentric circles. SPANISH corona circular Región comprendida entre dos círculos concéntricos. EXAMPLES apothem (p. 601) The perpendicular distance from the center of a regular polygon to a side of the polygon. apotema Distancia perpendicular desde el centro de un polígono regular hasta un lado del polígono. arc (p. 756) An unbroken part of a circle consisting of two points on the circle, called the endpoints, and all the points on the circle between them. arco Parte continua de un círculo formada por dos puntos del círculo denominados extremos y todos los puntos del círculo comprendidos entre éstos. arc length (p. 766) The distance along an arc measured in linear units. longitud de arco Distancia a lo largo de un arco medida en unidades lineales. arc marks (p. 22) Marks used on a figure to indicate congruent angles. marcas de arco Marcas utilizadas en una figura para indicar ángulos congruentes. m ⁀ CD = 5π ft area (p. 36) The number of nonoverlapping unit squares of a given size that will exactly cover the interior of a plane figure. área Cantidad de cuadrados unitarios de un determinado tamaño no superpuestos que cubren exactamente el interior de una figura plana. arrow notation (p. 50) A symbol used to describe a transformation. notación de flecha Símbolo utilizado para describir una transformación. The area is 10 square units. auxiliary line (p. 223) A line drawn in a figure to aid in a proof. recta auxiliar Recta dibujada en una figura como ayuda en una demostración. axiom (p. 7) See postulate. axioma Ver postulado. axis of a cone (p. 690) The segment with endpoints at the vertex and the center of the base. eje de un cono Segmento cuyos extremos se encuentran en el vértice y en el centro de la base. Glossary/Glosario S117 S117 �� ENGLISH axis of a cylinder (p. 681) The segment with endpoints at the centers of the two bases. SPANISH eje de un cilindro Segmentos cuyos extremos se encuentran en los centros de las dos bases. EXAMPLES axis of symmetry (p. 858) A line that divides a plane figure or a graph into two congruent reflected halves. eje de simetría Línea que divide una figura plana o una gráfica en dos mitades reflejadas congruentes. B base angle of a trapezoid (p. 429) One of a pair
of consecutive angles whose common side is a base of the trapezoid. ángulo base de un trapecio Uno de los dos ángulos consecutivos cuyo lado en común es la base del trapecio. base angle of an isosceles triangle (p. 273) One of the two angles that have the base of the triangle as a side. ángulo base de un triángulo isósceles Uno de los dos ángulos que tienen como lado la base del triángulo. base of a cone (p. 654) The circular face of the cone. base de un cono Cara circular del cono. base of a cylinder (p. 654) One of the two circular faces of the cylinder. base de un cilindro Una de las dos caras circulares del cilindro. base of a geometric figure (p. 429) A side of a polygon; a face of a three-dimensional figure by which the figure is measured or classified. base de una figura geométrica Lado de un polígono; cara de una figura tridimensional por la cual se mide o clasifica la figura. base of a prism (p. 654) One of the two congruent parallel faces of the prism. base de un prisma Una de las dos caras paralelas y congruentes del prisma. base of a pyramid (p. 654) The face of the pyramid that is opposite the vertex. base de una pirámide Cara de la pirámide opuesta al vértice. base of a trapezoid (p. 429) One of the two parallel sides of the trapezoid. base de un trapecio Uno de los dos lados paralelos del trapecio. base of a triangle (p. 36) Any side of a triangle. base de un triángulo Cualquier lado de un triángulo. S118 S118 Glossary/Glosario �� ENGLISH base of an isosceles triangle (p. 273) The side opposite the vertex angle. SPANISH base de un triángulo isósceles Lado opuesto al ángulo del vértice. EXAMPLES bearing (p. 252) Indicates direction. The number of degrees in the angle whose initial side is a line due north and whose terminal side is determined by a clockwise rotation. rumbo Indica dirección. La cantidad de grados en el ángulo cuyo lado inicial es una línea recta en dirección norte y cuyo lado terminal se determina por una rotación en el sentido de las agujas del reloj. between (p. 14) Given three points A, B, and C, B is between A and C if and only if all three of the points lie on the same line, and AB + BC = AC. entre Dados tres puntos A, B y C, B está entre A y C si y sólo si los tres puntos se encuentran en la misma línea y AB + BC = AC. biconditional statement (p. 96) A statement that can be written in the form “p if and only if q.” enunciado bicondicional Enunciado que puede expresarse en la forma “p si y sólo si q”. A figure is a triangle if and only if it is a three-sided polygon. bisect (p. 15) To divide into two congruent parts. trazar una bisectriz Dividir en dos partes congruentes. JK bisects ∠LJM.  C Cartesian coordinate system (p. 808) See coordinate plane. sistema de coordenadas cartesianas Ver plano cartesiano. center of a circle (p. 600) The point inside a circle that is the same distance from every point on the circle. centro de un círculo Punto dentro de un círculo que se encuentra a la misma distancia de todos los puntos del círculo. center of a regular polygon (p. 601) The point that is equidistant from all vertices of the regular polygon. centro de un polígono regular Punto equidistante de todos los vértices del polígono regular. center of a sphere (p. 714) The point inside a sphere that is the same distance from every point on the sphere. centro de una esfera Punto dentro de una esfera que está a la misma distancia de cualquier punto de la esfera. center of dilation (p. 873) The intersection of the lines that connect each point of the image with the corresponding point of the preimage. centro de dilatación Intersección de las líneas que conectan cada punto de la imagen con el punto correspondiente de la imagen original. Glossary/Glosario S119 S119 �� ENGLISH center of rotation (p. 840) The point around which a figure is rotated. SPANISH centro de rotación Punto alrededor del cual rota una figura. EXAMPLES central angle of a circle (p. 756) An angle whose vertex is the center of a circle. ángulo central de un círculo Ángulo cuyo vértice es el centro de un círculo. central angle of a regular polygon (p. 601) An angle whose vertex is the center of the regular polygon and whose sides pass through consecutive vertices. ángulo central de un polígono regular Ángulo cuyo vértice es el centro del polígono regular y cuyos lados pasan por vértices consecutivos. centroid of a triangle (p. 314) The point of concurrency of the three medians of a triangle. Also known as the center of gravity. centroide de un triángulo Punto donde se encuentran las tres medianas de un triángulo. También conocido como centro de gravedad. chord (p. 746) A segment whose endpoints lie on a circle. cuerda Segmento cuyos extremos se encuentran en un círculo. The centroid is P. circle (p. 600) The set of points in a plane that are a fixed distance from a given point called the center of the circle. circle graph (p. 755) A way to display data by using a circle divided into non-overlapping sectors. círculo Conjunto de puntos en un plano que se encuentran a una distancia fija de un punto determinado denominado centro del círculo. gráfica circular Forma de mostrar datos mediante un círculo dividido en sectores no superpuestos. circumcenter of a triangle (p. 307) The point of concurrency of the three perpendicular bisectors of a triangle. circuncentro de un triángulo Punto donde se cortan las tres mediatrices de un triángulo. circumference (p. 37) The distance around the circle. circunferencia Distancia alrededor del círculo. The circumcenter is P. S120 S120 Glossary/Glosario �� ENGLISH circumscribed circle (p. 308) Every vertex of the polygon lies on the circle. SPANISH círculo circunscrito Todos los vértices del polígono se encuentran sobre el círculo. EXAMPLES circumscribed polygon (p. 599) Each side of the polygon is tangent to the circle. polígono circunscrito Todos los lados del polígono son tangentes al círculo. coincide (p. 221) To correspond exactly; to be identical. coincidir Corresponder exactamente, ser idéntico. collinear (p. 6) Points that lie on the same line. colineal Puntos que se encuentran sobre la misma línea. common tangent (p. 748) A line that is tangent to two circles. tangente común Recta que es tangente a dos círculos. K, L, and M are collinear points. complement of an angle (p. 29) The sum of the measures of an angle and its complement is 90°. complemento de un ángulo La suma de las medidas de un ángulo y su complemento es 90°. The complement of a 53° angle is a 37° angle. complement of an event (p. 628) All outcomes in the sample space that are not in an ̶ E . event E, denoted complemento de un suceso Todos los resultados en el espacio muestral que no están en el ̶ E . suceso E y se expresan In the experiment of rolling a number cube, the complement of rolling a 3 is rolling a 1, 2, 4, 5, or 6. complementary angles (p. 29) Two angles whose measures have a sum of 90°. ángulos complementarios Dos ángulos cuyas medidas suman 90°. component form (p. 559) The form of a vector that lists the vertical and horizontal change from the initial point to the terminal point. forma de componente Forma de un vector que muestra el cambio horizontal y vertical desde el punto inicial hasta el punto terminal. The component form of  CD is 〈2, 3〉. composite figure (p. 600) A plane figure made up of triangles, rectangles, trapezoids, circles, and other simple shapes, or a three-dimensional figure made up of prisms, cones, pyramids, cylinders, and other simple threedimensional figures. figura compuesta Figura plana compuesta por triángulos, rectángulos, trapecios, círculos y otras formas simples, o figura tridimensional compuesta por prismas, conos, pirámides, cilindros y otras figuras tridimensionales simples. Glossary/Glosario S121 S121 �� ENGLISH SPANISH EXAMPLES composition of transformations (p. 848) One transformation followed by another transformation. composición de transformaciones Una transformación seguida de otra transformación. compound statement (p. 128) Two statements that are connected by the word and or or. enunciado compuesto Dos enunciados unidos por la palabra y u o. The sky is blue and the grass is green. I will drive to school or I will take the bus. concave polygon (p. 383) A polygon in which a diagonal can be drawn such that part of the diagonal contains points in the exterior of the polygon. polígono cóncavo Polígono en el cual se puede trazar una diagonal tal que parte de la diagonal contiene puntos ubicados fuera del polígono. concentric circles (p. 747) Coplanar circles with the same center. círculos concéntricos Círculos coplanares que comparten el mismo centro. conclusion (p. 81) The part of a conditional statement following the word then. conclusión Parte de un enunciado condicional que sigue a la palabra entonces. If x + 1 = 5, then x = 4. Conclusion concurrent (p. 307) Three or more lines that intersect at one point. concurrente Tres o más líneas rectas que se cortan en un punto. conditional statement (p. 81) A statement that can be written in the form “if p, then q,” where p is the hypothesis and q is the conclusion. enunciado condicional Enunciado que se puede expresar como “si p, entonces q”, donde p es la hipótesis y q es la conclusión. cone (p. 654) A three-dimensional figure with a circular base and a curved lateral surface that connects the base to a point called the vertex. cono Figura tridimensional con una base circular y una superficie lateral curva que conecta la base con un punto denominado vértice. congruence statement (p. 231) A statement that indicates that two polygons are congruent by listing the vertices in the order of correspondence. enunciado de congruencia Enunciado que
indica que dos polígonos son congruentes enumerando los vértices en orden de correspondencia. congruence transformation (p. 824) See isometry. transformación de congruencia Ver isometría. congruent (p. 13) Having the same size and shape, denoted by ≅. congruente Que tiene el mismo tamaño y forma, expresado por ≅. S122 S122 Glossary/Glosario If x + 1 = 5, then x = 4. Hypothesis Conclusion △HKL ≅ △YWK ̶̶ PQ ≅ ̶̶ SR �� ENGLISH congruent angles (p. 22) Angles that have the same measure. SPANISH ángulos congruentes Ángulos que tienen la misma medida. EXAMPLES ∠ABC ≅ ∠DEF congruent arcs (p. 757) Two arcs that are in the same or congruent circles and have the same measure. arcos congruentes Dos arcos que se encuentran en el mismo círculo o en círculos congruentes y que tienen la misma medida. congruent circles (p. 747) Two circles that have congruent radii. círculos congruentes Dos círculos que tienen radios congruentes. congruent polygons (p. 231) Two polygons whose corresponding sides and angles are congruent. polígonos congruentes Dos polígonos cuyos lados y ángulos correspondientes son congruentes. congruent segments (p. 13) Two segments that have the same length. segmentos congruentes Dos segmentos que tienen la misma longitud. conjecture (p. 74) A statement that is believed to be true. conjetura Enunciado que se supone verdadero. ̶̶ PQ ≅ ̶̶ SR A sequence begins with the terms 2, 4, 6, 8, 10. A reasonable conjecture is that the next term in the sequence is 12. conjunction (p. 128) A compound statement that uses the word and. conjunción Enunciado compuesto que contiene la palabra y. 3 is less than 5 AND greater than 0. consecutive interior angles (p. 147) See same-side interior angles. ángulos internos consecutivos Ver ángulos internos del mismo lado. construction (p. 14) A method of creating a figure that is considered to be mathematically precise. Figures may be constructed by using a compass and straightedge, geometry software, or paper folding. construcción Método para crear una figura que es considerado matemáticamente preciso. Se pueden construir figuras utilizando un compás y una regla, un programa de computación de geometría o plegando papeles. contraction (p. 873) See reduction. contracción Ver reducción. contrapositive (p. 83) The statement formed by both exchanging and negating the hypothesis and conclusion of a conditional statement. contrapuesto Enunciado que se forma al intercambiar y negar la hipótesis y la conclusión de un enunciado condicional. Statement: If n + 1 = 3, then n = 2 Contrapositive: If n ≠ 2, then n + 1 ≠ 3 converse (p. 83) The statement formed by exchanging the hypothesis and conclusion of a conditional statement. expresión recíproca Enunciado que se forma intercambiando la hipótesis y la conclusión de un enunciado condicional. Statement: If n + 1 = 3, then n = 2 Converse: If n = 2, then n + 1 = 3 Glossary/Glosario S123 S123 �� ENGLISH convex polygon (p. 383) A polygon in which no diagonal contains points in the exterior of the polygon. SPANISH EXAMPLES polígono convexo Polígono en el cual ninguna diagonal contiene puntos fuera del polígono. coordinate (p. 13) A number used to identify the location of a point. On a number line, one coordinate is used. On a coordinate plane, two coordinates are used, called the x-coordinate and the y-coordinate. In space, three coordinates are used, called the x-coordinate, the y-coordinate, and the z-coordinate. coordenada Número utilizado para identificar la ubicación de un punto. En una recta numérica se utiliza una coordenada. En un plano cartesiano se utilizan dos coordenadas, denominadas coordenada x y coordenada y. En el espacio se utilizan tres coordenadas, denominadas coordenada x, coordenada y y coordenada z. coordinate plane (p. 43) A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis. plano cartesiano Plano dividido en cuatro regiones por una línea horizontal denominada eje x y una línea vertical denominada eje y. The coordinate of point A is 3. The coordinates of point B are (1, 4) . coordinate proof (p. 267) A style of proof that uses coordinate geometry and algebra. prueba de coordenadas Tipo de demostración que utiliza geometría de coordenadas y álgebra. coplanar (p. 6) Points that lie in the same plane. coplanar Puntos que se encuentran en el mismo plano. corollary (p. 224) A theorem whose proof follows directly from another theorem. corolario Teorema cuya demostración proviene directamente de otro teorema. corresponding angles of lines intersected by a transversal (p. 147) For two lines intersected by a transversal, a pair of angles that lie on the same side of the transversal and on the same sides of the other two lines. ángulos correspondientes de líneas cortadas por una transversal Dadas dos rectas cortadas por una transversal, el par de ángulos ubicados en el mismo lado de la transversal y en los mismos lados de las otras dos rectas. ∠1 and ∠3 are corresponding. corresponding angles of polygons (p. 231) Angles in the same position in two different polygons that have the same number of angles. ángulos correspondientes de los polígonos Ángulos que tienen la misma posición en dos polígonos diferentes que tienen el mismo número de ángulos. corresponding sides of polygons (p. 231) Sides in the same position in two different polygons that have the same number of sides. lados correspondientes de los polígonos Lados que tienen la misma posición en dos polígonos diferentes que tienen el mismo número de lados. ∠A and ∠D are corresponding angles. ̶̶ AB and ̶̶ DE are corresponding sides. S124 S124 Glossary/Glosario �� ENGLISH SPANISH EXAMPLES cosecant (p. 532) In a right triangle, the cosecant of angle A is the ratio of the length of the hypotenuse to the length of the side opposite A. It is the reciprocal of the sine function. cosecante En un triángulo rectángulo, la cosecante del ángulo A es la razón entre la longitud de la hipotenusa y la longitud del cateto opuesto a A. Es la inversa de la función seno. cosine (p. 525) In a right triangle, the cosine of angle A is the ratio of the length of the leg adjacent to angle A to the length of the hypotenuse. It is the reciprocal of the secant function. coseno En un triángulo rectángulo, el coseno del ángulo A es la razón entre la longitud del cateto adyacente al ángulo A y la longitud de la hipotenusa. Es la inversa de la función secante. cotangent (p. 532) In a right triangle, the cotangent of angle A is the ratio of the length of the side adjacent to A to the length of the side opposite A. It is the reciprocal of the tangent function. cotangente En un triángulo rectángulo, la cotangente del ángulo A es la razón entre la longitud del cateto adyacente a A y la longitud del cateto opuesto a A. Es la inversa de la función tangente. counterexample (p. 75) An example that proves that a conjecture or statement is false. contraejemplo Ejemplo que demuestra que una conjetura o enunciado es falso. CPCTC (p. 260) An abbreviation for “Corresponding Parts of Congruent Triangles are Congruent,” which can be used as a justification in a proof after two triangles are proven congruent. PCTCC Abreviatura que significa “Las partes correspondientes de los triángulos congruentes son congruentes”, que se puede utilizar para justificar una demostración después de demostrar que dos triángulos son congruentes (CPCTC, por sus siglas en inglés). csc A = hypotenuse __ opposite = 1 _ sin A cos A = adjacent __ hypotenuse = 1 _ sec A cot A = adjacent _ opposite = 1 _ tan A cross products (p. 455) In the statement a __ = c __ , bc and ad are the d b cross products. productos cruzados En el enunciado a __ = c __ , bc y ad son los d b productos cruzados. = 3 _ 1 _ 6 2 Product of means: 2 · 3 = 6 Product of extremes: 1 · 6 = 6 cross section (p. 656) The intersection of a threedimensional figure and a plane. sección transversal Intersección de una figura tridimensional y un plano. cube (p. 654) A prism with six square faces. cubo Prisma con seis caras cuadradas. cylinder (p. 654) A threedimensional figure with two parallel circular bases and a curved lateral surface that connects the bases. cilindro Figura tridimensional con dos bases circulares congruentes paralelas y una superficie lateral curva que conecta las bases. Glossary/Glosario S125 S125 �� ENGLISH SPANISH EXAMPLES D decagon (p. 382) A ten-sided polygon. decágono Polígono de diez lados. deductive reasoning (p. 88) The process of using logic to draw conclusions. razonamiento deductivo Proceso en el que se utiliza la lógica para sacar conclusiones. definition (p. 97) A statement that describes a mathematical object and can be written as a true biconditional statement. definición Enunciado que describe un objeto matemático y se puede expresar como un enunciado bicondicional verdadero. degree (p. 20) A unit of angle measure; one degree is 1 ___ 360 of a circle. grado Unidad de medida de los ángulos; un grado es 1 ___ círculo. 360 de un denominator (p. 451) The bottom number of a fraction, which tells how many equal parts are in the whole. denominador El número inferior de una fracción, que indica la cantidad de partes iguales que hay en un entero. The denominator of 3 _ 7 is 7. diagonal of a polygon (p. 382) A segment connecting two nonconsecutive vertices of a polygon. diagonal de un polígono Segmento que conecta dos vértices no consecutivos de un polígono. diagonal of a polyhedron (p. 671) A segment whose endpoints are vertices of two different faces of a polyhedron. diagonal de un poliedro Segmento cuyos extremos son vértices de dos caras diferentes de un poliedro. diameter (p. 37) A segment that has endpoints on the circle and that passes through the center of the circle; also the length of that segment
. diámetro Segmento que atraviesa el centro de un círculo y cuyos extremos están sobre el círculo; longitud de dicho segmento. dilation (p. 495) A transformation in which the lines connecting every point P with its preimage P′ all intersect at a point C known as the center of dilation, and CP′ ___ CP is the same for every point P; a transformation that changes the size of a figure but not its shape. dilatación Transformación en la cual las rectas que conectan cada punto P con su imagen original P′ se cruzan en un punto C conocido como centro de dilatación, y CP′ ___ CP para cada punto P; transformación que cambia el tamaño de una figura pero no su forma. es igual direct reasoning (p. 332) The process of reasoning that begins with a true hypothesis and builds a logical argument to show that a conclusion is true. razonamiento directo Proceso de razonamiento que comienza con una hipótesis verdadera y elabora un argumento lógico para demostrar que una conclusión es verdadera. S126 S126 Glossary/Glosario �� ENGLISH SPANISH EXAMPLES direct variation (p. 501) A linear relationship between two variables, x and y, that can be written in the form y = kx, where k is a nonzero constant. direction of a vector (p. 560) The orientation of a vector, which is determined by the angle the vector makes with a horizontal line. variación directa Relación lineal entre dos variables, x e y, que puede expresarse en la forma y = kx, donde k es una constante distinta de cero. dirección de un vector Orientación de un vector, determinada por el ángulo que forma el vector con una recta horizontal. disjunction (p. 128) A compound statement that uses the word or. disyunción Enunciado compuesto que contiene la palabra o. John will walk to work or he will stay home. distance between two points (p. 13) The absolute value of the difference of the coordinates of the points. distancia entre dos puntos Valor absoluto de la diferencia entre las coordenadas de los puntos. distance from a point to a line (p. 172) The length of the perpendicular segment from the point to the line. distancia desde un punto hasta una línea Longitud del segmento perpendicular desde el punto hasta la línea. dodecagon (p. 382) A 12-sided polygon. dodecágono Polígono de 12 lados. dodecahedron (p. 669) A polyhedron with 12 faces. The faces of a regular dodecahedron are regular pentagons, with three faces meeting at each vertex. dodecaedro Poliedro con 12 caras. Las caras de un dodecaedro regular son pentágonos regulares, con tres caras que concurren en cada vértice. E edge of a graph (p. 95) A curve or segment that joins two vertices of the graph. arista de una gráfica Curva o segmento que une dos vértices de la gráfica. edge of a three-dimensional figure (p. 654) A segment that is the intersection of two faces of the figure. arista de una figura tridimensional Segmento que constituye la intersección de dos caras de la figura. endpoint (p. 7) A point at an end of a segment or the starting point of a ray. extremo Punto en el final de un segmento o punto de inicio de un rayo. enlargement (p. 873) A dilation with a scale factor greater than 1. In an enlargement, the image is larger than the preimage. agrandamiento Dilatación con un factor de escala mayor que 1. En un agrandamiento, la imagen es más grande que la imagen original. The distance from P to   AC is 5 units. Glossary/Glosario S127 S127 �� ENGLISH equal vectors (p. 561) Two vectors that have the same magnitude and the same direction. SPANISH EXAMPLES vectores iguales Dos vectores de la misma magnitud y con la misma dirección. equiangular polygon (p. 382) A polygon in which all angles are congruent. polígono equiangular Polígono cuyos ángulos son todos congruentes. equiangular triangle (p. 216) A triangle with three congruent angles. triángulo equiangular Triángulo con tres ángulos congruentes. equidistant (p. 300) The same distance from two or more objects. equidistante Igual distancia de dos o más objetos. equilateral polygon (p. 382) A polygon in which all sides are congruent. equilateral triangle (p. 217) A triangle with three congruent sides. polígono equilátero Polígono cuyos lados son todos congruentes. triángulo equilátero Triángulo con tres lados congruentes. Euclidean geometry (p. 726) The system of geometry described by Euclid. In particular, the system of Euclidean geometry satisfies the Parallel Postulate, which states that there is exactly one line through a given point parallel to a given line. geometría euclidiana Sistema geométrico desarrollado por Euclides. Específicamente, el sistema de la geometría euclidiana cumple con el postulado de las paralelas, que establece que por un punto dado se puede trazar una única recta paralela a una recta dada. Euler line (p. 321) The line containing the circumcenter (U ), centroid (C ), and orthocenter (O ) of a triangle. recta de Euler Recta que contiene el circuncentro (U ), el centroide (C ) y el ortocentro (O ) de un triángulo. X is equidistant from A and B. event (p. 628) An outcome or set of outcomes in a probability experiment. suceso Resultado o conjunto de resultados en un experimento de probabilidades. In the experiement of rolling a number cube, the event “an odd number” consists of the outcomes 1, 3, 5. expansion (p. 873) See enlargement. expansión Ver agrandamiento. experiment (p. 628) An operation, process, or activity in which outcomes can be used to estimate probability. experimento Una operación, proceso o actividad en la que se usan los resultados para estimar una probabilidad. Tossing a coin 10 times and noting the number of heads. S128 S128 Glossary/Glosario �� ENGLISH exterior of a circle (p. 746) The set of all points outside a circle. SPANISH exterior de un círculo Conjunto de todos los puntos que se encuentran fuera de un círculo. EXAMPLES �� exterior of an angle (p. 20) The set of all points outside an angle. exterior de un ángulo Conjunto de todos los puntos que se encuentran fuera de un ángulo. exterior of a polygon (p. 225) The set of all points outside a polygon. exterior de un polígono Conjunto de todos los puntos que se encuentran fuera de un polígono. exterior angle of a polygon (p. 225) An angle formed by one side of a polygon and the extension of an adjacent side. ángulo externo de un polígono Ángulo formado por un lado de un polígono y la prolongación del lado adyacente. external secant segment (p. 793) A segment of a secant that lies in the exterior of the circle with one endpoint on the circle. segmento secante externo Segmento de una secante que se encuentra en el exterior del círculo y tiene un extremo sobre el círculo. extremes of a proportion (p. 455) In the proportion = c __ a __ , a and d are the extremes. b d If the proportion is written as a:b = c:d, the extremes are in the first and last positions. valores extremos de una proporción En la proporción a __ = c __ , a y d son los d b valores extremos. Si la proporción se expresa como a:b = c:d, los extremos están en la primera y última posición. F face of a polyhedron (p. 654) A flat surface of the polyhedron. cara de un poliedro Superficie plana de un poliedro. ∠4 is an exterior angle. ̶̶̶ NM is an external secant segment. fair (p. 628) When all outcomes of an experiment are equally likely. justo Cuando todos los resultados de un experimento son igualmente probables. When tossing a fair coin, heads and tails are equally likely. Each has a probability of 1 __ . 2 Fibonacci sequence (p. 78) The infinite sequence of numbers beginning with 1, 1, … such that each term is the sum of the two previous terms. sucesión de Fibonacci Sucesión infinita de números que comienza con 1, 1, … de forma tal que cada término es la suma de los dos términos anteriores. 1, 1, 2, 3, 5, 8, 13, 21, … Glossary/Glosario S129 S129 �� ENGLISH flip (p. 50) See reflection. SPANISH inversión Ver reflexión. EXAMPLES flowchart proof (p. 118) A style of proof that uses boxes and arrows to show the structure of the proof. demostración con diagrama de flujo Tipo de demostración que se vale de cuadros y flechas para mostrar la estructura de la prueba. fractal (p. 882) A figure that is generated by iteration. fractal Figura generada por iteración. frieze pattern (p. 863) A pattern that has translation symmetry along a line. patrón de friso Patrón con simetría de traslación a lo largo de una línea. frustum of a cone (p. 668) A part of a cone with two parallel bases. tronco de cono Parte de un cono con dos bases paralelas. frustum of a pyramid (p. 696) A part of a pyramid with two parallel bases. tronco de pirámide Parte de una pirámide con dos bases paralelas. function (p. 389) A relation in which every input is paired with exactly one output. función Una relación en la que cada entrada corresponde exactamente a una salida. Function:  (0, 5) , (1, 3) , (2, 1) , (3, 3)  Not a Function:  (0, 1) , (0, 3) , (2, 1) , (2, 3)  G geometric mean (p. 519) For positive numbers a and b, the positive number x such that a __ x = x __ . In a geometric sequence, b a term that comes between two given nonconsecutive terms of the sequence. media geométrica Dados los números positivos a y b, el número positivo x tal que a __ x = x __ sucesión geométrica, un término que está entre dos términos no consecutivos dados de la sucesión. . En una b geometric probability (p. 630) A form of theoretical probability determined by a ratio of geometric measures such as lengths, areas, or volumes. probabilidad geométrica Método para calcular probabilidades basado en una medida geométrica como la longitud o el área. glide reflection (p. 848) A composition of a translation and a reflection across a line parallel to the translation vector. deslizamiento con inversión Composición de una traslación y una reflexión sobre una línea paralela al vector de traslación. S130
S130 Glossary/Glosario a x _ _ = x b x 2 = ab x = √  ab The probability of the pointer landing on red is 2 __ . 9 �� ENGLISH SPANISH EXAMPLES glide reflection symmetry (p. 863) A pattern has glide reflection symmetry if it coincides with its image after a glide reflection. simetría de deslizamiento con inversión Un patrón tiene simetría de deslizamiento con inversión si coincide con su imagen después de un deslizamiento con inversión. golden ratio (p. 460) If a segment is divided into two parts so that the ratio of the lengths of the whole segment to the longer part equals the ratio of the lengths of the longer part to the shorter part, then that ratio is called the golden ratio. The golden ratio is equal to 1 + √  5 ______ 2 ≈ 1.618. razón áurea Si se divide un segmento en dos partes de forma tal que la razón entre la longitud de todo el segmento y la de la parte más larga sea igual a la razón entre la longitud de la parte más larga y la de la parte más corta, entonces dicha razón se denomina razón áurea. La razón áurea es igual a 1 + √  5 ______ 2 ≈ 1.618. golden rectangle (p. 460) A rectangle in which the ratio of the lengths of the longer side to the shorter side is the golden ratio. rectángulo áureo Rectángulo en el cual la razón entre la longitud del lado más largo y la longitud del lado más corto es la razón áurea. great circle (p. 714) A circle on a sphere that divides the sphere into two hemispheres. círculo máximo En una esfera, círculo que divide la esfera en dos hemisferios. H head-to-tail method (p. 561) A method of adding two vectors by placing the tail of the second vector on the head of the first vector; the sum is the vector drawn from the tail of the first vector to the head of the second vector. height of a figure (p. 36) The length of an altitude of the figure. método de cola a punta Método para sumar dos vectores colocando la cola del segundo vector en la punta del primer vector. La suma es el vector trazado desde la cola del primer vector hasta la punta del segundo vector. altura de una figura Longitud de la altura de la figura. height of a triangle (p. 36) A segment from a vertex that forms a right angle with a line containing the base. altura de un triángulo Segmento que se extiende desde el vértice y forma un ángulo recto con la línea de la base. hemisphere (p. 714) Half of a sphere. hemisferio Mitad de una esfera. Golden ratio = AC _ AB = AB _ BC Create segment such that AC _ AB ≈ 1.62 and AB _ BC ≈ 1.62 A � D � E � F �� B C Glossary/Glosario S131 S131 ��ABC ENGLISH heptagon (p. 382) A seven-sided polygon. SPANISH heptágono Polígono de siete lados. EXAMPLES hexagon (p. 382) A six-sided polygon. hexágono Polígono de seis lados. horizon (p. 662) The horizontal line in a perspective drawing that contains the vanishing point(s). horizonte Línea horizontal en un dibujo en perspectiva que contiene el punto de fuga o los puntos de fuga. hypotenuse (p. 45) The side opposite the right angle in a right triangle. hipotenusa Lado opuesto al ángulo recto de un triángulo rectángulo. hypothesis (p. 81) The part of a conditional statement following the word if. hipótesis La parte de un enunciado condicional que sigue a la palabra si. If x + 1 = 5, then x = 4. Hypothesis I icosahedron (p. 669) A polyhedron with 20 faces. A regular icosahedron has equilateral triangles as faces, with 5 faces meeting at each vertex. icosaedro Poliedro con 20 caras. Las caras de un icosaedro regular son triángulos equiláteros y cada vértice es compartido por 5 caras. identity (p. 531) An equation that is true for all values of the variables. identidad Ecuación verdadera para todos los valores de las variables. 3 = 3 2 (x - 1) = 2x - 2 image (p. 50) A shape that results from a transformation of a figure known as the preimage. imagen Forma resultante de la transformación de una figura conocida como imagen original. incenter of a triangle (p. 309) The point of concurrency of the three angle bisectors of a triangle. incentro de un triángulo Punto donde se encuentran las tres bisectrices de los ángulos de un triángulo. P is the incenter. included angle (p. 242) The angle formed by two adjacent sides of a polygon. ángulo incluido Ángulo formado por dos lados adyacentes de un polígono. ∠B is the included ̶̶ angle between AB and ̶̶ BC . S132 S132 Glossary/Glosario ��HPGJMKL�� ENGLISH SPANISH EXAMPLES included side (p. 252) The common side of two consecutive angles of a polygon. lado incluido Lado común de dos ángulos consecutivos de un polígono. ̶̶ PQ is the included side between ∠P and ∠Q. indirect measurement (p. 488) A method of measurement that uses formulas, similar figures, and/or proportions. medición indirecta Método para medir objetos mediante fórmulas, figuras similares y/o proporciones. indirect proof (p. 332) A proof in which the statement to be proved is assumed to be false and a contradiction is shown. demostración indirecta Prueba en la que se supone que el enunciado a demostrar es falso y se muestra una contradicción. indirect reasoning (p. 332) See indirect proof. razonamiento indirecto Ver demostración indirecta. inductive reasoning (p. 74) The process of reasoning that a rule or statement is true because specific cases are true. razonamiento inductivo Proceso de razonamiento por el que se determina que una regla o enunciado son verdaderos porque ciertos casos específicos son verdaderos. inequality (p. 92) A statement that compares two expressions by using one of the following signs: <, >, ≤, ≥, or ≠. desigualdad Enunciado que compara dos expresiones utilizando uno de los siguientes signos: <, >, ≤, ≥ o ≠. x > 2 initial point of a vector (p. 559) The starting point of a vector. punto inicial de un vector Punto donde comienza un vector. initial side (p. 570) The ray that lies on the positive x-axis when an angle is drawn in standard position. lado inicial Rayo que se encuentra sobre el eje x positivo cuando se traza un ángulo en posición estándar. inscribed angle (p. 772) An angle whose vertex is on a circle and whose sides contain chords of the circle. ángulo inscrito Ángulo cuyo vértice se encuentra sobre un círculo y cuyos lados contienen cuerdas del círculo. inscribed circle (p. 309) A circle in which each side of the polygon is tangent to the circle. círculo inscrito Círculo en el que cada lado del polígono es tangente al círculo. inscribed polygon (p. 599) A polygon in which every vertex of the polygon lies on the circle. polígono inscrito Polígono cuyos vértices se encuentran sobre el círculo. Glossary/Glosario S133 S133 ��A��B��v��yx��DEF ENGLISH integer (p. 559) A member of the set of whole numbers and their opposites. SPANISH EXAMPLES entero Miembro del conjunto de números cabales y sus opuestos. … -3, -2, -1, 0, 1, 2, 3, … intercepted arc (p. 772) An arc that consists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between the endpoints. arco abarcado Arco cuyos extremos se encuentran en los lados de un ángulo inscrito y consta de todos los puntos del círculo ubicados entre dichos extremos. interior angle (p. 225) An angle formed by two sides of a polygon with a common vertex. ángulo interno Ángulo formado por dos lados de un polígono con un vértice común. interior of a circle (p. 746) The set of all points inside a circle. interior de un círculo Conjunto de todos los puntos que se encuentran dentro de un círculo. ⁀ DF is the intercepted arc. ∠1 is an interior angle. �� interior of an angle (p. 20) The set of all points between the sides of an angle. interior de un ángulo Conjunto de todos los puntos entre los lados de un ángulo. interior of a polygon (p. 225) The set of all points inside a polygon. interior de un polígono Conjunto de todos los puntos que se encuentran dentro de un polígono. inverse (p. 83) The statement formed by negating the hypothesis and conclusion of a conditional statement. inverse cosine (p. 534) The measure of an angle whose cosine ratio is known. inverse function (p. 533) The function that results from exchanging the input and output values of a one-to-one function. The inverse of f(x) is denoted f -1 (x). inverso Enunciado formado al negar la hipótesis y la conclusión de un enunciado condicional. coseno inverso Medida de un ángulo cuya razón coseno es conocida. Statement: If n + 1 = 3, then n = 2 Inverse: If n + 1 ≠ 3, then n ≠ 2 If cos A = x, then cos -1 x = m∠A. función inversa Función que resulta de intercambiar los valores de entrada y salida de una función uno a uno. La función inversa de f (x) se indica f -1 (x) . S134 S134 Glossary/Glosario The function y = 1 _ 2 inverse of the function y = 2x + 4. x - 2 is the DEF�� ENGLISH inverse sine (p. 534) The measure of an angle whose sine ratio is known. SPANISH seno inverso Medida de un ángulo cuya razón seno es conocida. EXAMPLES If sin A = x, then sin -1 x = m∠A. inverse tangent (p. 534) The measure of an angle whose tangent ratio is known. tangente inversa Medida de un ángulo cuya razón tangente es conocida. If tan A = x, then tan -1 x = m∠A. irrational number (p. 80) A real number that cannot be expressed as the ratio of two integers. número irracional Número real que no se puede expresar como una razón de dos enteros. √  2 , π, e irregular polygon (p. 382) A polygon that is not regular. polígono irregular Polígono que no es regular. isometric drawing (p. 662) A way of drawing three-dimensional figures using isometric dot paper, which has equally spaced dots in a repeating triangular pattern. dibujo isométrico Forma de dibujar figuras tridimensionales utilizando papel punteado isométrico, que
tiene puntos espaciados uniformemente en un patrón triangular que se repite. isometry (p. 824) A transformation that does not change the size or shape of a figure. isosceles trapezoid (p. 429) A trapezoid in which the legs are congruent. isometría Transformación que no cambia el tamaño ni la forma de una figura. Reflections, translations, and rotations are all examples of isometries. trapecio isósceles Trapecio cuyos lados no paralelos son congruentes. isosceles triangle (p. 217) A triangle with at least two congruent sides. triángulo isósceles Triángulo que tiene al menos dos lados congruentes. iteration (p. 882) The repetitive application of the same rule. iteración Aplicación repetitiva de la misma regla. K kite (p. 427) A quadrilateral with exactly two pairs of congruent consecutive sides. cometa o papalote Cuadrilátero con exactamente dos pares de lados congruentes consecutivos. Koch snowflake (p. 882) A fractal formed from a triangle by replacing the middle third of each segment with two segments that form a 60° angle. copo de nieve de Koch Fractal formado a partir de un triángulo sustituyendo el tercio central de cada segmento por dos segmentos que forman un ángulo de 60°. Glossary/Glosario S135 S135 BADC�� ENGLISH SPANISH EXAMPLES L lateral area (p. 680) The sum of the areas of the lateral faces of a prism or pyramid, or the area of the lateral surface of a cylinder or cone. área lateral Suma de las áreas de las caras laterales de un prisma o pirámide, o área de la superficie lateral de un cilindro o cono. lateral edge (p. 680) An edge of a prism or pyramid that is not an edge of a base. borde lateral Borde de un prisma o pirámide que no es el borde de una base. Lateral area = (28) (12) = 336 cm 2 lateral face (p. 680) A face of a prism or a pyramid that is not a base. cara lateral Cara de un prisma o pirámide que no es la base. lateral surface (p. 681) The curved surface of a cylinder or cone. superficie lateral Superficie curva de un cilindro o cono. leg of a right triangle (p. 45) One of the two sides of the right triangle that form the right angle. leg of a trapezoid (p. 429) One of the two nonparallel sides of the trapezoid. cateto de un triángulo rectángulo Uno de los dos lados de un triángulo rectángulo que forman un ángulo recto. cateto de un trapecio Uno de los dos lados no paralelos del trapecio. leg of an isosceles triangle (p. 273) One of the two congruent sides of the isosceles triangle. cateto de un triángulo isósceles Uno de los dos lados congruentes del triángulo isósceles. length (p. 13) The distance between the two endpoints of a segment. longitud Distancia entre los dos extremos de un segmento. line (p. 6) An undefined term in geometry, a line is a straight path that has no thickness and extends forever. línea Término indefinido en geometría; una línea es un trazo recto que no tiene grosor y se extiende infinitamente. S136 S136 Glossary/Glosario �� ENGLISH line of best fit (p. 198) The line that comes closest to all of the points in a data set. SPANISH línea de mejor ajuste Línea que más se acerca a todos los puntos de un conjunto de datos. EXAMPLES line of symmetry (p. 865) A line that divides a plane figure into two congruent reflected halves. eje de simetría Línea que divide una figura plana en dos mitades reflejas congruentes. line symmetry (p. 856) A figure that can be reflected across a line so that the image coincides with the preimage. simetría axial Figura que puede reflejarse sobre una línea de forma tal que la imagen coincida con la imagen original. linear pair (p. 28) A pair of adjacent angles whose noncommon sides are opposite rays. par lineal Par de ángulos adyacentes cuyos lados no comunes son rayos opuestos. literal equation (p. 588) An equation that contains two or more variables. locus (p. 300) A set of points that satisfies a given condition. ∠3 and ∠4 form a linear pair. ecuación literal Ecuación que contiene dos o más variables. d = rt ) lugar geométrico Conjunto de puntos que cumple con una condición determinada. logically equivalent statements (p. 83) Statements that have the same truth value. enunciados lógicamente equivalentes Enunciados que tienen el mismo valor de verdad. M magnitude (p. 560) The length of a vector, written ⎜  AB ⎟ or ⎜  v ⎟ . magnitud Longitud de un vector, que se expresa ⎜  AB ⎟ o ⎜  v ⎟ . major arc (p. 756) An arc of a circle whose points are on or in the exterior of a central angle. arco mayor Arco de un círculo cuyos puntos están sobre un ángulo central o en su exterior.  u ⎟ = 5 ⎜ ⁀ ADC is a major arc of the circle. Glossary/Glosario S137 S137 �� ENGLISH mapping (p. 50) An operation that matches each element of a set with another element, its image, in the same set. SPANISH EXAMPLES correspondencia Operación que establece una correlación entre cada elemento de un conjunto con otro elemento, su imagen, en el mismo conjunto. matrix (p. 846) A rectangular array of numbers. matriz Arreglo rectangular de números. ⎡ 1 ⎢ -2 7 ⎣ 0 2 -6 ⎤ 3 ⎥ -5 3 ⎦ means of a proportion (p. 455) In the proportion a __ = c __ , b and c are d b the means. If the proportion is written as a:b = c:d, the means are in the two middle positions. valores medios de una proporción En la proporción a __ = c __ , b y c son los d b valores medios. Si la proporción se expresa como a:b = c:d, los valores medios están en las dos posiciones del medio. measure of an angle (p. 20) Angles are measured in degrees. A degree is 1 ___ 360 of a complete circle. medida de un ángulo Los ángulos se miden en grados. Un grado es 1 ___ 360 de un círculo completo. measure of a major arc (p. 756) The difference of 360° and the measure of the associated minor arc. medida de un arco mayor Diferencia entre 360° y la medida del arco menor asociado. measure of a minor arc (p. 756) The measure of its central angle. medida de un arco menor Medida de su ángulo central. m∠M = 26.8° m ⁀ ADC = 360° - x° m ⁀ AC = x° median of a triangle (p. 314) A segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. mediana de un triángulo Segmento cuyos extremos son un vértice del triángulo y el punto medio del lado opuesto. midpoint (p. 15) The point that divides a segment into two congruent segments. punto medio Punto que divide un segmento en dos segmentos congruentes. B is the midpoint of ̶̶ AC . midsegment of a trapezoid (p. 431) The segment whose endpoints are the midpoints of the legs of the trapezoid. segmento medio de un trapecio Segmento cuyos extremos son los puntos medios de los catetos del trapecio. midsegment of a triangle (p. 322) A segment that joins the midpoints of two sides of the triangle. segmento medio de un triángulo Segmento que une los puntos medios de dos lados del triángulo. S138 S138 Glossary/Glosario �� ENGLISH SPANISH EXAMPLES midsegment triangle (p. 322) The triangle formed by the three midsegments of a triangle. triángulo de segmentos medios Triángulo formado por los tres segmentos medios de un triángulo. minor arc (p. 756) An arc of a circle whose points are on or in the interior of a central angle. arco menor Arco de un círculo cuyos puntos están sobre un ángulo central o en su interior. N natural number (p. 80) A counting number. ⁀ AC is a minor arc of the circle. número natural Número de conteo. 1, 2, 3, 4, 5, 6, … negation (p. 82) The negation of statement p is “not p,” written as ∼p. negación La negación de un enunciado p es “no p”, que se escribe p. negation of a vector (p. 566) The vector obtained by negating each component of a given vector. negación de un vector Vector que se obtiene por la negación de cada componente de un vector dado. The negation of 〈3, -2〉 is 〈-3, 2〉. net (p. 655) A diagram of the faces of a three-dimensional figure arranged in such a way that the diagram can be folded to form the three-dimensional figure. plantilla Diagrama de las caras y superficies de una figura tridimensional que se puede plegar para formar la figura tridimensional. network (p. 95) A diagram of vertices and edges. red Diagrama de vértices y aristas. n-gon (p. 382) An n-sided polygon. nonagon (p. 382) A nine-sided polygon. n-ágono Polígono de n lados. nonágono Polígono de nueve lados. noncollinear (p. 6) Points that do not lie on the same line. no colineal Puntos que no se encuentran sobre la misma línea. non-Euclidean geometry (p. 726) A system of geometry in which the Parallel Postulate, which states that there is exactly one line through a given point parallel to a given line, does not hold. geometría no euclidiana Sistema de geometría en el cual no se cumple el postulado de las paralelas, que establece que por un punto dado se puede trazar una única recta paralela a una recta dada. Points A, B, and D are not collinear. In spherical geometry, there are no parallel lines. The sum of the angles in a triangle is always greater than 180°. Glossary/Glosario S139 S139 �� ENGLISH noncoplanar (p. 6) Points that do not lie on the same plane. SPANISH EXAMPLES no coplanar Puntos que no se encuentran en el mismo plano. numerator (p. 451) The top number of a fraction, which tells how many parts of a whole are being considered. numerador El número superior de una fracción, que indica la cantidad de partes de un entero que se consideran. T, U, V, and S are not coplanar. The numerator of 3 _ 7 is 3. O oblique cone (p. 690) A cone whose axis is not perpendicular to the base. cono oblicuo Cono cuyo eje no es perpendicular a la base. oblique cylinder (p. 681) A cylinder whose axis is not perpendicular to the bases. cilindro oblicuo Cilindro cuyo eje no es perpendicular a las bases. oblique prism (p. 680) A prism that has at least one nonrectangular
lateral face. prisma oblicuo Prisma que tiene por lo menos una cara lateral no rectangular. obtuse angle (p. 21) An angle that measures greater than 90° and less than 180°. ángulo obtuso Ángulo que mide más de 90° y menos de 180°. obtuse triangle (p. 216) A triangle with one obtuse angle. triángulo obtusángulo Triángulo con un ángulo obtuso. octagon (p. 382) An eight-sided polygon. octágono Polígono de ocho lados. octahedron (p. 669) A polyhedron with eight faces. octaedro Poliedro con ocho caras. one-point perspective (p. 662) A perspective drawing with one vanishing point. perspectiva de un punto Dibujo en perspectiva con un punto de fuga. opposite rays (p. 7) Two rays that have a common endpoint and form a line. rayos opuestos Dos rayos que tienen un extremo común y forman una recta.  EF and  EG are opposite rays. S140 S140 Glossary/Glosario �� ENGLISH opposite reciprocal (p. 184) The opposite of the reciprocal of a number. The opposite reciprocal of a is - 1 __ a . order of rotational symmetry (p. 857) The number of times a figure with rotational symmetry coincides with itself as it rotates 360°. SPANISH EXAMPLES recíproco opuesto Opuesto del recíproco de un número. El recíproco opuesto de a es - 1 __ a . The opposite reciprocal of 2 _ 3 is -3 _ 2 orden de simetría de rotación Cantidad de veces que una figura con simetría de rotación coincide consigo misma cuando rota 360°. Order of rotational symmetry: 4 ordered pair (p. 42) A pair of numbers (x, y) that can be used to locate a point on a coordinate plane. The first number x indicates the distance to the left or right of the origin, and the second number y indicates the distance above or below the origin. par ordenado Par de números (x, y) que se pueden utilizar para ubicar un punto en un plano cartesiano. El primer número indica la distancia a la izquierda o derecha del origen y el segundo número indica la distancia hacia arriba o hacia abajo del origen. ordered triple (p. 671) A set of three numbers that can be used to locate a point (x, y, z) in a threedimensional coordinate system. tripleta ordenada Conjunto de tres números que se pueden utilizar para ubicar un punto (x, y, z) en un sistema de coordenadas tridimensional. origin (p. 42) The intersection of the x- and y-axes in a coordinate plane. The coordinates of the origin are (0, 0) . origen Intersección de los ejes x e y en un plano cartesiano. Las coordenadas de origen son (0, 0) . orthocenter of a triangle (p. 316) The point of concurrency of the three altitudes of a triangle. ortocentro de un triángulo Punto de intersección de las tres alturas de un triángulo. P is the orthocenter. orthographic drawing (p. 661) A drawing that shows a threedimensional object in which the line of sight for each view is perpendicular to the plane of the picture. dibujo ortográfico Dibujo que muestra un objeto tridimensional en el que la línea visual para cada vista es perpendicular al plano de la imagen. Glossary/Glosario S141 S141 �� ENGLISH outcome (p. 628) A possible result of a probability experiment. SPANISH resultado Resultado posible de un experimento de probabilidades. EXAMPLES In the experiment of rolling a number cube, the possible outcomes are 1, 2, 3, 4, 5, and 6. P paragraph proof (p. 120) A style of proof in which the statements and reasons are presented in paragraph form. demostración con párrafos Tipo de demostración en la cual los enunciados y las razones se presentan en forma de párrafo. parallel lines (p. 146) Lines in the same plane that do not intersect. líneas paralelas Líneas rectas en el mismo plano que no se cruzan. parallel planes (p. 146) Planes that do not intersect. planos paralelos Planos que no se cruzan. r ǁ s parallel vectors (p. 561) Vectors with the same or opposite direction. vectores paralelos Vectores con dirección igual u opuesta. Plane AEF and plane CGH are parallel planes. parallelogram (p. 391) A quadrilateral with two pairs of parallel sides. parallelogram method (p. 561) A method of adding two vectors by drawing a parallelogram using the vectors as two of the consecutive sides; the sum is a vector along the diagonal of the parallelogram. paralelogramo Cuadrilátero con dos pares de lados paralelos. método del paralelogramo Método mediante el cual se suman dos vectores dibujando un paralelogramo, utilizando los vectores como dos de los lados consecutivos; el resultado de la suma es un vector a lo largo de la diagonal del paralelogramo. parent function (p. 221) The simplest function with the defining characteristics of the family. Functions in the same family are transformations of their parent function. función madre La función más básica que tiene las características distintivas de una familia. Las funciones de la misma familia son transformaciones de su función madre. Pascal’s triangle (p. 883) A triangular arrangement of numbers in which every row starts and ends with 1 and each other number is the sum of the two numbers above it. triángulo de Pascal Arreglo triangular de números en el cual cada fila comienza y termina con 1 y cada uno de los otros números es la suma de los dos números que están encima de él. S142 S142 Glossary/Glosario f (x) = x 2 is the parent function for g (x) = x 2 + 4 and h (x) = 5 (x + 2) 2 - 3. �� ENGLISH pentagon (p. 382) A five-sided polygon. SPANISH pentágono Polígono de cinco lados. EXAMPLES perimeter (p. 36) The sum of the side lengths of a closed plane figure. perímetro Suma de las longitudes de los lados de una figura plana cerrada. perpendicular (p. 146) Intersecting to form 90° angles, denoted by ⊥. perpendicular Que se cruza para formar ángulos de 90°, expresado por ⊥. Perimeter = 18 + 6 + 18 + 6 = 48 ft m ⊥ n perpendicular bisector of a segment (p. 172) A line perpendicular to a segment at the segment’s midpoint. mediatriz de un segmento Línea perpendicular a un segmento en el punto medio del segmento. perpendicular lines (p. 146) Lines that intersect at 90° angles. líneas perpendiculares Líneas que se cruzan en ángulos de 90°. ℓ is the perpendicular bisector of ̶̶ AB . m ⊥ n perspective drawing (p. 662) A drawing in which nonvertical parallel lines meet at a point called a vanishing point. Perspective drawings can have one or two vanishing points. dibujo en perspectiva Dibujo en el cual las líneas paralelas no verticales se encuentran en un punto denominado punto de fuga. Los dibujos en perspectiva pueden tener uno o dos puntos de fuga. pi (p. 37) The ratio of the circumference of a circle to its diameter, denoted by the Greek letter π (pi). The value of π is irrational, often approximated by 3.14 or 22 __ 7 . pi Razón entre la circunferencia de un círculo y su diámetro, expresado por la letra griega π (pi). El valor de π es irracional y por lo general se aproxima a 3.14 ó 22 __ 7 . If a circle has a diameter of 5 inches and a circumference of C inches, then C __ = π, or C = 5π inches, or 5 about 15.7 inches. plane (p. 6) An undefined term in geometry, it is a flat surface that has no thickness and extends forever. plano Término indefinido en geometría; un plano es una superficie plana que no tiene grosor y se extiende infinitamente. plane symmetry (p. 858) A threedimensional figure that can be divided into two congruent reflected halves by a plane has plane symmetry. simetría de plano Una figura tridimensional que se puede dividir en dos mitades congruentes reflejadas por un plano tiene simetría de plano. Platonic solid (p. 669) One of the five regular polyhedra: a tetrahedron, a cube, an octahedron, a dodecahedron, or an icosahedron. sólido platónico Uno de los cinco poliedros regulares: tetraedro, cubo, octaedro, dodecaedro o icosaedro. plane R or plane ABC Glossary/Glosario S143 S143 �� ENGLISH point (p. 6) An undefined term in geometry, it names a location and has no size. SPANISH EXAMPLES punto Término indefinido de la geometría que denomina una ubicación y no tiene tamaño. point P point matrix (p. 846) A matrix that represents the coordinates of the vertices of a polygon. The first row of the matrix consists of the xcoordinates of the points, and the second row consists of the y-coordinates. matriz de puntos Matriz que representa las coordenadas de los vértices de un polígono. La primera fila de la matriz contiene las coordenadas x de los puntos y la segunda fila contiene las coordenadas y. ⎡ ⎢ ⎣ 1 2 -2 0 3 -4 ⎤ ⎥ ⎦ point of concurrency (p. 307) A point where three or more lines coincide. punto de concurrencia Punto donde se cruzan tres o más líneas. point of tangency (p. 746) The point of intersection of a circle or sphere with a tangent line or plane. punto de tangencia Punto de intersección de un círculo o esfera con una línea o plano tangente. point-slope form (p. 190) y - y 1 = m (x - x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line. polar axis (p. 808) In a polar coordinate system, the horizontal ray with the pole as its endpoint that lies along the positive x-axis. forma de punto y pendiente (y - y 1 ) = m (x - x 1 ) , donde m es la pendiente y ( x 1 , y 1 ) es un punto en la línea. eje polar En un sistema de coordenadas polares, el rayo horizontal, cuyo extremo es el polo, que se encuentra a lo largo del eje x positivo. polar coordinate system (p. 808) A system in which a point in a plane is located by its distance r from a point called the pole, and by the measure of a central angle θ. sistema de coordenadas polares Sistema en el cual un punto en un plano se ubica por su distancia r de un punto denominado polo y por la medida de un ángulo central θ. S144 S144 Glossary/Glosario �� ENGLISH pole (p. 808) The point from whi
ch distances are measured in a polar coordinate system. SPANISH polo Punto desde el que se miden las distancias en un sistema de coordenadas polares. EXAMPLES polygon (p. 98) A closed plane figure formed by three or more segments such that each segment intersects exactly two other segments only at their endpoints and no two segments with a common endpoint are collinear. polígono Figura plana cerrada formada por tres o más segmentos tal que cada segmento se cruza únicamente con otros dos segmentos sólo en sus extremos y ningún segmento con un extremo común a otro es colineal con éste. polyhedron (p. 670) A closed three-dimensional figure formed by four or more polygons that intersect only at their edges. poliedro Figura tridimensional cerrada formada por cuatro o más polígonos que se cruzan sólo en sus aristas. postulate (p. 7) A statement that is accepted as true without proof. Also called an axiom. postulado Enunciado que se acepta como verdadero sin demostración. También denominado axioma. preimage (p. 50) The original figure in a transformation. imagen original Figura original en una transformación. primes (p. 50) Symbols used to label the image in a transformation. apóstrofos Símbolos utilizados para identificar la imagen en una transformación. A′B′C′ prism (p. 713) A polyhedron formed by two parallel congruent polygonal bases connected by lateral faces that are parallelograms. probability (p. 237) A number from 0 to 1 (or 0% to 100%) that is the measure of how likely an event is to occur. prisma Poliedro formado por dos bases poligonales congruentes paralelas conectadas por caras laterales que son paralelogramos. probabilidad Número entre 0 y 1 (o entre 0% y 100%) que describe cuán probable es que ocurra un suceso. A bag contains 3 red marbles and 4 blue marbles. The probability of randomly choosing a red marble is 3 __ . 7 proof (p. 104) An argument that uses logic to show that a conclusion is true. demostración Argumento que se vale de la lógica para probar que una conclusión es verdadera. proof by contradiction (p. 322) See indirect proof. demostración por contradicción Ver demostración indirecta. Glossary/Glosario S145 S145 �� ENGLISH proportion (p. 455) A statement = c __ that two ratios are equal; a __ . d b SPANISH proporción Ecuación que establece = c __ que dos razones son iguales; a __ . d b EXAMPLES = 4 _ 2 _ 6 3 pyramid (p. 654) A polyhedron formed by a polygonal base and triangular lateral faces that meet at a common vertex. pirámide Poliedro formado por una base poligonal y caras laterales triangulares que se encuentran en un vértice común. Pythagorean triple (p. 349) A set of three nonzero whole numbers a, b, and c such that a 2 + b 2 = c 2 . Tripleta de Pitágoras Conjunto de tres números cabales distintos de cero a, b y c tal que 3, 4, 5 quadrant (p. 42) One of the four regions into which the x- and y-axes divide the coordinate plane. cuadrante Una de las cuatro regiones en las que los ejes x e y dividen el plano cartesiano. quadrilateral (p. 98) A four-sided polygon. cuadrilátero Polígono de cuatro lados. R radial symmetry (p. 857) See rotational symmetry. simetría radial Ver simetría de rotación. radical symbol (p. 346) The symbol √  used to denote a root. The symbol is used alone to indicate a square root or with an index, n √  , to indicate the nth root. símbolo de radical Símbolo √  que se utiliza para expresar una raíz. Puede utilizarse solo para indicar una raíz cuadrada, o con un índice, n √  , para indicar la enésima raíz. √  36 = 6 3 √  27 = 3 radicand (p. 346) The expression under a radical sign. radicando Número o expresión debajo del signo de radical. Expression: √  x + 3 Radicand: x + 3 radius of a circle (p. 37) A segment whose endpoints are the center of a circle and a point on the circle; the distance from the center of a circle to any point on the circle. radio de un círculo Segmento cuyos extremos son el centro y un punto del círculo; distancia desde el centro de un círculo hasta cualquier punto de éste. radius of a cone (p. 681) The distance from the center of the base of the cone to any point on the base. radio de un cono Distancia desde el centro de la base del cono hasta un punto cualquiera de la base. S146 S146 Glossary/Glosario �� EXAMPLES ENGLISH radius of a cylinder (p. 690) The distance from the center of the base of the cylinder to any point on the base. SPANISH radio de un cilindro Distancia desde el centro de la base del cilindro hasta un punto cualquiera de la base. radius of a sphere (p. 714) A segment whose endpoints are the center of a sphere and any point on the sphere; the distance from the center of a sphere to any point on the sphere. radio de una esfera Segmento cuyos extremos son el centro de una esfera y cualquier punto sobre la esfera; distancia desde el centro de una esfera hasta cualquier punto sobre la esfera. rate of change (p. 97) A ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. tasa de cambio Razón que compara la cantidad de cambio de la variable dependiente con la cantidad de cambio de la variable independiente. Rate of change = change in y __ change in x = 6 _ 4 = 3 _ 2 ratio (p. 454) A comparison of two quantities by division. razón Comparación de dos cantidades mediante una división. rational number (p. 80) A number that can be written in the form a __ , b where a and b are integers and b ≠ 0. ray (p. 7) A part of a line that starts at an endpoint and extends forever in one direction. número racional Número que se puede expresar como a __ , donde a y b b son números enteros y b ≠ 0. ̶ 3 , - 2 _ 3, 1.75, 0. 3 , 0 rayo Parte de una recta que comienza en un extremo y se extiende infinitamente en una dirección. rectangle (p. 408) A quadrilateral with four right angles. rectángulo Cuadrilátero con cuatro ángulos rectos. reduction (p. 873) A dilation with a scale factor greater than 0 but less than 1. In a reduction, the image is smaller than the preimage. reducción Dilatación con un factor de escala mayor que 0 pero menor que 1. En una reducción, la imagen es más pequeña que la imagen original. reference angle (p. 570) For an angle in standard position, the reference angle is the positive acute angle formed by the terminal side of the angle and the x-axis. ángulo de referencia Dado un ángulo en posición estándar, el ángulo de referencia es el ángulo agudo positivo formado por el lado terminal del ángulo y el eje x. Glossary/Glosario S147 S147 �� ENGLISH reflection (p. 50) A transformation across a line, called the line of reflection, such that the line of reflection is the perpendicular bisector of each segment joining each point and its image. SPANISH EXAMPLES reflexión Transformación sobre una línea, denominada la línea de reflexión. La línea de reflexión es la mediatriz de cada segmento que une un punto con su imagen. reflection symmetry (p. 856) See line symmetry. simetría de reflexión Ver simetría axial. regular polygon (p. 382) A polygon that is both equilateral and equiangular. polígono regular Polígono equilátero de ángulos iguales. regular polyhedron (p. 669) A polyhedron in which all faces are congruent regular polygons and the same number of faces meet at each vertex. See also Platonic solid. poliedro regular Poliedro cuyas caras son todas polígonos regulares congruentes y en el que el mismo número de caras se encuentran en cada vértice. Ver también sólido platónico. regular pyramid (p. 689) A pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles. pirámide regular Pirámide cuya base es un polígono regular y cuyas caras laterales son triángulos isósceles congruentes. regular tessellation (p. 864) A repeating pattern of congruent regular polygons that completely covers a plane with no gaps or overlaps. teselado regular Patrón que se repite formado por polígonos regulares congruentes que cubren completamente un plano sin dejar espacios y sin superponerse. relation (p. 389) A set of ordered pairs. relación Conjunto de pares ordenados.  (0, 5) , (0, 4) , (2, 3) , (4, 0)  remote interior angle (p. 225) An interior angle of a polygon that is not adjacent to the exterior angle. ángulo interno remoto Ángulo interno de un polígono que no es adyacente al ángulo externo. The remote interior angles of ∠4 are ∠1 and ∠2 resultant vector (p. 561) The vector that represents the sum of two given vectors. vector resultante Vector que representa la suma de dos vectores dados. rhombus (p. 409) A quadrilateral with four congruent sides. rombo Cuadrilátero con cuatro lados congruentes. right angle (p. 21) An angle that measures 90°. ángulo recto Ángulo que mide 90°. S148 S148 Glossary/Glosario �� ENGLISH right cone (p. 690) A cone whose axis is perpendicular to the base. SPANISH EXAMPLES cono regular Cono cuyo eje es perpendicular a la base. right cylinder (p. 681) A cylinder whose axis is perpendicular to its bases. cilindro regular Cilindro cuyo eje es perpendicular a sus bases. right prism (p. 680) A prism whose lateral faces are all rectangles. prisma regular Prisma cuyas caras laterales son todas rectángulos. right triangle (p. 216) A triangle with one right angle. triángulo rectángulo Triángulo con un ángulo recto. rigid transformation (p. 824) See isometry. rise (p. 182) The difference in the y-values of two points on a line. transformación rígida Ver isometría. distancia vertical Diferencia entre los valores de y de dos puntos de una línea. For the points (3, -1) and (6, 5) , the rise is 5 - (-1) = 6. rotation (p. 50) A transformation about a point P, also known as the center of rotation, such that each point and its image are the same distance from P. All of the angles with vertex P formed by a point and its image are congruen
t. rotación Transformación sobre un punto P, también conocido como el centro de rotación, tal que cada punto y su imagen estén a la misma distancia de P. Todos los ángulos con vértice P formados por un punto y su imagen son congruentes. rotational symmetry (p. 857) A figure that can be rotated about a point by an angle less than 360° so that the image that coincides with the preimage has rotational symmetry. simetría de rotación Una figura que puede rotarse alrededor de un punto en un ángulo menor de 360° de forma tal que la imagen coincide con la imagen original que tenga simetría de rotación. run (p. 182) The difference in the x-values of two points on a line. distancia horizontal Diferencia entre los valores de x de dos puntos de una línea. S same-side interior angles (p. 147) For two lines intersected by a transversal, a pair of angles that lie on the same side of the transversal and between the two lines. ángulos internos del mismo lado Dadas dos rectas cortadas por una transversal, el par de ángulos ubicados en el mismo lado de la transversal y entre las dos rectas. Order of rotational symmetry: 4 For the points (3, -1) and (6, 5) , the run is 6 - 3 = 3. ∠2 and ∠3 are same-side interior angles. Glossary/Glosario S149 S149 �� ENGLISH SPANISH sample space (p. 628) The set of all possible outcomes of a probability experiment. espacio muestral Conjunto de todos los resultados posibles de un experimento de probabilidades. EXAMPLES in the experiment of rolling a number cube, the sample space is {1, 2, 3, 4, 5, 6}. scalar multiplication of a vector (p. 566) The process of multiplying a vector by a constant. multiplicación escalar de un vector Proceso por el cual se multiplica un vector por una constante. 3〈-8, 1〉 = 〈-24, 3〉 scale (p. 489) The ratio between two corresponding measurements. escala Razón entre dos medidas correspondientes. 1 cm : 5 mi A blueprint is an example of a scale drawing. Scale factor: 2 scale drawing (p. 489) A drawing that uses a scale to represent an object as smaller or larger than the actual object. dibujo a escala Dibujo que utiliza una escala para representar un objeto como más pequeño o más grande que el objeto original. scale factor (p. 495) The multiplier used on each dimension to change one figure into a similar figure. factor de escala El multiplicador utilizado en cada dimensión para transformar una figura en una figura semejante. scale model (p. 456) A threedimensional model that uses a scale to represent an object as smaller or larger than the actual object. modelo a escala Modelo tridimensional que utiliza una escala para representar un objeto como más pequeño o más grande que el objeto real. scalene triangle (p. 217) A triangle with no congruent sides. triángulo escaleno Triángulo sin lados congruentes. scatter plot (p. 198) A graph with points plotted to show a possible relationship between two sets of data. diagrama de dispersión Gráfica con puntos dispersos para demostrar una relación posible entre dos conjuntos de datos. secant of a circle (p. 746) A line that intersects a circle at two points. secante de un círculo Línea que corta un círculo en dos puntos. S150 S150 Glossary/Glosario �� ENGLISH SPANISH EXAMPLES secant of an angle (p. 532) In a right triangle, the ratio of the length of the hypotenuse to the length of the side adjacent to angle A. It is the reciprocal of the cosine function. secante de un ángulo En un triángulo rectángulo, la razón entre la longitud de la hipotenusa y la longitud del cateto adyacente al ángulo A. Es la inversa de la función coseno. secant segment (p. 793) A segment of a secant with at least one endpoint on the circle. segmento secante Segmento de una secante que tiene al menos un extremo sobre el círculo. sector of a circle (p. 764) A region inside a circle bounded by two radii of the circle and their intercepted arc. sector de un círculo Región dentro de un círculo delimitado por dos radios del círculo y por su arco abarcado. segment bisector (p. 16) A line, ray, or segment that divides a segment into two congruent segments. bisectriz de un segmento Línea, rayo o segmento que divide un segmento en dos segmentos congruentes. segment of a circle (p. 765) A region inside a circle bounded by a chord and an arc. segmento de un círculo Región dentro de un círculo delimitada por una cuerda y un arco. segment of a line (p. 7) A part of a line consisting of two endpoints and all points between them. segmento de una línea Parte de una línea que consiste en dos extremos y todos los puntos entre éstos. self-similar (p. 882) A figure that can be divided into parts, each of which is similar to the entire figure. autosemejante Figura que se puede dividir en partes, cada una de las cuales es semejante a la figura entera. semicircle (p. 756) An arc of a circle whose endpoints lie on a diameter. semicírculo Arco de un círculo cuyos extremos se encuentran sobre un diámetro. semiregular tessellation (p. 864) A repeating pattern formed by two or more regular polygons in which the same number of each polygon occur in the same order at every vertex and completely cover a plane with no gaps or overlaps. teselado semirregular Patrón que se repite formado por dos o más polígonos regulares en los que el mismo número de cada polígono se presenta en el mismo orden en cada vértice y que cubren un plano completamente sin dejar espacios vacíos ni superponerse. sec A = hypotenuse __ adjacent = 1 _ cos A ̶̶̶ NM is an external secant segment. ̶̶ JK is an internal secant segment. Glossary/Glosario S151 S151 �� ENGLISH side of a polygon (p. 382) One of the segments that form a polygon. SPANISH EXAMPLES lado de un polígono Uno de los segmentos que forman un polígono. side of an angle (p. 20) One of the two rays that form an angle. lado de un ángulo Uno de los dos rayos que forman un ángulo.  AB are  AC and sides of ∠CAB. Sierpinski triangle (p. 882) A fractal formed from a triangle by removing triangles with vertices at the midpoints of the sides of each remaining triangle. triángulo de Sierpinski Fractal formado a partir de un triángulo al cual se le recortan triángulos cuyos vértices se encuentran en los puntos medios de los lados de cada triángulo restante. similar (p. 462) Two figures are similar if they have the same shape but not necessarily the same size. similar polygons (p. 462) Two polygons whose corresponding angles are congruent and whose corresponding sides are proportional. similarity ratio (p. 463) The ratio of two corresponding linear measurements in a pair of similar figures. similarity statement (p. 463) A statement that indicates that two polygons are similar by listing the vertices in the order of correspondence. semejantes Dos figuras con la misma forma pero no necesariamente del mismo tamaño. polígonos semejantes Dos polígonos cuyos ángulos correspondientes son congruentes y cuyos lados correspondientes son proporcionales. razón de semejanza Razón de dos medidas lineales correspondientes en un par de figuras semejantes. Similarity ratio: 3.5 _ 2.1 = 5 _ 3 enunciado de semejanza Enunciado que indica que dos polígonos son similares enumerando los vértices en orden de correspondencia. quadrilateral ABCD ∼ quadrilateral EFGH sine (p. 525) In a right triangle, the ratio of the length of the leg opposite ∠A to the length of the hypotenuse. seno En un triángulo rectángulo, razón entre la longitud del cateto opuesto a ∠A y la longitud de la hipotenusa. skew lines (p. 146) Lines that are not coplanar. líneas oblicuas Líneas que no son coplanares. S152 S152 Glossary/Glosario sin A = opposite __ hypotenuse   AE and   CD are skew lines. �� ENGLISH slant height of a regular pyramid (p. 689) The distance from the vertex of a regular pyramid to the midpoint of an edge of the base. SPANISH EXAMPLES altura inclinada de una pirámide regular Distancia desde el vértice de una pirámide regular hasta el punto medio de una arista de la base. slant height of a right cone (p. 690) The distance from the vertex of a right cone to a point on the edge of the base. altura inclinada de un cono regular Distancia desde el vértice de un cono regular hasta un punto en el borde de la base. slide (p. 50) See translation. deslizamiento Ver traslación. slope (p. 182) A measure of the steepness of a line. If ( x 1 , y 1 ) and ( x 2 , y 2 ) are any two points on the line, the slope of the line, known as m, is represented by the y 2 - y 1 _____ equation m = x 2 - x 1 . pendiente Medida de la inclinación de una línea. Dados dos puntos ( ) en una línea, la pendiente de la línea, denominada m, se representa por la ecuación m = y 2 - y 1 _____ x 2 - x 1 . slope-intercept form (p. 190) The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. forma de pendiente-intersección La forma de pendiente-intersección de una ecuación lineal es y = mx + b, donde m es la pendiente y b es la intersección con el eje y. solid (p. 654) A three-dimensional figure. cuerpo geométrico Figura tridimensional. solving a triangle (p. 535) Using given measures to find unknown angle measures or side lengths of a triangle. resolución de un triángulo Utilizar medidas dadas para descubrir las medidas desconocidas de los ángulos o las longitudes laterales de un triángulo. space (p. 671) The set of all points in three dimensions. espacio Conjunto de todos los puntos en tres dimensiones. special parallelogram (p. 410) A rectangle, rhombus, or square. paralelogramo especial Un rectángulo, rombo o cuadrado. special quadrilateral (p. 391) A parallelogram, rectangle, rhombus, square, kite, or trapezoid. cuadrilátero especial Un paralelogramo, rectángulo, rombo, cuadrado, cometa o trapecio. special right triangle (p. 356) A 45°-45°-90
° triangle or a 30°-60°-90° triangle. triángulo rectángulo especial Triángulo de 45°-45°-90° o triángulo de 30°-60°-90°. y = -2x + 4 The slope is -2. The y-intercept is 4. Glossary/Glosario S153 S153 �� ENGLISH sphere (p. 714) The set of points in space that are a fixed distance from a given point called the center of the sphere. SPANISH EXAMPLES esfera Conjunto de puntos en el espacio que se encuentran a una distancia fija de un punto determinado denominado centro de la esfera. spherical geometry (p. 726) A system of geometry defined on a sphere. A line is defined as a great circle of the sphere, and there are no parallel lines. geometría esférica Sistema de geometría definido sobre una esfera. Una línea se define como un gran círculo de la esfera y no existen líneas paralelas. square (p. 410) A quadrilateral with four congruent sides and four right angles. cuadrado Cuadrilátero con cuatro lados congruentes y cuatro ángulos rectos. standard position (p. 687) An angle in standard position has its vertex at the origin and its initial side on the positive x-axis. posición estándar Ángulo cuyo vértice se encuentra en el origen y cuyo lado inicial se encuentra sobre el eje x positivo. straight angle (p. 21) A 180° angle. ángulo llano Ángulo que mide 180°. subtend (p. 772) A segment or arc subtends an angle if the endpoints of the segment or arc lie on the sides of the angle. subtender Un segmento o arco subtiende un ángulo si los extremos del segmento o arco se encuentran sobre los lados del ángulo. If D and F are the endpoints of an arc or chord, and E is a point ̶̶ ̶̶ DF , then ⁀ DF or DF is said not on to subtend ∠DEF. supplementary angles (p. 29) Two angles whose measures have a sum of 180°. ángulos suplementarios Dos ángulos cuyas medidas suman 180°. ∠3 and ∠4 are supplementary angles. surface area (p. 680) The total area of all faces and curved surfaces of a three-dimensional figure. área total Área total de todas las caras y superficies curvas de una figura tridimensional. Surface area = 2 (8) (12) + 2 (8) (6) + 2 (12) (6) = 432 cm 2 symmetry (p. 856) In the transformation of a figure such that the image coincides with the preimage, the image and preimage have symmetry. simetría En la transformación de una figura tal que la imagen coincide con la imagen original, la imagen y la imagen original tienen simetría. S154 S154 Glossary/Glosario �� ENGLISH symmetry about an axis (p. 858) In the transformation of a figure such that there is a line about which a three-dimensional figure can be rotated by an angle greater than 0° and less than 360° so that the image coincides with the preimage, the image and preimage have symmetry about an axis. SPANISH simetría axial En la transformación de una figura tal que existe una línea sobre la cual se puede rotar una figura tridimensional a un ángulo mayor que 0° y menor que 360° de forma que la imagen coincida con la imagen original, la imagen y la imagen original tienen simetría axial. EXAMPLES system of equations (p. 421) A set of two or more equations that have two or more variables. sistema de ecuaciones Conjunto de dos o más ecuaciones que contienen dos o más variables. 2x + 3y = -1 3x - 3y = 4 T tangent circles (p. 747) Two coplanar circles that intersect at exactly one point. If one circle is contained inside the other, they are internally tangent. If not, they are externally tangent. círculos tangentes Dos círculos coplanares que se cruzan únicamente en un punto. Si un círculo contiene a otro, son tangentes internamente. De lo contrario, son tangentes externamente. tangent of an angle (p. 525) In a right triangle, the ratio of the length of the leg opposite ∠A to the length of the leg adjacent to ∠A. tangente de un ángulo En un triángulo rectángulo, razón entre la longitud del cateto opuesto a ∠A y la longitud del cateto adyacente a ∠A. tangent segment (p. 794) A segment of a tangent with one endpoint on the circle. segmento tangente Segmento de una tangente con un extremo en el círculo. tangent of a circle (p. 746) A line that is in the same plane as a circle and intersects the circle at exactly one point. tangente de un círculo Línea que se encuentra en el mismo plano que un círculo y lo cruza únicamente en un punto. tangent of a sphere (p. 805) A line that intersects the sphere at exactly one point. tangente de una esfera Línea que toca la esfera únicamente en un punto. tan A = opposite _ adjacent ̶̶ BC is a tangent segment. Glossary/Glosario S155 S155 �� ENGLISH terminal point of a vector (p. 559) The endpoint of a vector. SPANISH EXAMPLES punto terminal de un vector Extremo de un vector. terminal side (p. 570) For an angle in standard position, the ray that is rotated relative to the positive x-axis. lado terminal Para un ángulo en posición estándar, el rayo que se rota en relación con el eje x positivo. tessellation (p. 863) A repeating pattern of plane figures that completely covers a plane with no gaps or overlaps. teselado Patrón que se repite formado por figuras planas que cubren completamente un plano sin dejar espacios libres y sin superponerse. tetrahedron (p. 669) A polyhedron with four faces. A regular tetrahedron has equilateral triangles as faces, with three faces meeting at each vertex. tetraedro Poliedro con cuatro caras. Las caras de un tetraedro regular son triángulos equiláteros y cada vértice es compartido por tres caras. theorem (p. 110) A statement that has been proven. teorema Enunciado que ha sido demostrado. theoretical probability (p. 214) The ratio of the number of equally likely outcomes in an event to the total number of possible outcomes. probabilidad teórica Razón entre el número de resultados igualmente probables de un suceso y el número total de resultados posibles. In the experiment of rolling a number cube, the theoretical probability of rolling an odd number is 3 __ 6 = 1 __ . 2 three-dimensional coordinate system (p. 671) A space that is divided into eight regions by an x-axis, a y-axis, and a z-axis. The locations, or coordinates, of points are given by ordered triples. sistema de coordenadas tridimensional Espacio dividido en ocho regiones por un eje x, un eje y un eje z. Las ubicaciones, o coordenadas, de los puntos son dadas por tripletas ordenadas. tick marks (p. 13) Marks used on a figure to indicate congruent segments. marcas “\|” Marcas utilizadas en una figura para indicar segmentos congruentes. tiling (p. 862) See tessellation. teselación Ver teselado transformation (p. 50) A change in the position, size, or shape of a figure or graph. transformación Cambio en la posición, tamaño o forma de una figura o gráfica. S156 S156 Glossary/Glosario �� ENGLISH SPANISH EXAMPLES translation (p. 50) A transformation that shifts or slides every point of a figure or graph the same distance in the same direction. translation symmetry (p. 863) A figure has translation symmetry if it can be translated along a vector so that the image coincides with the preimage. traslación Transformación en la que todos los puntos de una figura o gráfica se mueven la misma distancia en la misma dirección. simetría de traslación Una figura tiene simetría de traslación si se puede trasladar a lo largo de un vector de forma tal que la imagen coincida con la imagen original. transversal (p. 147) A line that intersects two coplanar lines at two different points. transversal Línea que corta dos líneas coplanares en dos puntos diferentes. trapezoid (p. 429) A quadrilateral with exactly one pair of parallel sides. triangle (p. 98) A three-sided polygon. trapecio Cuadrilátero con sólo un par de lados paralelos. triángulo Polígono de tres lados. triangle rigidity (p. 242) A property of triangles that states that if the side lengths of a triangle are fixed, the triangle can have only one shape. rigidez del triángulo Propiedad de los triángulos que establece que, si las longitudes de los lados de un triángulo son fijas, el triángulo puede tener sólo una forma. triangulation (p. 223) The method for finding the distance between two points by using them as vertices of a triangle in which one side has a known, or measurable, length. triangulación Método para calcular la distancia entre dos puntos utilizándolos como vértices de un triángulo en el cual un lado tiene una longitud conocida o medible. trigonometric ratio (p. 525) A ratio of two sides of a right triangle. razón trigonométrica Razón entre dos lados de un triángulo rectángulo. trigonometry (p. 514) The study of the measurement of triangles and of trigonometric functions and their applications. trigonometría Estudio de la medición de los triángulos y de las funciones trigonométricas y sus aplicaciones. trisect (p. 25) To divide into three equal parts. trisecar Dividir en tres partes iguales. truth table (p. 128) A table that lists all possible combinations of truth values for a statement and its components. tabla de verdad Tabla en la que se enumeran todas las combinaciones posibles de valores de verdad para un enunciado y sus componentes. sin A = a _ c ; cos A = b _ c ; tan A = a _ b ̶̶̶ AD is trisected. Glossary/Glosario S157 S157 �� ENGLISH SPANISH EXAMPLES truth value (p. 82) A statement can have a truth value of true (T) or false (F). valor de verdad Un enunciado puede tener un valor de verdad verdadero (V) o falso (F). turn (p. 50) See rotation. giro Ver rotación. two-column proof (p. 111) A style of proof in which the statements are written in the left-hand column and the reasons are written in the right-hand column. demostración a dos columnas Estilo de demostración en la que los enunciados se escriben en la columna de la izquierda y las razones en la columna de la derecha. two-point perspective (p. 662) A perspective drawing with two vanishing points. persp
ectiva de dos puntos Dibujo en perspectiva con dos puntos de fuga. U undefined term (p. 6) A basic figure that is not defined in terms of other figures. The undefined terms in geometry are point, line, and plane. término indefinido Figura básica que no está definida en función de otras figuras. Los términos indefinidos en geometría son el punto, la línea y el plano. unit circle (p. 570) A circle with a radius of 1, centered at the origin. círculo unitario Círculo con un radio de 1, centrado en el origen. V vanishing point (p. 662) In a perspective drawing, a point on the horizon where parallel lines appear to meet. punto de fuga En un dibujo en perspectiva, punto en el horizonte donde todas las líneas paralelas parecen encontrarse. vector (p. 559) A quantity that has both magnitude and direction. vector Cantidad que tiene magnitud y dirección. Venn diagram (p. 80) A diagram used to show relationships between sets. diagrama de Venn Diagrama utilizado para mostrar la relación entre conjuntos. S158 S158 Glossary/Glosario �� ENGLISH vertex angle of an isosceles triangle (p. 273) The angle formed by the legs of an isosceles triangle. SPANISH ángulo del vértice de un triángulo isósceles Ángulo formado por los catetos de un triángulo isósceles. EXAMPLES vertex of a cone (p. 654) The point opposite the base of the cone. vértice de un cono Punto opuesto a la base del cono. vertex of a graph (p. 95) A point on a graph. vértice de una gráfica Punto en una gráfica. vertex of a polygon (p. 382) The intersection of two sides of the polygon. vértice de un polígono La intersección de dos lados del polígono. A, B, C, D, and E are vertices of the polygon. vertex of a pyramid (p. 689) The point opposite the base of the pyramid. vértice de una pirámide Punto opuesto a la base de la pirámide. vertex of a three-dimensional figure (p. 654) The point that is the intersection of three or more faces of the figure. vertex of a triangle (p. 216) The intersection of two sides of the triangle. vértice de una figura tridimensional Punto que representa la intersección de tres o más caras de la figura. vértice de un triángulo Intersección de dos lados del triángulo. vertex of an angle (p. 20) The common endpoint of the sides of the angle. vértice de un ángulo Extremo común de los lados del ángulo. vertical angles (p. 30) The nonadjacent angles formed by two intersecting lines. ángulos opuestos por el vértice Ángulos no adyacentes formados por dos rectas que se cruzan. A, B, and C are vertices of △ABC. A is the vertex of ∠CAB. ∠1 and ∠3 are vertical angles. ∠2 and ∠4 are vertical angles. volume (p. 697) The number of nonoverlapping unit cubes of a given size that will exactly fill the interior of a three-dimensional figure. volumen Cantidad de cubos unitarios no superpuestos de un determinado tamaño que llenan exactamente el interior de una figura tridimensional. Volume = (3) (4) (12) = 144 ft 3 Glossary/Glosario S159 S159 �� ENGLISH SPANISH EXAMPLES W whole number (p. 80) The set of natural numbers and zero. número cabal Conjunto de los números naturales y cero. 0, 1, 2, 3, 4, 5, … X x-axis (p. 42) The horizontal axis in a coordinate plane. eje x Eje horizontal en un plano cartesiano. Y y-axis (p. 42) The vertical axis in a coordinate plane. eje y Eje vertical en un plano cartesiano. Z z-axis (p. 671) The third axis in a three-dimensional coordinate system. eje z Tercer eje en un sistema de coordenadas tridimensional. S160 S160 Glossary/Glosario �� Index A AA (angle-angle) similarity, 470 AAS (angle-angle-side) congruence, 254 proof of, 254 Absolute error, S73 Absolute value, 21, S61 equations, see Equations expressions, see Expressions Accuracy, S72 Acute angle, 21 Acute triangle, 216 Addition of vectors, 561–562 Addition Property of Equality, 104 Addition Property of Inequality, 330 Adjacent angles, 28 Adjacent arcs, 757 Adjustable parallels, 836 Advertising, 499 Agriculture, 765 Ahmes Papyrus, 41 Algebra, 40, 100, 109, 115, 124, 156, 159, 162, 164, 217, 224–225, 244, 249, 261, 274, 301–303, 308, 312, 316, 318, 325–326, 341, 355, 384–385, 387, 393, 396, 399, 403, 409, 415, 424, 430, 433–434, 523, 531–532, 590, 597, 605, 620–621, 627, 634–635, 675, 677, 704, 749, 753, 758–759, 762, 775, 777, 788, 792–794, 798–799, 803, 805, 860, S56–S69 The review and development of algebra skills is found throughout this book. absolute value, 19, 21, S61 equations, see Equations expressions, see Expressions binomials, multiplying, 40 coordinate plane, 42, 43, 361, 393, 397, 400, 402, 405, 410, 420–423, 434, 435, S56 area in the, 616–619 circles in the, 799–801 dilations in the, 495–497, 874 distance in the, 43–46 graphing in the, 42 lines in the, 190–193 midpoint in the, 43–46 parallelograms in the, 393 perimeter in the, 616–619 reflections in the, 826 rotations in the, 840 similarity in the, 495–497 strategies for positioning figures in the, 267 transformations in the, 50–52 translations in the, 832 determining whether lines are parallel, perpendicular, or neither, 184 direct variation, 161, 501, S62 equation(s) of circles, 799–805 finding, 805 of a horizontal line, 190 of a line, 303, 304, 305, 306, 308, 311, 312, 313, 315–318, 339 literal, 588, 590, S59 quadratic, 266 solving, 25 linear, 11, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 29, 31, 32, 33, 34, 38, 39, 40, 41, 44, 104–109, 124, 155, 156, 158, 159, 219, 220, 221, 227, 228, 230, 235, 236, 237, 245, 246, 249, 259, 264, 265, 272, 276, 277, 301, 302, 304, 305, 318, 320, 325, 337, 349, 352, 353, 354, 355, 384, 385, 386, 387, 388, 392, 393, 395, 396, 397, 403, 405, 406, 409, 410, 412, 423, 425, 430, 432, 433, 434, 751, 753, 760, 761, 762, 776, 777, 778, 779, 786, 787, 788, 789, 795, 796, 797, 798, 807, 814, S58 literal, 41, 169, S59 quadratic, 27, 228, 230, 235, 237, 246, 259, 277, 326, 349, 350, 352, 353, 354, 355, 388, 415, 430, 432, 433, 434, 494, 752, 771, 777, 796, 797, 798, 804, 814, S66 radical, 49, S68 systems of, 125, 158, 159, 176, 177, 194, 805 by elimination, 152–153, 157, 193, S67 by graphing, 8, 193, 195, 196 by substitution, 316–318, 396, S67 of a vertical line, 190 writing, 11, 18, 25 linear, 19, 31, 32, 33, 34, 38, 39, 40, 41 of lines, 190–197 literal, 41 expressions, 40 evaluating, 19, 162, 164–167, 334, 336, 384, 392, 393, 395, 399, 402, 405, 408–410, 412, 429, 432, 433 simplifying, 19, 36 writing, 229, 788, S57 factoring, to solve quadratic equations, S66 finding slope, see Slope functions, 11, 41, 49, 389, 789 evaluating, 150 factoring to find the zeros of, 55 graphing, 425 identifying, 11, 49, 389, S61 inverse, 533, S62 inverse trigonometric, 533, 534 quadratic, S65 transformations of, 838, S63 graphing functions, see Functions graphing lines, 191, 197 point-slope form, 190, 191, 194, 198, 199 proof of, 190 slope-intercept form, 188, 190, 191, 194 proof of, 196 inequalities compound, 126 graphing, linear, 249, S59 properties of, 330 solving, 805 compound, 330 linear, 26, 109, 172, 175, 176, 249, 338, 341, 343, 345, 435, S60 systems, S68 triangle, 331 in two triangles, 340–342 writing, S59 intercepts x-intercept, 187, 191 finding, 523 identifying, 259 y-intercept, 187–189, 191 finding, 523 identifying, 259 inverse variation, 161 linear equations, see Equations linear inequalities, see Inequalities lines of best fit, 199 literal equations, 169 matrices, S69 monomials, S64 On Track for TAKS, 42, 152–153, 266, 330, 346, 389, 501, 533, 588, 713, 838 ordered pair, 11, 49, S56, see also Coordinate plane polynomial, degree of, S64 quadratic equations, see Equations radical equations, 49, S68 radicals, simplifying, 44, 519–521 rate of change, 182, see also Slope relations, S61 regression, see Lines of best fit sequences, 558 simplifying expressions, see Expressions slope(s), 182–185, 188, 322, 324, 539 finding, 279 formula, 182, 183, 185, 186, 199 of parallel lines, 184–186, 188, 306 of perpendicular lines, 184–186, 189, 306, 617 point-slope form, 303, 305 through two points, 182, 183, 185, 186, 558 of vertical lines, 182 solving equations, see Equations solving inequalities, see Inequalities systems of equations, see Equations x-intercept, see Intercepts y-intercept, see Intercepts Index S161 S161 Algebraic proof, 104–107 Alhambra, 50 Alice’s Adventures in Wonderland, Angle Bisector Theorem, 301 Converse of the, 301 Angle bisectors, 23, 300–303 102 Alternate exterior angles, 147 Alternate Exterior Angles Theorem, constructing, 23 of a triangle, 480 Angle measures, triangle classification 156 Converse of the, 163 proof of the, 164 proof of the, 159 Alternate interior angles, 147 Alternate Interior Angles Theorem, 156 Converse of the, 163 proof of the, 168 proof of the, 156 Altitude of cones, 690 of prisms or cylinders, 680 of pyramids, 689 of triangles, 314–317 Ames room, 149 Andersen, Hans Christian, 167 Angle(s), 20 acute, 21 adjacent, 28 alternate exterior, 147 alternate interior, 147 base, see Base angles central, see Central angles complementary, 29 congruent to a given angle, constructing, 22 corresponding, 147, 231 of depression, 544–546 of elevation, 544–546 exterior, 225, 384 exterior of an, 20 formed by parallel lines and transversals, 155–157 included, 242 inscribed, see Inscribed angles interior, 225 interior of an, 20 measure of an, 20 measuring and constructing, 20–24 naming, 20 obtuse, 21 opposite, of quadrilaterals, 391 pairs of, 28–31 reference, 570 remote interior, 225 right, 21 of rotational symmetry, 857 same-side interior, 147 straight, 21 supplementary, 29 trisecting, 25 types of, 21 vertex, 273 vertical, 30 Angle-angle-side (AAS) congruence, 254 proof of, 254 by, 216 Angle relationships in circles, 780–786 in triangles, 223–226 Angle-side-angle (ASA) congruence, 252 Animation, 53, 105, 835, 842 Annulus, 612 Answers, choosing combinations of, 816–817 Anthropology, 802 Anti
kythera, 792 Antique speakers, 692 Apothem, 601 Applications Advertising, 499 Agriculture, 765 Animation, 53, 835, 842 Anthropology, 802 Archaeology, 262, 787, 793 Architecture, 47, 159, 166, 220, 324, 457, 467, 485, 529, 658, 667, 695, 706, 767, 843, 859, 875 Art, 10, 32, 167, 465, 483, 557, 593, 657, 668, 834, 849, 860, 863, 873, 876 Art History, 52 Astronomy, 227, 274, 494, 720, 752, 844, 877 Aviation, 229, 277, 546, 547, 564 Bicycles, 337 Biology, 75, 77, 83, 100, 185, 604, 685, 715, 784, 857 Bird Watching, 401 Building, 538 Business, 108, 194, 312, 625 Carpentry, 18, 168, 304, 325, 408, 412, 418, 434, 555, 710, 836 Cars, 396, 425 Chemistry, 100, 683, 828, 868 City Planning, 305, 827 Communication, 634, 802 Community, 310 Computer Graphics, 495 Computers, 352 Conservation, 271 Consumer, 48, 684, 760 Crafts, 37, 38, 219, 357, 408, 422, 432, 594 Cycling, 538 Design, 311, 313, 317, 318, 336, 360, 403, 433 Drama, 610 Ecology, 108, 248 Electronics, 692 Engineering, 115, 233, 234, 243, 260, 412, 472, 554, 795, 841, 845 Entertainment, 149, 341, 360, 624, Angle-angle (AA) similarity, 470 625, 803, 833 S162 S162 Index Finance, 108, 522 Food, 195, 603, 656, 701–703, 718, 721 Football, 566 Forestry, 548 Games, 592, 852 Gardening, 422, 597 Geography, 39, 177, 186, 610, 626, 720, 729, 861 Geology, 86, 547, 709, 796, 804 Graphic Design, 498, 752 Health, 343 History, 48, 413, 531, 703, 778 Hobbies, 235, 464, 466, 596, 773 Home Improvement, 596 Indirect Measurement, 323 Industrial Arts, 77 Industry, 344 Interior Decorating, 609, 867 Jewelry, 719 Landscaping, 607, 686, 702 Manufacturing, 38, 754 Marine Biology, 698, 720 Math History, 25, 78, 493, 566, 768 Measurement, 404, 488, 491, 520–522, 531, 547, 596, 605 Mechanics, 434 Media, 88 Meteorology, 85, 476, 675, 703, 797, 801 Movie Rentals, 107 Music, 24, 157, 176, 218, 601 Navigation, 228, 271, 278, 402, 558, 567, 729, 767 Nutrition, 107 Oceanography, 174 Optics, 868 Optometry, 877 Orienteering, 556 Parachute, 302 Parking, 159 Pets, 361 Photography, 385, 459, 475 Physical Fitness, 79 Physics, 25, 565, 861, 867 Political Science, 79, 93 Problem-Solving Applications, 30, 105, 193, 252–253, 315–316, 428, 456, 528, 618, 749, 825 Racing, 392 Real Estate, 486 Recreation, 15, 92, 108, 271, 476, 564, 636, 673, 674, 828, 850 Safety, 349, 353, 386, 395, 530 Sailing, 245 Science, 786 Shipping, 395 Shuffleboard, 305 Social Studies, 403 Space Exploration, 354, 491, 492, 751 Space Shuttle, 548 Sports, 17, 19, 40, 46, 149, 165, 175, 259, 458, 492, 530, 562, 603, 635, 720, 729, 761, 851 Surveying, 25, 224, 256, 257, 263, 276, Assessment Test Prep 353, 474, 547, 556 Technology, 92, 809 Textiles, 125 Theater, 246 Transportation, 183, 194, 360, 620, 631, 633, 866 Travel, 17, 54, 84, 335, 458, 484 Appropriate methods choosing, 372, 373, 616, 619, 620 Appropriate units choosing, 596 Approximating, 37, 335, 360, 460–461, 484, 491–492, 549, 577–579, 796 Arc intercepted, 772 major, 756 minor, 756 Arc Addition Postulate, 757 Arc length, 766 Arc marks, 22 Archaeology, 262, 787, 793 Archery, 635 Archimedes, 599, 703 Architecture, 47, 159, 166, 220, 324, 457, 467, 485, 529, 658, 667, 695, 706, 767, 843, 859, 875 Arcs, 756 adjacent, 757 chords and, 756–759 congruent, 757 measure, 756 Are You Ready?, 3, 71, 143, 213, 297, 377, 451, 515, 585, 651, 743, 821 Area(s), 36, 754, 815, 818 of circles, 37, 600 in the coordinate plane, 616–619 under curves, estimating, 621 of kites, 591 lateral, see Lateral area of lattice polygons, developing Pick’s Theorem for, 613 of parallelograms, 589 perimeter and, 36 proportional, 490 of regular polygons, 601 of rhombuses, 591 of sectors, 764–766 of segments, 765 of spherical triangles, 727 surface, see Surface area of trapezoids, 590 of triangles, 590 Area Addition Postulate, 589 Area ratio, 490 Argument, convincing, writing a, 379 Armstrong, Lance, 337 Arrow notation, 50 Art, 10, 32, 167, 465, 483, 577, 593, 657, 668, 834, 849, 860, 863, 873, 876 Art History, 52 Artifacts, 26 ASA (angle-side-angle) congruence, 252 Chapter Test, 64, 134, 206, 288, 370, 442, 508, 576, 644, 734, 814, 888 College Entrance Exam Practice ACT, 207, 289, 889 SAT, 65, 443, 509 SAT Mathematics Subject Tests, 135, 371, 577, 735, 815 SAT Student-Produced Responses, 645 Cumulative Assessment, 68–69, 138–139, 210–211, 292–293, 374–375, 446–447, 512–513, 580–581, 648–649, 738–739, 818–819, 892–893 Multi-Step TAKS Prep, 34, 58, 102, 126, 180, 200, 238, 280, 328, 364, 406, 436, 478, 502, 542, 568, 614, 638, 678, 724, 770, 806, 854, 880 Multi-Step TAKS Prep questions are also found in every exercise set. Some examples are: 10, 18, 26, 32, 39 Ready to Go On?, 35, 59, 103, 127, 181, 201, 239, 281, 329, 365, 407, 437, 479, 503, 543, 569, 615, 639, 679, 725, 771, 807, 855, 881 Standardized Test Prep, 69, 139, 211, 293, 375, 447, 513, 581, 649, 739, 819, 893 Study Guide: Preview, 21, 72, 144, 214, 298, 378, 452, 516, 586, 652, 744, 822 Study Guide: Review, 60–63, 130–133, 202–205, 284–287, 366–369, 438–441, 504–507, 572–575, 640–643, 730–733, 810–813, 884–887 TAKS Prep, 68–69, 138–139, 210–211, 292–293, 374–375, 446–447, 512–513, 580–581, 648–649, 738–739, 818–819, 892–893 TAKS Tackler Any Question Type Check with a Different Method, 372–373 Estimate, 578–579 Highlight Main Ideas, 890–891 Identify Key Words and Context Clues, 290–291 Interpret Coordinate Graphs, 208–209 Interpret a Diagram, 510–511 Measure to Solve Problems, 736–737 Use a Formula Sheet, 646–647 Gridded Response: Record Your Answer, 136–137 Multiple Choice, Eliminate Answer Choices, 444–445 Recognize Distracters, 816–817 Work Backward, 66–67 Test Prep questions are found in every exercise set. Some examples are: 11, 19, 26, 33, 40 Asterism, 227 Astronomy, 227, 274, 494, 720, 752, 844, 877 Auxiliary line, 223 Aviation, 229, 277, 546, 547, 564 Axioms, see Postulates Axis of a cone, 690 of a cylinder, 681 polar, 808 of symmetry, 362, 856 symmetry about an, 858 B Bar graph, S78 Bascule bridges, 895 Base(s) of cones, 654 of cylinders, 654 of isosceles trapezoids, 426 of isosceles triangles, 273 of prisms, 654 of pyramids, 654 of trapezoids, 429 of triangles, 36 Base angles, 273 of isosceles triangles, 273 of trapezoids, 429 Base edges, 680 Baseball fields, 43 Bathysphere, 720 Bearing of a vector, 560 Berg, Bryan, 146 Between, 12, 14 Biconditional statements, 96–98 Bicycles, 337 Big Bend National Park, 626, 678 Big Tex, 518, 520 Binomials, multiplication of, 40, 592 Biology, 75, 77, 83, 100, 185, 604, 685, 715, 784, 857 Bird-Watching, 401 Bisector angle, see Angle bisectors perpendicular, see Perpendicular bisectors Bisects, 15 Blood, 100 Blood sugar, 687 Bob Bullock Texas State History Museum, 212 Box-and-Whisker Plot, S80 Bridges, 115, 895 Broken Obelisk, 296 Building, 538 Burnham, Daniel, 220 Business, 108, 194, 312, 625 Butterflies, 81, 820 Index S163S163 circumscribe an equilateral triangle describe, 8, 37, 122, 148, 174, 193, C Calculator graphing, see Graphing calculator Card structures, 146 Career Path Desktop Publisher, 87 Electrician, 320 Emergency Medical Services Program, 237 Furniture Maker, 805 Photogrammetrist, 494 Technical writer, 612 Careers, xviii Carpentry, 18, 168, 304, 325, 408, 412, 418, 434, 555, 710, 836 Cars, 396, 425 Caution!, 13, 90, 128, 184, 223, 243, 268, 333, 419, 526, 527, 545, 561, 571, 681, 726 Cavalieri’s principle, 697, 699 Cavanaugh Flight Museum, 294 Celsius (C) degrees, 105 Center of a circle, 600 constructing, 774 of dilation, 872 of gravity, 314 of regular polygons, 601 of a sphere, 714 Central angles, 755, 756 of regular polygons, 601 Central Library, 6 Central tendency, measures of, 477, S76 Centroid of a triangle, 314 constructing, 314 Centroid Theorem, 314 Ceva, Giovanni, 318 Changing dimensions, see Effects of changing dimensions Chapter Test, 64, 134, 206, 288, 370, 442, 508, 576, 644, 734, 814, 888, see also Assessment Checking solutions, 125 Chemistry, 100, 683, 828, 868 Chess, 848, 852 Chiral molecules, 828 Chokwe design, 860 Choosing appropriate methods, 372, 373, 616, 619, 620 about a, 779 circumscribed, 308, 313 concentric, 747 constructing a tangent to a from an exterior point, 779 at a point, 748 in the coordinate plane, 799–801 developing formulas for, 600–602 equations of, see Equations of circles exterior of a, 746 graphing, 800–805 great, see Great circle inscribed, 309, 313 interior of a, 746 lines that intersect, 746–750 sector of a, 764 segment of a, 765 segment relationships in, 790–795 segments that intersect, 746 tangent, 747 through three noncollinear points, constructing, 763 unit, see Unit circle Circle graphs, 26, 755, S80 Circumcenter of a triangle, 307 constructing, 307 Circumcenter Theorem, 308 proof of the, 308 Circumference, 37, 600 and area of a circle, 37 of a great circle of a sphere, 769 Circumscribe, 308 a circle about a triangle, constructing, 313, 778 an equilateral triangle about a circle, constructing, 779 Circumscribed circle, 308, 313 City Planning, 305, 827 Classifying pairs of lines, 192 triangles, 230 Clinometer, 550 Clouds, 675 CN Tower, 843 Coinciding lines, 192 Coins, 741 College Entrance Exam Practice, see also Assessment ACT, 207, 289, 889 SAT, 65, 443, 509 SAT Mathematics Subject Tests, 135, 371, 577, 735, 815 appropriate units, 596 combinations of answers, 816–817 Chord-Chord Product Theorem, 792 SAT Student-Produced Responses, 645 Collinear points, 6 Common Angles Theorem, 117 proof of the, 797 Chords, 746 arcs and, 756–759 Circle(s), 600, 742–819 angle relationships in, 780–786 area of, 37, 600 centers of, 600 constructing, 774 circumference of a, 37, 600 circumscribe a, about a triangle, 778 proof of the, 117 Common Segments Theorem, 118 Converse of the, 119 proof of the, 119 proof of the, 118 Common tangent, 748 Communication, 634, 802 compare, 98, 148, 165, 185, 245, 593, 656, 700, 717 S164 S164 Index 218, 245, 342, 359, 537, 563, 602, 608, 619, 664, 683, 841, 850, 858, 874 discuss, 624 explain, 8, 16, 24, 31, 46, 52, 76, 90, 107, 122, 157, 165, 174
, 185, 193, 218, 226, 245, 255, 262, 269, 276, 303, 310, 324, 335, 342, 352, 359, 385, 421, 431, 457, 473, 490, 497, 520, 546, 563, 593, 602, 608, 619, 633, 673, 683, 692, 700, 708, 717, 750, 766, 775, 786, 801, 826, 833, 850, 858, 866, 874 justify, 394, 801 list, 113, 148, 352 name, 8, 24, 218, 233, 593 sketch (draw), 8, 16, 24, 31, 46, 52, 218, 226, 255, 269, 276, 310, 317, 359, 385, 394, 401, 421, 464, 473, 484, 490, 528, 546, 554, 593, 750, 759, 786, 801, 850, 858 summarize, 352 write, 16, 31, 37, 46, 255, 262, 303, 324, 342, 401, 411, 421, 431, 457, 464, 473, 484, 490, 497, 520, 528, 537, 546, 619, 683, 692, 700, 750, 759, 775, 786, 801 Community, 310 Comparing surface areas and volumes, 722–723 Comparison Property of Inequality, 330 Compass, 14, F47 Compass and straightedge, see Construction(s), using compass and straightedge Complement of an event, 628 Complementary angles, 29 Complements, 29 Component form of a vector, 559 Composite figures, 606–608, 818 measuring, 611 Compositions of transformations, 848–850 Compound inequalities, 126 solving, 330 Compound statements, 128 Computer-animated films, 835 Computer Graphics, 495 Computers, 352 Concave polygons, 383 Concentric circles, 747 Conclusion, 81 Concurrency, point of, 307 Concurrent lines, 307 Conditional statements, 81–84 Conditionals, related, 83 Conditions for Parallelograms, 398, 399 proof of the, 398 Conditions for Rectangles, 418 Conditions for Rhombuses, 419 Cones, 654 altitude of, 690 axis of, 690 double, 660 drawing, 653 frustum of, 668, 696 oblique, 690 right, see Right cones surface area of, 689–692 vertex of, 690 volume of, 705–708 Congruence properties of, 106 triangle, see Triangle congruence Congruence transformations, 824, 854 Congruent angles, 22 Congruent arcs, 757 Congruent Complements Theorem, 112 proof of the, 112 Congruent polygons, 231 properties of, 231 Congruent segments, 13 Congruent Supplements Theorem, 111 proof of the, 111 Congruent triangles, 231–233 constructing using ASA, 253 using SAS, 243 properties of, 231 Conjecture, 74, 171, 188, 189, 222, 241, 250, 251, 278 making a, 321, 331, 381, 390, 416, 417, 426, 613, 669, 676, 781, 790, 847 using deductive reasoning to verify a, 88–90 using inductive reasoning to make a, 74–76 Conjunction, 128 Conservation, 271 Constant of variation, 501, S62 Construction(s), 14, 17, 79, 177, 248, 258, 306, 313, 404, 424, 487 For a complete list, see page S87 angle bisector, 23 perpendicular lines, 179 proving valid, 282–283 segment bisector, 16 using compass and straightedge angle congruent to a given angle, 22 center of a circle, 774 centroid of a triangle, 314 circle through three noncollinear points, 763 circumcenter of a triangle, 307 circumscribe a circle about a triangle, 778 midsegment of a triangle, 327 orthocenter of a triangle, 320 parallel lines, 163, 170–171, 179 parallelogram, 404 perpendicular bisector of a segment, 172 perpendicular lines, 179 reflections, 829 regular polygons, 380–381 decagon, 381 dodecagon, 380 hexagon, 380 octagon, 380 pentagon, 381 square, 380 rhombus, 415 right triangle, 258 rotations, 844 segment congruent to a given segment, 14 segment of given length, 18 tangent to a circle at a point, 748 tangent to a circle from an exterior point, 779 translations, 836 using geometry software, 154, 480, 781 midpoint, 12 special points in triangles, 321 transformations, 56–57 congruent triangles, 249 similar triangles, 468–469 using patty paper midpoint, 16 parallel lines, 171 reflect a figure, 824 rotate a figure, 839 translate a figure, 831 Consumer Application, 48, 684, 760 Contraction, 873 Contradiction, proof by, 332 Contrapositive, 83 Law of, 83 Converse, 83 of a theorem, 162 Converse of the Alternate Exterior Angles Theorem, 163 proof of the, 164 Converse of the Alternate Interior Angles Theorem, 163 proof of the, 168 Converse of the Angle Bisector Theorem, 301 Converse of the Common Segments Theorem, 119 proof of the, 118 Converse of the Corresponding circumscribe an equilateral triangle Angles Postulate, 162 about a circle, 779 congruent triangles using ASA, 253 congruent triangles using SAS, 243 dilations, 872, 878 equilateral triangle, 220 incenter of a triangle, 313 irrational numbers, 363 kites, 435 line parallel to side of triangle, 481 Converse of the Hinge Theorem, 340 proof of the, 340 Converse of the Isosceles Triangle Theorem, 273 Converse of the Perpendicular Bisector Theorem, 300 Converse of the Pythagorean Theorem, 350 Converse of the Same-Side Interior Angles Theorem, 163 proof of the, 168 Converse of the Triangle Proportionality Theorem, 482 Convex polygons, 383 Convincing argument, writing a, 379 Coordinate plane, 42, 43, 361, 393, 397, 400, 402, 405, 410, 420–423, 434, 435, S56 area in the, 616–619 circles in the, 799–801 dilations in the, 495–497, 874 distance in the, 43–46 graphing in the, 42, 208–209 lines in the, 190–193 midpoint in the, 43–46 parallelograms in the, 393 perimeter in the, 616–619 reflections in the, 826 rotations in the, 840 similarity in the, 495–497 strategies for positioning figures in the, 267 transformations in the, 50–52 translations in the, 832 Coordinate proof, 267–269, 275, 313, 319, 355, 434 Coordinates, 3, 13, 317, 319, 397 finding, 753 polar, 808–809 Coplanar points, 6 Cornell system of note taking, 145 Corollaries, 224, 228, 274–275, 519, 773, 778 For a complete list, see pages S82–S87 Correlation, 397 Corresponding angles, 147, 231 Corresponding Angles Postulate, 155 Converse of the, 162 Corresponding Parts of Congruent Triangles are Congruent (CPCTC), 260–262 Corresponding sides, 231 Cosecant, 531 Cosine, 525, 841 Cosines, Law of, 551–554 proof of the, 557 Cotangent, 531 Countdown to TAKS, TX4–TX27 Counterclockwise, 42, 55 Counterexamples, 75 CPCTC (Corresponding Parts of Congruent Triangles are Congruent), 260–262 Crafts, 37, 38, 219, 357, 408, 422, 432, 594, 660, 693, 776 Critical Thinking Critical Thinking questions appear in every exercise set. Some examples: 10, 11, 19, 26, 32 Cross products, 455 Cross Products Property, 455 Cross section, 656 Index S165S165 Cubes, 654, 669 surface area of, 680 Cumulative Assessment, see Assessment Curves, area under, estimating, 621 Customary system of measurement, S70, back cover Cycling, 538 Cylinders, 654 altitude of, 680 axis of, 681 drawing, 653 lateral surface of, 681 oblique, see Oblique cylinders right, see Right cylinders surface area of, 680–683 volume of, 697–700 D Data, 756 displaying, S78–S81 misleading, S81 organizing and describing, S47, S76, S78–S81 Data Analysis, 26, 860, S76, S80 On Track for TAKS, 198–199, 755 Data Collection, 196, 199, 761 Decagons, 382 regular, 381 Deductive reasoning, 88 using, to verify conjectures, 88–90 Definitions, 96–98 Degrees, 20 DeMorgan’s Laws, 129 Denominator, 451 Dentzel, Gustav, 803 Depression, angles of, 544–546 Deriving formulas, 37, 39, 220, 541, 696 the Pythagorean Theorem, 522 Design, 311, 313, 317, 318, 336, 360, 403, 433 Detachment, Law of, 89 Diagonal, 48 of the polygon, 382 of a right rectangular prism, 671 Diagrams, 73, S40 interpreting, 510–511 Diameter, 37, 747 Dilations, 495, 872–874 center of, 872 in the coordinate plane, 495–497, 874 of figures, constructing, 878 Dimensions changing, see Effects of changing dimensions three, see Three dimensions Direct reasoning, 332 Direct variation, 161, 501, S62 Direction of a vector, 560 Disjunction, 128 Disjunctive Inference, Law of, 129 Displacement, 703 Distance, 13 solving, 25 in the coordinate plane, 43–46 between a point and a line, 301 from a point to a line, 172 Distance Formula, 44 proof of the, 354 in three dimensions, 672 Distance Function, using, 263, 271, 272 Distributive Property, 104, S51 Division Property of Equality, 104 Division Property of Inequality, 109, 330 Dodecagons, 382 regular, 380 Dodecahedron, 669 Domain, 41, 389, 405, 533, 547 Double cone, 660 Drama, 610 Drawing(s), 17 diagram that represents information, 19 isometric, 662 one- and two-point perspective, 668 orthographic, 661 perspective, 662 segments, 14 Dual of a tessellation, 868 Dulac, Edmund, 167 E Earthquakes, 804 Ecology, 108, 248 Edge base, 680 lateral, 680 of a three-dimensional figure, 654 Effects of changing dimensions, 683, 691, 700, 708, 713, 716 proportionally, 622–624 Egypt, ancient, 353 Electronics, 692 The Elements, 257 Elevation, angles of, 544–546 Elimination, solving systems of equations by, 152–153, 157, 193, S67 Endpoints, 7, 9 Engineering, 115, 233, 234, 243, 260, 412, 472, 554, 795, 841, 845 Enlargement, 495, 873 Entertainment, 149, 341, 360, 624, 625, 803, 833 Epicenter, 804 Equal vectors, 561 Equality, properties of, 104 Equation of a Circle Theorem, 799 Equations of circles, 799 finding, 805 of a horizontal line, 190 of lines, 303–306, 308, 311–313, 315–318, 339 literal, 588, 590 quadratic, 266 linear, 11, 15–19, 22–26, 29, 31–34, 38–41, 44, 104–109, 124, 155, 156, 158, 159, 219–221, 227, 228, 230, 235–237, 245, 246, 249, 259, 264, 265, 272, 276, 277, 301, 302, 304, 305, 318, 320, 325, 337, 349, 352–355, 384–388, 392, 393, 395–397, 403, 405, 406, 409, 410, 412, 423, 425, 430, 432, 433, 434, 751, 753, 760–762, 776–779, 786–789, 795–798, 807, 814, S58 literal, 41, 169 quadratic, 27, 228, 230, 235, 237, 246, 259, 277, 326, 349, 350, 352, 355, 388, 415, 430, 432–434, 494, 752, 771, 777, 796–798, 804, 814, S66 radical, 49 systems of, 125, 158, 159, 176, 177, 194, 805 by elimination, 152–153, 157, 193, S67 by graphing, 8, 193, 195, 196 by substitution, 316–318, 396, S67 of a vertical line, 190 writing, 11, 18, 25 linear, 19, 31–34, 38–41 of lines, 190–197 literal, 41 Equiangular Triangle Corollary, 275 Equiangular triangles, 216 Equidistant, 300, 746, 799 Equilateral triangle, circumscribed about a circle, 779 Equilateral Triangle Corollary, 274 Equilateral triangles, 217, 273–276 constructing, 220 Error Analysis, 18, 39, 92, 124, 160, 195, 236, 258, 325, 353, 387, 413, 423, 476, 499, 522, 540, 557, 604, 634, 659, 711, 762, 797, 852, 877 Escher, M. C., 861, 868, 8
76, 880 Estimating area under curves, 621 Estimation, 25, 37, 41, 77, 108, 177, 195, 229, 278, 325, 361, 387, 433, 466, 492, 493, 538, 565, 611, 621, 676, 719, 768, 803, 844, 877, S52 rounding and, S52 Estimation strategies, 578–579 Euclid, 257, 460 Euler, Leonhard, 78 Euler line, 321 Euler’s Formula, 670 Event, 628 complement of an, 628 Exam, final, preparing for, 823 Expansion, 873 Experiment, 628 fair, 628 Experimental probability, 798 Exponents, S53 properties of, S54 S166 S166 Index Expressions, 3 evaluating, 19, 162, 164–167, 334, 336, 384, 392, 393, 395, 399, 402, 405, 408–410, 412, 429, 432, 433, S57 simplifying, 19, 36, S50 writing, 229, 788, S57 Extended Response, 69, 139, 178, 211, 230, 259, 293, 355, 375, 425, 447, 487, 513, 581, 621, 649, 739, 819, 893 write extended responses, 290–291 Extension Introduction to Symbolic Logic, 128–129 Polar Coordinates, 808–809 Proving Constructions Valid, 282–283 Spherical Geometry, 726–727 Trigonometry and the Unit Circle, 570–571 Using Patterns to Generate Fractals, 882 Exterior, 225 of an angle, 20 of a circle, 746 Exterior Angle Theorem, 225 Exterior angles, 225, 384 Exterior point, constructing tangent to a circle from an, 779 External secant segment, 793 Extremes, 455 F Face lateral, 680 of a three-dimensional figure, 654 Factoring to find the zeros of each function, 55 solving by, 388, S66 using, 279 Fahrenheit (F) degrees, 105 Fair experiment, 628 Fair Park, 803 Fiber-optic cables, 28 Fibonacci sequence, 78, 461 Figures composite, see Composite figures three-dimensional, see Three- dimensional figures Final exam, preparing for your, 823 Finance, 108, 522 Finding slope, see Slope Fitness Link, 539 Flash cards, 587 Flatiron Building, 220 Flips, see Reflections Flowchart proofs, see Proofs, flowchart FOIL, 592 Food, 195, 603, 656, 701–703, 718, 721 Football, 566 Forbidden City, 48 Forestry, 548 Formula sheet, using a, 646–647 Formulas, see back cover deriving, 39, 541, 696 developing, 589–591, 600, 601 for circles, 600 for regular polygons, 601 for triangles, 590 for quadrilaterals, 589–591 functional relationships in, 713, S63 memorize, 587 in three dimensions, 670–673 using, 36–37 45°-45°-90° triangle, 356 Four Chromatic Gates, 650 Fractals, 882 Freescale Marathon, 140 Fresnel, Augustine, 894 Fresnel lenses, 894 Frieze pattern, 863 Frustum of a cone, 668, 696 of a pyramid, 696 Functional relationships in formulas, 713, S63 Functions, 11, 41, 49, 389, 789 evaluating, 150 factoring to find the zeros of, 55 graphing, 425 identifying, 11, 49, 389, S61 inverse, 533, S62 inverse trigonometric, 533, 534 quadratic, S65 transformations of, 838, S63 G Games, 592, 852 Gardening, 422, 597 Garfield, James, 595 Gears, 845 General translations in the coordinate plane, 832 Geodesic dome, 234 Geography, 39, 177, 186, 610, 626, 720, 729, 861 Geology, 86, 547, 709, 796, 804 Geometric mean, 819 Geometric Means Corollaries, 519, 540 Geometric probability, 630–633 using, to estimate π, 637 Geometric proof, 110–113 Geometry hyperbolic, 729 non-Euclidean, 726 spherical, see Spherical geometry using formulas in, 36–37 Geometry Lab, see also Technology lab Construct Parallel Lines, 170–171 Construct Perpendicular Lines, 179 Construct Regular Polygons, 380–381 Design Plans for Proofs, 117 Develop π, 598–599 Develop Pick’s Theorem for Area of Lattice Polygons, 613 Develop the Triangle Sum Theorem, 222 Explore Properties of Parallelograms, 390 Explore SSS and SAS Triangle Congruence, 240–241 Explore Triangle Inequalities, 331 Graph Irrational Numbers, 363 Hands-on Proof of the Pythagorean Theorem, 347 Indirect Measurement Using Trigonometry, 550 Model Right and Oblique Cylinders, 688 Solve Logic Puzzles, 94–95 Use Geometric Probability to Estimate π, 637 Use Nets to Create Polyhedrons, 669 Use Transformations to Extend Tessellations, 870–871 Geometry software, 12, 27, 56–57, 154, 230, 416–417, 668, 879, F47, see also Construction(s) Geometry symbols, reading, 215 Get Organized, see Graphic organizers Given statement, 111 Glide reflection, 848, 851 Glide reflection symmetry, 863 Global Positioning System, 556 Glossary, S115–S160 go.hrw.com, see Online Resources Goldbach, 78 Golden Ratio, 460–461 Graphic Design, 498, 752 Graphic organizers Graphic Organizers are available for every lesson. Some examples are: 8, 16, 24, 31, 37 Graphing in the coordinate plane, 42 irrational numbers, 363 Graphing calculator, 188–189, 196, 199, 761, 800, 846–847, 879 Graphing functions, see Functions Graphing lines, 191, 197, 259 point-slope form, 190, 191, 194, 198, 199 proof of, 190, 196 slope-intercept form, 188, 190, 191, 194 proof of, 196 Graphs, bar, S78 circle, 755 histogram, S78 misleading, S81 Gravity, center of, 314 Great circle, 714, 726 of a sphere, circumference of a, 769 Great Texas Balloon Race, 295 Gridded Response, 11, 69, 93, 109, 136–137, 139, 169, 211, 221, 279, 293, 313, 355, 362, 373, 375, 397, 435, 447, 477, 500, 510, 511, 513, 540, 567, 578, 579, 581, 597, 627, 647, 649, 687, 712, 739, 763, 769, 789, 819, 845, 878, 890, 891, 893 Record Your Answer, 136–137 Index S167S167 H Hands-on proof of the Pythagorean Theorem, 347 Hayes, Joanna, 19 Head-to-tail vector addition method, 561 Health, 343 Height of triangle, 36 Helpful Hint, 6, 43, 83, 98, 105, 110, 112, 119, 146, 147, 156, 226, 231, 232, 261, 307, 316, 332, 334, 391, 400, 401, 410, 464, 483, 488, 495, 520, 535, 546, 551, 553, 554, 601, 623, 631, 663, 749, 764, 784, 800, 840, 863, 872 Hemisphere, 714 Henry VIII, 305 Heptagon, 382 Hexagon, 382 regular, 380, 819 Highlighting main ideas, 890–891 Hinge Theorem, 340 Converse of the, 340 proof of the, 340 Hippocrates, 611 Histogram, 677, S78 History, 48, 413, 531, 566, 595, 703, 778 math, see Math History HL (hypotenuse-leg) congruence, 255 Hobbies, 235, 464, 466, 596, 773 Hogan, 803 Home Improvement, 596 Homework Help Online Homework Help Online is available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 9, 17, 24, 31, 38 Horizon, 662 Horizontal line, equation of a, 190 Horizontal translations in the coordinate plane, 832 of parent functions, 838 Hot Tip, 65, 135, 207, 289, 371, 443, 509, 577, 645, 735, 815, 889 How to Study Geometry, xx Hurricanes, 476 Hypatia, 768 Hyperbolic geometry, 729 Hypotenuse, 45 Hypotenuse-leg (HL) congruence, 255 Hypothesis, 81 I Icosahedron, 669 Identity, Pythagorean, 531–532 Image, 50 Incenter of a triangle, 309 Incenter Theorem, 309 S168 S168 Index Included angles, 242 Included sides, 252 Indirect measurement, 323, 488 using trigonometry, 550 Indirect proof, 332–335, 339 Inductive reasoning, 74, 75 using, to make conjectures, 74–76 Industrial Arts, 77 Industry, 344 Inequalities compound, 126 graphing, linear, 249, S59 properties of, 330 solving, 805 compound, 330 linear, 26, 109, 172, 175, 176, 249, 338, 341, 343, 345, 435, S60 systems, S68 triangle, exploring, 331 in two triangles, 340–342 writing, S59 Information not enough, 247, 248, 250, 405, 420, 422, 423, 425, 437, 440, 442, 446, 473, 512, 554, 556 too much, 209 Inscribed Angle Theorem, 772 proof of the, 772, 778 Inscribed angles, 772–775 Inscribed circle, 309, 313 Inscribed polygons, 380 Integers, 80, S53 Intercepted arc, 772 Intercepts x-intercept, 187, 191 finding, 523 identifying, 259 y-intercept, 187–189, 191 finding, 523 identifying, 259 Interior, 225 of an angle, 20 of a circle, 746 Interior angles, 225 Interior Decorating, 609, 867 Interpreting diagrams, 510–511 Intersecting lines, 192 Intersections, 8 of lines and planes, 8 Inverse, 83 Inverse functions, 533, S62 Inverse trigonometric functions, 533, 534 Inverse variation, 161 Irrational numbers, 37, 80, S53 graphing, 363 Irregular polygons, 382 Isle of Man, 861 Isometric drawings, 662 Isometry, 824 Isosceles trapezoids, 426, 429 bases of, 426 legs of, 426 Isosceles Triangle Theorem, 273 Converse of the, 273 proof of the, 273 Isosceles triangles, 217, 273–276 legs of, 273 Iteration, 882 J Jewelry, 719 Johnson, Lyndon B., 583 Johnson Space Center, 548 K Kaleidoscope, 868 Kite, 427, 433 area of, 591 constructing, 435 properties of, 427–429 proof of, 427 Know It Note Know-It Notes are found throughout this book. Some examples: 6, 7, 8, 13, 14 Koch snowflake fractal, 882 L Lady Bird Johnson Wildflower Center, 607 Land Development, 160 Landscaping, 31, 607, 686, 702 Lateral area, 818 of prisms, 680 of regular pyramids, 689 of right cones, 690 of right cylinders, 681 of right prisms, 680 Lateral edge, 680 Lateral face, 680 Lateral surface of a cylinder, 681 Lattice polygons, 613 area of, developing Pick’s Theorem for, 613 Law of Contrapositive, 83 Law of Cosines, 551–554 proof of the, 557 Law of Detachment, 89 Law of Disjunctive Inference, 129 Law of Sines, 551–554 ambiguous case of the, 556 Law of Syllogism, 89 Learning math vocabulary, 299 Legs of isosceles triangles, 273 of right triangles, 45 of trapezoids, 426, 429 Length, 13 arc, see Arc length Lift bridges, 895 Lighthouse Rock, 450 Lincoln Memorial, 489 Line(s), 6, 7 auxiliary, 223 of best fit, 199 concurrent, 307 in the coordinate plane, 190–193 distance from a point to a, 172, 301 graphing, see Graphing lines pairs of, classifying, 192 parallel, see Parallel lines parallel to side of triangle, constructing, 481 perpendicular, see Perpendicular lines and planes, intersection of, 8 proving parallel, 162–165 skew, 146 of symmetry, 318, 856 that intersect circles, 746–750 Line symmetry, 856 Linear equations, see Equations Linear inequalities, see Inequalities Linear pair, 28 Linear Pair Theorem, 110 proof of the, 111 Linear units, 36 Link, xix, see also Texas Link Animation, 835 Architecture, 159, 220, 695 Astronomy, 752 Biology, 100, 604 Chemistry, 828 Conservation, 271 Design, 313 Ecology, 248 Electronics, 692 Entertainment, 149 Fitness, 539 Food, 195 Geology, 86, 804 History, 413, 531, 566, 595 Marine Biology, 720 Math History, 41, 78, 257, 318, 493, 611, 703, 768 Measurement, 404 Mechanics, 434 Meteorology, 675, 797 Monument, 466 Naviga
tion, 278 Recreation, 92 Shuffleboard, 305 Sports, 19, 635 Surveying, 353, 556 Travel, 458 Literal equations, 41, 169, 588, 590 Locus, 300, 302, 304, 306, 600, 714, 743, 804 Logic puzzles, solving, 94–95 Logically equivalent statements, 83 Lune, 611 Lunette, 767 Luxor Hotel, 159 M Madurodam, 458 Magnitude of a vector, 560 Main ideas, highlighting, 890–891 Major arc, 756 Make a Conjecture, 321, 331, 381, 390, 416, 417, 426, 613, 669, 676, 781, 790, 847, see also Conjecture Manufacturing, 38, 754 Mapping, 50 Marine Biology, 698, 720 Math Builders, xxiii–xxvii Math History, 25, 41, 78, 257, 318, 493, 566, 611, 703, 768 Math vocabulary, learning, 299 Matrices operations, S69 point, 846 transformations with, 846–847 McDonald Observatory, 234 Mean, 11, 43, S76 geometric, 819 Means, 455 Measurement, 404, 488, 491, 520–522, 531, 547, 585, 596, 599, 605, 611 back cover absolute error, S73 indirect, see Indirect measurement accuracy, precision, and tolerance, S72 choose appropriate units, S74 customary system of, S70, back cover metric system of, 3, S70, back cover nonstandard units, S74 relative error, S73 rates, S70 significant digits, S73 tools of, choose appropriate, S75 units, S70–S71 Measures of central tendency, 477, S76 Measuring to solve problems, 736–737 Mechanics, 434 Media, 88 Median(s), 11, S76 of triangles, 314–317 Memorize formulas, 587 Meteorology, 85, 476, 675, 703, 797, 801 Meters, 36 Metric system of measurement, 3, S70, back cover Metronome, 24 Midpoint, 12, 15 constructing, 16 in the coordinate plane, 43–46 Midpoint Formula, 43 in three dimensions, 672 Midsegment triangle, 322 Midsegments of trapezoids, 431 of triangles, constructing, 327 Migration patterns, 74 Minor arc, 756 Minute Maid Park, 43 Minutes (in degrees), 27 Möbius, August Ferdinand, 566 Mode, 11, 345, S76 Model make a, S41 probability, 630 Modeling oblique cylinders, 688 right cylinders, 688 Mohs’ scale, 86 Monument Link, 466 Mosaic, 376, 876 Motions, rigid, 824 Moveable bridges, 895 Movie Rentals, 107 Multi-Step Multi-Step questions appear in every exercise set. Some examples: 11, 17, 24, 25, 26 Multi-Step TAKS Prep, 34, 58, 102, 126, 180, 200, 238, 280, 328, 364, 406, 436, 478, 502, 542, 568, 614, 638, 678, 724, 770, 806, 854, 880, see also Assessment Multi-Step TAKS Prep questions are also found in every exercise set. Some examples are: 10, 18, 26, 32, 39 Multiple Choice, 66–69, 138–139, 210–211, 372–374, 444–447, 510–513, 578–581, 646–649, 736, 738–739, 816–819, 891–893 Choose Combinations of Answers, 816–817 Eliminate Answer Choices, 444–445 Work Backward, 66–67 Multiple Representations, 6, 7, 21, 50, 80–81, 83, 128, 173, 226, 255, 330, 350, 429, 455, 462, 525, 528, 561, 630, 669, 681, 690, 746–748, 756, 764–766, 785 Multiplication of binomials, 40, 592 scalar, 566 Multiplication Property of Equality, 104 Multiplication Property of Inequality, 109, 330 Municipal Marina, 142 Music, 24, 157, 176, 218, 601 Musical triangles, 218 Mystery spots, 150, 180 N n-gons, 382 Naming angles, 20 Natural numbers, 41, 80, S53 Navigation, 228, 271, 278, 402, 558, 567, 729, 767 Negation, 82 of a vector, 566 Nets, 655, 669 Network, 95 New Madrid earthquake, 804 Non-Euclidean geometry, 726 Nonagons, 382 Index S169S169 Noncollinear points, 6 three, constructing circle through, 763 Noncoplanar points, 6 Normal curve, standard, 860 Not enough information, 247, 248, 250, 405, 420, 422, 423, 425, 437, 440, 442, 446, 473, 512, 554, 556 Note taking Strategies, see Reading and Writing Math Number Theory, On Track for TAKS, 80 Numbers classifying, S53 estimating, S52 irrational, see Irrational numbers natural, 41, 80, S53 properties of, S51 rational, 80, S53 real, S50, S53 rounding, S52 whole, 80, S53 Numerator, 451 Nutrition, 107 O Oblique cones, 690 Oblique cylinders, 681 modeling, 688 Oblique prism, 680 Obtuse angles, 21 Obtuse triangles, 216 Oceanography, 174 Octagon, 382 regular, 380 Octahedron, 669 Olympic Games, 2004, 19 Olympus Mons, 752 On Track for TAKS Parent Resources Online is available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 9, 17, 24, 31, 38 TAKS Practice Online, 68, 138, 210, 292, 374, 446, 512, 580, 648, 738, 818, 892 Op art, 860 Opposite angles of quadrilaterals, 391 Opposite rays, 7 Opposite reciprocals, 184 Opposite sides of quadrilaterals, 391 Optics, 868 Optometry, 877 Order of operations, 3, S50 of rotational symmetry, 857 Ordered pair, 11, 49, S56, see also Coordinate plane Ordered triples, 671 Orienteering, 252, 556 Origami, 238, 594 Origin, 42, S56 Orthocenter of a triangle, 316 constructing, 320 Orthographic drawings, 661 Outcome, 628 P Pairs of angles, 28 of circles, 747 of lines, 192 Algebra, 42, 152–153, 266, 330, 346, classifying, 192 389, 501, 533, 588, 713, 838 Data Analysis, 198–199, 755 Number Theory, 80 Probability, 628–629 One-point perspective, 662 drawing figures in, 668 One-to-one correspondence, 20 Online Resources Career Resources Online, 87, 237, 320, 494, 612, 805 Chapter Project Online, 2, 70, 142, 212, 296, 376, 450, 514, 584, 650, 742, 820 Homework Help Online Homework Help Online is available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 9, 17, 24, 31, 38 Lab Resources Online, 56, 154, 188, 250, 321, 426, 460, 468, 480, 524, 780, 790, 846 Parent Resources Online Parabola, S65 Parachute, 302 Paragraph proofs, see Proofs, paragraph Parallel lines, 142–211 constructing, 163, 170–171, 179 defined, 146 exploring, 154, 188–189 proving, 162–165 slopes of, 184–186, 188, 192, 306 and transversals, 155–157 Parallel Lines Theorem, 184 Parallel planes, 146 Parallel Postulate, 163 Parallel rays, 146 Parallel segments, 146 Parallel vectors, 561 Parallelogram lift, 396 Parallelogram mount, 398 Parallelogram vector addition method, 561 Parallelograms, 390 area of, 589 conditions for, 398–401 constructing, 404 properties of, 390–394 special, see Special parallelograms Parallels, adjustable, 836 Parent functions, 221 horizontal translations of, 838 reflections of, 838 transformations of, 838 vertical translations of, 838 Parent Resources Online Parent Resources Online is available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 9, 17, 24, 31, 38 Parking, 159 Pascal’s triangle, 883 Pasteur, Louis, 828 Patterns, 327, S44 frieze, 863 looking for, 763 using, to generate fractals, 882 Patty paper, see Construction(s), using patty paper Peaucellier cell, 434 Pendulum, 25 Penny-farthing bicycle, 768 Pentagon building, 859 Pentagon, 382 regular, 381 Percent grade, 534, 536, 539, 540 Percents, 27, 41 Perimeter, 36 in the coordinate plane, 616–619 proportional, 490 ratio, 490 Perpendicular Bisector Theorem, 300 Converse of the, 300 proof of the, 300 Perpendicular bisector, 300–303 of a segment, 172 constructing, 172 Perpendicular lines, 142–211 constructing, 179 defined, 146 exploring, 188–189 proving, 173 slopes of, 184–186, 189, 306, 617 Perpendicular Lines Theorem, 184 Perpendicular rays, 146 Perpendicular segments, 146 Perpendicular Transversal Theorem, 173 proof of the, 173 Perspective, 481 Perspective drawings, 662 Pets, 361 pH, 96, 761, S74 Photography, 385, 459, 475 Physical Fitness, 79 Physics, 25, 565, 861, 867 Pi (π), 37 developing, 598–599 using geometric probability to estimate, 637 Pi (π) calculator key, 601 Piano strings, 155 S170 S170 Index Pick’s Theorem, 613 for area of lattice polygons, developing, 613 Pizza, 195 Plane symmetry, 858 Planes, 6, 7 intersection of lines and, 8 Platonic solids, 669 Plumb bob, 168 Point(s), 6, 7, 12 collinear, 6 of concurrency, 307 constructing a tangent to a circle at a, 748 coplanar, 6 equidistant, 300, 746, 799 exterior, constructing a tangent to a circle from an, 779 and a line, distance between a, 172, 301 noncollinear, 6 noncoplanar, 6 special, in triangles, 321 of tangency, 746 three noncollinear, constructing a circle through, 763 two, slope of a line through, 558 vanishing, 662 Point Isabel Lighthouse, 894 Point matrix, 846 Point-slope form, 190, 191, 194, 198, 199, 303, 305 proof of, 190 Pointillism, 10 Polar axis, 808 Polar coordinate system, 808–809 Polar coordinates, 808–809 Polaris, 844 Pole, 808 Political Science, 79, 93 Polygon(s), 98 concave, 383 congruent, 231 convex, 383 diagonal of the, 382 inscribing, 380 irregular, 382 lattice, 613 properties and attributes of, 382–385 quadrilaterals and, 376–449 regular, see Regular polygons sides of, 382 similar, 462–464 vertex of the, 382 Polygon Angle Sum Theorem, 383 Polygon Exterior Angle Sum Theorem, 384 Polyhedrons, 669, 670 creating, by using nets, 669 regular, 669 Polymer, 868 Pompeii, 413 Positioning figures in the coordinate plane, strategies for, 267 Postulates, 7 For a complete list, see pages S82–S87 Precision, 596, S72 Predicting, 634 conditions for special parallelograms, 416–417 other triangle congruence relationships, 250–251 triangle similarity relationships, 468–469 Preimage, 50 Preparing for your final exam, 823 Preparing for TAKS, TX2–TX3 Prime number, 78 Primes, 50 Prisms, 654 altitude of, 680 drawing, 653 lateral area of, 680 oblique, 680 right, see Right prisms right rectangular, diagonals of, 671 surface area of, 680–683 volume of, 697–700 Probability, 10, 32, 85, 230, 237, 339, 459, 467, 565, 628–629, 702, 769, 798, 835, S77 experimental, 798 geometric, see Geometric probability model, 630 On Track for TAKS, 628–629 theoretical, 628 Problem-Solving Applications, 30, 105, 193, 252–253, 315–316, 428, 456, 528, 618, 749, 825 Problem-Solving Handbook, S40–S49, see also Problem-Solving Strategies Problem Solving on Location, xxix Cavanaugh Flight Museum, Addison, 294 The Freescale Marathon, Austin, 140 The Great Texas Balloon Race, Longview, Estes, 295 Lyndon B. Johnson’s Birthplace, Johnson City, 583 Moveable Bridges, Quintana Island, Tule Lake, Rio Hondo, Corpus Christi, 895 Point Isabel Lighthouse, Port Isabel, 894 Reliant Stadium, Houston, 740 Reun

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