Skills Practice Skills Practice for Lesson.1 Name Date Get Radical or (Be)! Radicals and the Pythagorean Theorem Vocabulary Write the term that best completes each statement. 1. An expression that includes a symbol such as A or A is called a(n) radical expression.. A quantity that is enclosed by a radical symbol is called the radicand.. The process of eliminating a radical from the denominator is called rationalizing the denominator. 4. The Pythagorean Theorem states that a b c, where a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse of the triangle. 009 Carnegie Learning, Inc. Problem Set Simplify each radical expression completely. 1. 7. 50 9 9 5 5 5. 18 4. 11 64 64 8 16 7 16 7 4 7 5. 4 18 6. 6 7 4 9 4 9 1 6 6 6 6 6 7. a b 5 a b 4 b a b 4 b ab b 8. x 4 y x 4 y y x 4 y y x y y Chapter l Skills Practice 51
9. 9x 7 y 9x 6 xy 9 x 6 x y x y x 10. 4x y 5 (4)(6)x xy 4 y 4 6 x x y 4 y xy 6xy Rationalize the denominator to simplify each radical expression completely. 11. 5 5 5 5 5 5 5 1. 1. 5 6 5 5 6 6 6 5 6 6 6 6 6 6 6 14. 15 5 15 15 5 5 5 15 5 5 5 5 15. 6 6 6 6 6 ()() () 6 6 6 6 6 16. 8 8 8 8 4 8 8 17. ab b a ab b ab a b a a ab a a ab ()()() 8 8 8 1 a 18. ab 5a ab ab 5a 5a 5a a b 5 a a b 5 5b 5a 5a 5a 5 009 Carnegie Learning, Inc. 5 Chapter l Skills Practice
Name Date Use the Pythagorean Theorem to answer each question. 19. The interior of a small moving van has a height of 9 feet. A couch that is 8 feet long and.5 feet high is tipped on its end to fit in the van. Can the couch be set back on its feet while inside the moving van? a b c (8) (.5) c 64 6.5 c 70.5 c 70.5 c c 8.8 Because 8.8 feet is less than 9 feet, the couch can be set back on its feet while inside the moving van. 0. A firefighter has a -foot ladder. If he stands the bottom of the ladder 7 feet from the base of a building, will the ladder be long enough to reach a window 19 feet from the ground? a b c (19) (7) c 61 49 c 410 c 410 c 009 Carnegie Learning, Inc. c 0.48 Because 0.48 feet is less than feet, the ladder will be long enough to reach the window. 1. A baseball diamond is a square with sides of 90 feet. The first base player stands on first base and throws a ball to third base. To the nearest foot, what distance does the ball travel? a b c The ball travels about 17 feet. (90) (90) c 8100 8100 c 16,00 c 16,00 c c 17.79 Chapter l Skills Practice 5
. A natural gas line is buried diagonally across a rectangular lot. The lot measures 40 feet by 16 feet. To the nearest foot, how much gas line is buried in the lot? a b c About 90 feet of gas line is buried in the lot. (40) (16) c 57,600 6,44 c 8,844 c 8,844 c c 89.558. A guy wire is connected to the top of a 16-meter pole and to a point on the ground 8 meters from the bottom of the pole. To the nearest meter, what length is the guy wire? a b c (16) (8) c The guy wire is about 18 meters long. 56 64 c 0 c 0 c c 17.889 4. Peter lives 10 miles directly north of a tower that broadcasts a wireless signal for his computer. If he drives directly east from his home, he can keep the wireless connection for 7 miles. About how many miles is the broadcasting range of the tower? a b c The broadcasting range of the tower is about 1 miles. 009 Carnegie Learning, Inc. (10) (7) c 100 49 c 149 c 149 c c 1.07 54 Chapter l Skills Practice
Skills Practice Skills Practice for Lesson. Name Date The Pythagorean Theorem Disguised as the Distance Formula! The Distance Formula and Midpoint Formula Vocabulary Write each formula, explain how it is found, and explain why it is used. 1. the distance formula The distance formula is used to find the distance between two points. It is derived from the Pythagorean Theorem by drawing a right triangle with the two points as endpoints of the hypotenuse. Using the length of each leg, the length of the hypotenuse can then be found. The distance formula states that the distance between the two points (x 1, y 1 ) and (x, y ) is d (x x 1 ) ( y y 1 ).. the midpoint formula The midpoint formula is used to find the center point between two points on a coordinate plane. It is derived by finding the mean of the x-coordinates and the mean of the y-coordinates. The midpoint formula states that the midpoint between the two points (x 1, y 1 ) and (x, y ) is ( x x 1, y y 1 ). 009 Carnegie Learning, Inc. Chapter l Skills Practice 55
Problem Set Plot each pair of points and connect them with a line segment. Draw a right triangle with this line segment as the hypotenuse. Label the length of all three sides of the right triangle to the nearest tenth. 1. (0, 1) and (8, 5). (, 4) and (6, 8) y 10 y 10 8 6 8 6 5.7 4 4 8.9 4 4 4 10 8 6 4 0 8 4 x 6 8 10 10 8 6 4 0 4 x 6 8 10 4 4 6 8 10 6 8 10. ( 1, ) and (5, 1) 4. ( 6, 0) and (, 4) y 10 y 10 8 8 6 6 10 8 6 4 4 0 4 6 8 6. 6 4 x 6 8 10 10 8 6 4 8 4 0 8.9 4 6 8 4 4 x 6 8 10 009 Carnegie Learning, Inc. 10 10 56 Chapter l Skills Practice
Name Date Use the distance formula to calculate the distance between each pair of points. Round decimal answers to the nearest thousandth. 5. (0, 1) and (8, 5) 6. (, 4) and (6, 8) d (x x 1 ) ( y y 1 ) d (x x 1 ) ( y y 1 ) d (8 0) (5 1) d (6 ) (8 4) d 64 16 d 16 16 d 80 d d 4 5 8.944 d 4 5.657 7. ( 1, ) and (5, 1) 8. ( 6, 0) and (, 4) d (x x 1 ) ( y y 1 ) d (x x 1 ) ( y y 1 ) d (5 ( 1)) (1 ) d ( ( 6)) ( 4 0) d 6 4 d 64 16 d 40 d 80 d 10 6.5 d 4 5 8.944 009 Carnegie Learning, Inc. 9. ( 6, 6) and ( 6, 1) 10. (, 7) and (0, 5) d (x x 1 ) ( y y 1 ) d (x x 1 ) ( y y 1 ) d ( 6 ( 6)) ( 1 ( 6)) d (0 ( )) (5 7) d (0) ( 6) d 9 4 d 6 d 1.606 d 6 Use the given information to solve for y. 11. The distance between (6, 0) and (, y) is 10. d (x x 1 ) ( y y 1 ) 10 ( 6) ( y 0) 10 81 y 10 81 y 49 y y 7 Chapter l Skills Practice 57
1. The distance between ( 5, 0) and (8, y) is 69. d (x x 1 ) ( y y 1 ) 69 (8 ( 5)) ( y 0) 69 169 y 69 169 y 100 y y 10 1. The distance between ( 4, 0) and (6, y) is 116. d (x x 1 ) ( y y 1 ) 116 (6 ( 4)) ( y 0) 116 100 y 116 100 y 16 y y 4 14. The distance between (8, 0) and (1, y) is 11. d (x x 1 ) ( y y 1 ) 11 (1 8) ( y 0) 11 49 y 11 49 y 009 Carnegie Learning, Inc. 64 y y 8 58 Chapter l Skills Practice
Name Date Use the midpoint formula to calculate the midpoint between each pair of points. 15. (0, 1) and (8, 5) ( x x 1, y y 1 16. (, 4) and (6, 8) ( x x 1, y y 1 17. ( 1, ) and (5, 1) ( x x 1, y y 1 ) ( 0 8 ) ( 6 ) ( 18. ( 6, 0) and (, 4) ( x x 1, y y 1 ) (, 1 5, 4 8 ) ( 8 ) ( 8 1 5, 1 ) ( 4 6 0, ( 4), 6 ) (4, ), 1 ) (4, 6), 4 ) (, ) ) ( 4 19. ( 1, 16) and ( 4, 9) ( x x 1, y y 1 ) 1 ( 4) 16 (, ( 9) 0. ( 9, 8) and (0, 10) ( x x 1, y y 1 ) 9 0 ( 8, ( 10), 4 ) (, ) ) ( 16 ) ( 9, 5 ) ( 8, 1.5), 18 ) ( 4.5, 9) Use the given information to solve for x. 009 Carnegie Learning, Inc. 1. The point (, 0) is the midpoint of a line segment with endpoints (7, 4) and (x, 4). ( x x 1, y y 1 ) ( 7 x 4, ( 4) ) ( 7 x, 0 ) ( 7 x, 0 ) (, 0) 7 x 7 x 6 x 1. The point ( 6, ) is the midpoint of a line segment with endpoints (1, 1) and (x, 5). ( x x 1, y y 1 ) ( 1 x 1, ( 5) ) ( 1 x, 4 ) ( 1 x, ) ( 6, ) 1 x 6 1 x 1 x 1 Chapter l Skills Practice 59
. The point (5, 7) is the midpoint of a line segment with endpoints (0, ) and (x, 16). ( x x 1, y y 1 ) ( 0 x, ( 16) ) ( x, 14 ) ( x, 7 ) (5, 7) x 5 x 10 4. The point ( 8, 4) is the midpoint of a line segment with endpoints ( 1, ) and (x, 10). ( x x 1, y y 1 ) 1 x (, 10 ) 1 x (, 8 ) 1 x (, 4 ) ( 8, 4) 1 x 8 1 x 16 x 009 Carnegie Learning, Inc. 60 Chapter l Skills Practice
Skills Practice Skills Practice for Lesson. Name Date Drafting Equipment Properties of 45º 45º 90º Triangles Vocabulary Explain how to find each of the following. 1. the leg length of a 45º 45º 90º triangle when you know the length of the other leg A 45º 45º 90º triangle is an isosceles triangle. So, both legs are the same length. When you know the length of one leg of a 45º 45º 90º triangle, then you also know the other.. the leg length of a 45º 45º 90º triangle when you know the length of the hypotenuse The 45º 45º 90º Triangle Theorem says that the length of the hypotenuse in a 45º 45º 90º triangle is times the length of a leg. When you know the length of the hypotenuse and want to find the length of a leg, divide the length of the hypotenuse by.. the hypotenuse of a 45º 45º 90º triangle when you know the length of a leg 009 Carnegie Learning, Inc. The 45º 45º 90º Triangle Theorem says that the length of the hypotenuse in a 45º 45º 90º triangle is times the length of a leg. When you know the length of a leg and want to find the length of the hypotenuse, multiply the length of the leg by. 4. the area of a 45º 45º 90º triangle You can find the area of any triangle by using the formula A 1 bh, where b is the base length and h is the height. In a 45º 45º 90º triangle, the base and height are the legs, which are the same length. So, to find the area, calculate 1 l where l is the length of a leg. Chapter l Skills Practice 61
Problem Set Determine the unknown side length of each triangle. 1.. m c 4 cm c m 4 cm c ( ) m c 4 ( ) 4() 8 cm. 45 4. 45 18 inches a 11 feet a 45 a 18 a 11 a 18 18 18 9 inches a 11 45 11 11 feet Use the 45º 45º 90º Triangle Theorem to calculate the indicated length. 5. What is the leg length of an isosceles right triangle with a hypotenuse of 6 inches? Let a leg length and c hypotenuse length. a c a 6 a 6 6 6 The leg length is inches. 009 Carnegie Learning, Inc. 6 Chapter l Skills Practice
Name Date 6. What is the leg length of an isosceles right triangle with a hypotenuse of 8 centimeters? Let a leg length and c hypotenuse length. a c a 8 a 8 8 8 14 The leg length is 14 centimeters. 7. What is the length of the hypotenuse of an isosceles right triangle with a leg length of 7 feet? Let a leg length and c hypotenuse length. c a c 7 The length of the hypotenuse is 7 feet. 8. What is the length of the hypotenuse of an isosceles right triangle with a leg length of meters? 009 Carnegie Learning, Inc. Let a leg length and c hypotenuse length. c a c The length of the hypotenuse is meters. 9. The side length of a square is 4.5 feet. What is the length of the diagonal? Let a side length and c diagonal length. c a c 4.5 The length of the diagonal is 4.5 feet. Chapter l Skills Practice 6
10. The side length of a square is feet. What is the length of the diagonal? Let a side length and c diagonal length. c a c ( )( ) c () 4 The length of the diagonal is 4 feet. 11. The length of the diagonal of a square is feet. What is the length of each side? Let a side length and c diagonal length. a c a a The length of each side is feet. 1. The length of the diagonal of a square is 80 millimeters. What is the length of each side? Let a side length and c diagonal length. a c a 80 a 80 80 80 40 The length of each side is 40 millimeters. 009 Carnegie Learning, Inc. 64 Chapter l Skills Practice
Name Date Calculate the area of each triangle. 1. 14. 45 45 5 feet 45 1.4 meters A 1 bh A 1 bh A 1 (5)(5) A 1 (1.4)(1.4) A 1 (5) A 1 (15.76) A 1.5 square feet A 76.88 square meters 15. 45 16. 45 4.8 inches 15 feet 45 009 Carnegie Learning, Inc. (side length) 15 side length 15 15 15 feet A 1 bh A 1 ( 15 )( 15 ) A (15 )(15 ) ()()() A (15)(15) ()() 5 56.5 4 A 56.5 square feet (side length) 4.8 side length 4.8 4.8 4.8 A 1 bh.4 inches A 1 (.4 )(.4 ) A (.4 )(.4 ) () A (.4)(.4)() (.4)(.4) 5.76 () A 5.76 square inches Chapter l Skills Practice 65
Calculate the area of each square. 17. 18. 6. meters 1 inches A s A s A (6.) A (1) A 8.44 square meters A 144 square inches 19. 0. 5 cm 17 mm (side length) (diagonal length) s 5 s 5 5 5 A s A ( 5 ) A ( 5 )( 5 ) A 5 1.5 (side length) (diagonal length) 17 s 17 17 17 A s A ( 17 ) A ( 17 )( 17 ) A 89 144.5 009 Carnegie Learning, Inc. A 1.5 square centimeters A 144.5 square millimeters 66 Chapter l Skills Practice
Skills Practice Skills Practice for Lesson.4 Name Date Finishing Concrete Properties of 0º 60º 90º Triangles Vocabulary Answer each question in your own words. 1. How are the side lengths of a 0º 60º 90º triangle related? In a 0º 60º 90º triangle with a shorter leg of length a, the hypotenuse has a length of a. The longer leg has a length of a.. How is a 0º 60º 90º triangle used to find the altitude of an equilateral triangle? When an altitude is drawn in an equilateral triangle, it forms two congruent 0º 60º 90º triangles. The side length of the equilateral triangle becomes the length of the hypotenuse in the 0º 60º 90º triangle. The 0º 60º 90º triangle side length relationship can be used to find the altitude. The altitude is equal to 1 (side length)( ). Problem Set Use the Pythagorean Theorem to calculate the missing side length of each triangle. Leave radicals in simplest form. 009 Carnegie Learning, Inc. 1.. 8 c 8 7 cm a 8, b 8 b 7, c 14 a b c a b c (8) (8 ) c a (7 ) (14) 64 64() c a (14) (7 ) a 14 56 c a 196 147 16 c a 49 c 16 units a 7 centimeters Chapter l Skills Practice 67
. A right triangle has a hypotenuse length of 40 units and one leg length of 0 units. Find the length of the other leg. Let a 0, c 40. a b c (0) b (40) 400 b 1600 b 1600 400 b 100 b 100 b 400 b 0 The other leg is 0 units long. 4. A right triangle has a hypotenuse length of 1 units and one leg length of 6 units. Find the length of the other leg. Let a 6, c 1. a b c (6) b (1) 6 b 144 b 144 6 b 108 b 108 b 6 b 6 The other leg is 6 units long. 009 Carnegie Learning, Inc. 68 Chapter l Skills Practice
Name Date Use the 0º 60º 90º Triangle Theorem to calculate the missing side lengths in each triangle. Leave radicals in simplest form. 5. 6. b 0 4 cm 0 a 18 m b a shorter leg: a 4 shorter leg: a 18 a cm a 9 m longer leg: b a cm longer leg: b a 9 m 7. 8. a c a c 009 Carnegie Learning, Inc. 0 0 6 ft in shorter leg: a b shorter leg: a b a 6 a a 6 ( 6 )( ) ()()() ft a ( )( ) 6 6 in hypotenuse: c a hypotenuse: c a c ft c 6 in Chapter l Skills Practice 69
9. A 0º 60º 90º triangle has a shorter leg that measures 0 inches. Find the length of the longer leg and the length of the hypotenuse. a 0 longer leg: b a 0 inches hypotenuse: c a c (0) 60 inches 10. A 0º 60º 90º triangle has a shorter leg that measures 14 inches. Find the length of the longer leg and the length of the hypotenuse. a 14 longer leg: b a 14 inches hypotenuse: c a c (14) 8 inches 009 Carnegie Learning, Inc. 70 Chapter l Skills Practice
Name Date Use the 0º 60º 90º Triangle Theorem to calculate the missing measurement of each equilateral triangle. Leave radicals in simplest form. 11. An equilateral triangle has a side length of 100 inches. What is the measurement of the altitude? The altitude divides the equilateral triangle into two 0º 60º 90º triangles. The side length is the hypotenuse and the altitude is the longer leg. Find the shorter leg first. hypotenuse: c a 100 a a 50 longer leg: b a 50 inches The measurement of the altitude is 50 inches. 1. An equilateral triangle has a side length of inches. What is the measurement of the altitude? The altitude divides the equilateral triangle into two 0º 60º 90º triangles. The side length is the hypotenuse and the altitude is the longer leg. Find the shorter leg first. hypotenuse: c a 009 Carnegie Learning, Inc. a a 11 longer leg: b a 11 inches The measurement of the altitude is 11 inches. Chapter l Skills Practice 71
1. The altitude of an equilateral triangle has a measurement of 4 feet. What is the side length? The altitude divides the equilateral triangle into two 0º 60º 90º triangles. The side length is the hypotenuse and the altitude is the longer leg. Find the shorter leg first. shorter leg: a b a 4 a 4 hypotenuse: c a c (4) 8 The side length is 8 feet. 14. The altitude of an equilateral triangle has a measurement of 4 millimeters. What is the side length? The altitude divides the equilateral triangle into two 0º 60º 90º triangles. The side length is the hypotenuse and the altitude is the longer leg. Find the shorter leg first. shorter leg: a b a 4 a 4 hypotenuse: c a c (4) 84 The side length is 84 millimeters. 009 Carnegie Learning, Inc. 7 Chapter l Skills Practice
Name Date Calculate the area of each triangle. 15. 16. 8 in 7 ft 0 0 height 4 inches height 7 feet base 4 inches base 7 feet area 1 bh area 1 bh 1 (4 )(4) 1 (7 )(7) 8 square inches 1 (49 ) 49 square feet 17. 18. cm 70 mm 009 Carnegie Learning, Inc. Draw an altitude. This is the height. Draw an altitude. This is the height. base centimeters base 70 millimeters height 1 () height 1 (70) 5 area 1 bh area 1 bh 1 ( () ) 1 (70)(5 ) ()( ) 15 square millimeters ()() 9 4 9 square centimeters 4 Chapter l Skills Practice 7
Calculate the surface area and volume of each triangular prism. Round decimals to the nearest tenth. 19. in 0 1 in Triangular base: a, b, c 4 B 1 ()( ) P 4 6 S ( ) (6 )(1) 4 7 4 7 8 10.5 square inches V 1 ()( )(1) 4 41.6 cubic inches 0. 6 cm 0 cm Half of triangular base: a, b, c 6 B 1 (6)( ) 9 P 6 6 6 18 S (9 ) (18)(0) 18 60 91. square centimeters V 1 (6)( )(0) 9 (0) 180 009 Carnegie Learning, Inc. 11.8 cubic centimeters 74 Chapter l Skills Practice
Name Date 1. 18 ft 15 ft 0 0 a 7.5, b 7.5, c 15 B 1 (15)(7.5 ) 56.5 P 15 15 15 45 S (56.5 ) 45(18) 11.5 810 1004.9 square feet V 1 (15)(7.5 )(18) 101.5 175.7 cubic feet 009 Carnegie Learning, Inc.. 9 m m a 9, b 9, c 18 B 1 (9)(9 ) 40.5 P 9 9 18 7 9 S (40.5 ) (7 9 )() 81 864 88 69 864 150.1 square meters V 1 (9)(9 )() 196 44.7 cubic meters Chapter l Skills Practice 75
009 Carnegie Learning, Inc. 76 Chapter l Skills Practice