Lesson 6.1 Skills Practice


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1 Lesson 6.1 Skills Practice Name Date Soon You Will Determine the Right Triangle Connection The Pythagorean Theorem Vocabulary Match each definition to its corresponding term. 1. A mathematical statement that can be proven using definitions, a. diagonal of a postulates, and other theorems. square 2. Either of the two shorter sides of a right triangle. b. right triangle 3. An angle that has a measure of 90 and is indicated by a c. Pythagorean square drawn at the corner formed by the angle. Theorem 4. A series of steps used to prove the validity of an ifthen d. right angle statement. 5. A line segment connecting opposite vertices of a square. e. theorem 6. If a and b are the lengths of the legs of a right triangle and c is f. leg the length of the hypotenuse, then a 2 1 b 2 5 c A mathematical statement that cannot be proven but is g. postulate considered to be true. 8. A triangle with a right angle. h. hypotenuse Chapter 6 Skills Practice 533
2 Lesson 6.1 Skills Practice page 2 9. The longest side of a right triangle. This side is always i. proof opposite the right angle in a right triangle. Problem Set The side lengths of a right triangle are given. Determine which length is the hypotenuse. Use the Pythagorean Theorem to verify each length. 1. 9, 12, 15 The length of the hypotenuse is , 26, , 12, , 8, Chapter 6 Skills Practice
3 Lesson 6.1 Skills Practice page 3 Name Date 5. 25, 15, , 36, 39 Calculate the length of the hypotenuse of each given triangle c 2 5 a 2 1 b 2 c c c c c 5 30 Chapter 6 Skills Practice 535
4 Lesson 6.1 Skills Practice page Chapter 6 Skills Practice
5 Lesson 6.1 Skills Practice page 5 Name Date Answer each question using the scenario. 13. Clayton is responsible for changing the broken light bulb in a streetlamp. The streetlamp is 12 feet high. Clayton places the base of his ladder 4 feet from the base of the streetlamp. Clayton can extend his ladder from 10 feet to 14 feet. How long must his ladder be to reach the top of the streetlamp? Round your answer to the nearest hundredth. 12 ft 4 ft c 2 5 a 2 1 b 2 c c c c < Clayton must extend his ladder about feet. Chapter 6 Skills Practice 537
6 Lesson 6.1 Skills Practice page Jada is helping to build a swing set at the community park. The swing bar at the top of the set should be 8 feet from the ground. The base of the support beam extends 3 feet from the plane of the swing bar. How long should each support beam be? Round your answer to the nearest tenth. 8 ft 3 ft 538 Chapter 6 Skills Practice
7 Lesson 6.1 Skills Practice page 7 Name Date 15. Perry wants to replace the net on his basketball hoop. The hoop is 10 feet high. Perry places his ladder 4 feet from the base of the hoop. How long must his ladder be to reach the hoop? Round your answer to the nearest hundredth. 10 ft 4 ft Chapter 6 Skills Practice 539
8 Lesson 6.1 Skills Practice page Ling wants to create a diagonal path through her flower garden using stepping stones. She would like to place one stone every 2 feet. How many stepping stones does she need? 12 ft 16 ft 540 Chapter 6 Skills Practice
9 Lesson 6.1 Skills Practice page 9 Name Date Calculate the length of the missing side of each given triangle c 2 5 a 2 1 b b b b < b Chapter 6 Skills Practice 541
10 Lesson 6.1 Skills Practice page Chapter 6 Skills Practice
11 Lesson 6.2 Skills Practice Name Date Can That Be Right? The Converse of the Pythagorean Theorem Vocabulary Write the term that best completes the statement. 1. The states: If a 2 1 b 2 5 c 2, then the triangle is a right triangle. 2. The of a theorem is created when the ifthen parts of the theorem are exchanged. 3. A set of three positive integers a, b, and c that satisfy the equation a 2 1 b 2 5 c 2 is a(n). Problem Set Determine whether each triangle with the given side lengths is a right triangle. 1. 8, 15, 17 c 2 5 a 2 1 b , 9, This is a right triangle. Chapter 6 Skills Practice 543
12 Lesson 6.2 Skills Practice page , 15, , 12, , 8, , 12, Chapter 6 Skills Practice
13 Lesson 6.2 Skills Practice page 3 Name Date Answer each question using the scenario. 7. A computer monitor is sold by the diagonal length of the screen. A computer monitor has a 15inch screen. The screen has a width of 13 inches. What is the height of the screen? Round your answer to the nearest tenth. a 2 1 b 2 5 c 2 a a a a 5 56 a < 7.5 The height of the computer monitor screen is about 7.5 inches. Chapter 6 Skills Practice 545
14 Lesson 6.2 Skills Practice page 4 8. Luisa is building a sand box in her backyard. She places four pieces of wood in a rectangle to form the frame. The rectangle is 4 feet long and 3 feet wide. How can she use a measuring tape to make sure that the corners of the frame will be right angles? 546 Chapter 6 Skills Practice
15 Lesson 6.2 Skills Practice page 5 Name Date 9. Firefighters need to cross from the roof of a 25feettall building to the roof of a 35feettall building by using a ladder. The buildings are 20 feet apart. What minimum length does the ladder need to be in order to span the two buildings? Ladder 35 ft 25 ft 20 ft Chapter 6 Skills Practice 547
16 Lesson 6.2 Skills Practice page Chen is building a ramp for his remote control car. He wants the end of the ramp to extend 4 feet from the base of the ramp. The base of the ramp is 18 inches high. How long should the piece of wood for the ramp be? Round your answer to the nearest tenth. 18 in. 4 ft 548 Chapter 6 Skills Practice
17 Lesson 6.2 Skills Practice page 7 Name Date 11. Perry wants to use a 12foot ladder to reach a shelf that is 11 feet above the ground. How far from the wall should Perry place the base of the ladder so that the top of the ladder reaches the shelf? Round your answer to the nearest tenth. 12. Lea walks to soccer practice on Saturday. She leaves her home and walks 6 blocks north. Lea then turns east and walks 4 blocks to the soccer field. How far is the soccer field from Lea s home? Round your answer to the nearest whole number. Chapter 6 Skills Practice 549
18 Lesson 6.2 Skills Practice page 8 Calculate the length of the segment that connects the points in each. Write your answer as a radical if necessary a 2 1 b 2 5 c c c c c 5 5 c 550 Chapter 6 Skills Practice
19 Lesson 6.2 Skills Practice page 9 Name Date Chapter 6 Skills Practice 551
20 Lesson 6.2 Skills Practice page Chapter 6 Skills Practice
21 Lesson 6.3 Skills Practice Name Date Pythagoras to the Rescue Solving for Unknown Lengths Problem Set Determine the length of the hypotenuse of each given triangle c 24 c 6 c c c c c 5 26 The length of the hypotenuse is 26 units. Chapter 6 Skills Practice 553
22 Lesson 6.3 Skills Practice page c 4 c Chapter 6 Skills Practice
23 Lesson 6.3 Skills Practice page 3 Name Date c c Chapter 6 Skills Practice 555
24 Lesson 6.3 Skills Practice page 4 Determine each unknown leg length a b b b b b b b 5 16 The length of the leg is 16 units. 556 Chapter 6 Skills Practice
25 Lesson 6.3 Skills Practice page 5 Name Date b a Chapter 6 Skills Practice 557
26 Lesson 6.3 Skills Practice page b a 558 Chapter 6 Skills Practice
27 Lesson 6.3 Skills Practice page 7 Name Date Use the Pythagorean Theorem to determine whether each given triangle is a right triangle fi 289 The triangle is not a right triangle Chapter 6 Skills Practice 559
28 Lesson 6.3 Skills Practice page Chapter 6 Skills Practice
29 Lesson 6.3 Skills Practice page 9 Name Date Use the Pythagorean Theorem to calculate each unknown length. 19. The design for a bridge truss is shown. The distance between the horizontal beams is 24 feet. The distance between the vertical beams is 18 feet. Determine the length (x) of each diagonal brace. 18 ft 24 ft x x x x x 30 5 x Each diagonal brace is 30 feet long. Chapter 6 Skills Practice 561
30 Lesson 6.3 Skills Practice page The Archery Team is practicing on the basketball court in the gymnasium. The court is 50 feet wide and 94 feet long. The archers are shooting at a target placed at one corner of the court while they stand in the corner diagonally across the court. Determine the distance of each practice shot. 562 Chapter 6 Skills Practice
31 Lesson 6.3 Skills Practice page 11 Name Date 21. The water company installed a 40yard diagonal brace on a water tower between two vertical beams that are 12 yards apart as shown. Determine the height of each vertical beam. 40 yd 12 yd Chapter 6 Skills Practice 563
32 Lesson 6.3 Skills Practice page The lengths of the legs of a right triangle are 15 meters each. Determine the length of the hypotenuse. 23. The length of the hypotenuse of a right triangle is 50 inches. Determine the length of the legs if each leg is the same length. 564 Chapter 6 Skills Practice
33 Lesson 6.3 Skills Practice page 13 Name Date 24. A rescue boat leaves Walker Dock and travels 18 miles due north to haul in a sailing vessel stranded in the middle of a lake. After attaching a cable, the rescue boat hauls the sailing vessel 80 miles due east to Blue Haven Dock. Determine the direct distance from Walker Dock to Blue Haven Dock. Chapter 6 Skills Practice 565
34 566 Chapter 6 Skills Practice
35 Lesson 6.4 Skills Practice Name Date Meeting Friends The Distance Between Two Points in a Coordinate System Problem Set Determine the distance between each given pair of points by graphing and connecting the points, creating a right triangle, and applying the Pythagorean Theorem. 1. (2, 2) and (8, 5) y c 2 5 a 2 1 b c c x c c c < The distance between (2, 2) and (8, 5) 8 is approximately 6.71 units. Chapter 6 Skills Practice 567
36 Lesson 6.4 Skills Practice page 2 2. (3, 7) and (7, 3) y x (26, 8) and (6, 3) y x Chapter 6 Skills Practice
37 Lesson 6.4 Skills Practice page 3 Name Date 4. (7, 5) and (3, 23) y x (24, 24) and (5, 8) y x 6 8 Chapter 6 Skills Practice 569
38 Lesson 6.4 Skills Practice page 4 6. (29, 3) and (7, 5) y x (27, 3) and (8, 25) y x Chapter 6 Skills Practice
39 Lesson 6.4 Skills Practice page 5 Name Date 8. (29, 6) and (8, 1) y x Archaeologists map each item they find at a dig on a 1foot by 1foot coordinate grid. Calculate the distance between the given pair of objects on the coordinate grid. 9. Determine the distance between the spindle and the beads. 9 8 y c 2 5 a 2 1 b 2 c spindle c c c 5 25 c beads x The distance between the spindle and the beads is 5 feet. Chapter 6 Skills Practice 571
40 Lesson 6.4 Skills Practice page Determine the distance between the pottery shard and the axe head y axe head pottery shard x 11. Determine the distance between the coins and the beads. 9 y beads coins x 572 Chapter 6 Skills Practice
41 Lesson 6.4 Skills Practice page 7 Name Date 12. Determine the distance between the coins and the axe head y axe head 5 4 coins x 13. Determine the distance between the mask and the beads. 9 y mask beads x Chapter 6 Skills Practice 573
42 Lesson 6.4 Skills Practice page Determine the distance between the pottery shard and the beads. 9 y pottery shard beads x 15. Determine the distance between the spindle and the axe head y axe head spindle x 574 Chapter 6 Skills Practice
43 Lesson 6.4 Skills Practice page 9 Name Date 16. Determine the distance between the mask and the coins. 9 y mask coins x Chapter 6 Skills Practice 575
44 576 Chapter 6 Skills Practice
45 Lesson 6.5 Skills Practice Name Date Diagonally Diagonals in Two Dimensions Problem Set Determine the length of the diagonals in each given quadrilateral. 1. The quadrilateral is a square. A B c 2 5 a 2 1 b 2 15 ft c c D C c c c < The length of diagonal AC is approximately feet. The length of diagonal BD is approximately feet. Chapter 6 Skills Practice 577
46 Lesson 6.5 Skills Practice page 2 2. The quadrilateral is a rectangle. E F 10 in. H 18 in. G 578 Chapter 6 Skills Practice
47 Lesson 6.5 Skills Practice page 3 Name Date 3. The quadrilateral is a parallelogram. 11 m J K 6 m N 8 m M Chapter 6 Skills Practice 579
48 Lesson 6.5 Skills Practice page 4 4. The quadrilateral is a trapezoid. 9 y P Q 3 2 S R x 580 Chapter 6 Skills Practice
49 Lesson 6.5 Skills Practice page 5 Name Date 5. The quadrilateral is an isosceles trapezoid. 9 y W X Z Y x Chapter 6 Skills Practice 581
50 Lesson 6.5 Skills Practice page 6 6. The quadrilateral is a rhombus. 9 y B G 3 2 M K x 582 Chapter 6 Skills Practice
51 Lesson 6.5 Skills Practice page 7 Name Date Calculate the area of each shaded region. 7. The figure is composed of a circle and a rectangle. The diagonal of the rectangle is the same length as the diameter of the circle. 4 in. The area of the rectangle is: A 5 bh A 5 (4)(9) 9 in. A 5 36 in. 2 The length of the rectangle s diagonal is: c 2 5 a 2 1 b 2 c c c c 5 97 c < 9.85 in. The area of the circle is: A 5 πr 2 A < (3.14)(4.93) 2 A < in. 2 The area of the shaded region is approximately in. 2. Chapter 6 Skills Practice 583
52 Lesson 6.5 Skills Practice page 8 8. The figure is composed of two squares. The length of the diagonal of the smaller square is equal to the width of the larger square. 10 ft 584 Chapter 6 Skills Practice
53 Lesson 6.5 Skills Practice page 9 Name Date 9. The figure is composed of a right triangle and a circle. The hypotenuse of the right triangle is the same length as the diameter of the circle. 5 m 12 m Chapter 6 Skills Practice 585
54 Lesson 6.5 Skills Practice page The figure is composed of a right triangle and a square. The hypotenuse of the right triangle is one side of the square. 15 yd 20 yd 586 Chapter 6 Skills Practice
55 Lesson 6.5 Skills Practice page 11 Name Date 11. The figure is composed of a right triangle and a semicircle. The hypotenuse of the right triangle is the same length as the diameter of the semicircle. 5 ft 5 ft Chapter 6 Skills Practice 587
56 Lesson 6.5 Skills Practice page The figure is composed of two right triangles. The hypotenuse of one right triangle is the leg of the other right triangle. 3 cm 3 cm 4 cm 588 Chapter 6 Skills Practice
57 Lesson 6.6 Skills Practice Name Date Two Dimensions Meet Three Dimensions Diagonals in Three Dimensions Problem Set Draw all of the edges you cannot see in each rectangular solid using dotted lines. Then draw a threedimensional diagonal using a solid line Chapter 6 Skills Practice 589
58 Lesson 6.6 Skills Practice page Chapter 6 Skills Practice
59 Lesson 6.6 Skills Practice page 3 Name Date Determine the length of the threedimensional diagonal in the given rectangular solid using each Pythagorean Theorem m 8 m 8 m Length of second leg: Length of diagonal: c d 2 < c d 2 < c c d 2 < d < c < d < The length of the threedimensional diagonal in the rectangular solid is approximately meters. Chapter 6 Skills Practice 591
60 Lesson 6.6 Skills Practice page in. 4 in. 1 in. 592 Chapter 6 Skills Practice
61 Lesson 6.6 Skills Practice page 5 Name Date cm 6 cm 3 cm Chapter 6 Skills Practice 593
62 Lesson 6.6 Skills Practice page m 4 m 3 m 594 Chapter 6 Skills Practice
63 Lesson 6.6 Skills Practice page 7 Name Date ft 12 ft 5 ft Chapter 6 Skills Practice 595
64 Lesson 6.6 Skills Practice page in. 7 in. 13 in. 596 Chapter 6 Skills Practice
65 Lesson 6.6 Skills Practice page 9 Name Date Use the diagonals across the front face, the side face, and the top face of each given solid to determine the length of the threedimensional diagonal. Use a formula " 8" 6" d ( ) d ( ) 2 d (109) d d d < 7.38 The length of the threedimensional diagonal is approximately 7.38 inches. Chapter 6 Skills Practice 597
66 Lesson 6.6 Skills Practice page m 3 m 10 m 598 Chapter 6 Skills Practice
67 Lesson 6.6 Skills Practice page 11 Name Date ft 10 ft 12 ft Chapter 6 Skills Practice 599
68 Lesson 6.6 Skills Practice page m 5 m 6 m 600 Chapter 6 Skills Practice
69 Lesson 6.6 Skills Practice page 13 Name Date yd 4 yd 10 yd Chapter 6 Skills Practice 601
70 Lesson 6.6 Skills Practice page " 15" 13" 602 Chapter 6 Skills Practice
71 Lesson 6.6 Skills Practice page 15 Name Date Use a formula to answer each question. 19. A packing company is in the planning stages of creating a box that includes a threedimensional diagonal support inside the box. The box has a width of 5 feet, a length of 6 feet, and a height of 8 feet. How long will the diagonal support need to be? d d d d d < The diagonal support will need to be approximately feet. 20. A plumber needs to transport a 12foot pipe to a jobsite. The interior of his van is 90 inches in length, 40 inches in width, and 40 inches in height. Will the pipe fit inside his van? Chapter 6 Skills Practice 603
72 Lesson 6.6 Skills Practice page You are landscaping the flower beds in your front yard. You choose to plant a tree that measures 5 feet from the root ball to the top. The interior of your car is 60 inches in length, 45 inches in width, and 40 inches in height. Will the tree fit inside your car? 22. Julian is constructing a box for actors to stand on during a school play. To make the box stronger he decides to include diagonals on all sides of the box and a threedimensional diagonal through the center of the box. The diagonals across the front and back of the box are each 2 feet, the diagonals across the sides of the box are each 3 feet, and the diagonals across the top and bottom of the box are each 7 feet. How long is the diagonal through the center of the box? 604 Chapter 6 Skills Practice
73 Lesson 6.6 Skills Practice page 17 Name Date 23. Carmen has a cardboard box. The length of the diagonal across the front of the box is 9 inches. The length of the diagonal across the side of the box is 7 inches. The length of the diagonal across the top of the box is 5 inches. Carmen wants to place a 10inch stick into the box and be able to close the lid. Will the stick fit inside the box? Chapter 6 Skills Practice 605
74 Lesson 6.6 Skills Practice page A technician needs to pack a television in a cardboard box. The length of the diagonal across the front of the box is 17 inches. The length of the diagonal across the side of the box is 19 inches. The length of the diagonal across the top of the box is 20 inches. The threedimensional diagonal of the television is 24 inches. Will the television fit in the box? Determine each unknown measurement. 25. A rectangular box has a length of 8 inches and a width of 5 inches. The length of the threedimensional diagonal of the box is 12 inches. What is the height of the box? d 2 5 l 2 1 w 2 1 h h h h h 7.42 < h The height of the box is approximately 7.42 inches. 606 Chapter 6 Skills Practice
75 Lesson 6.6 Skills Practice page 19 Name Date 26. The length of the diagonal across the front of a rectangular box is 6 feet, and the length of the diagonal across the top of the box is 9 feet. The length of the threedimensional diagonal is 14 feet. What is the length of the diagonal across the side of the box? 27. A rectangular box has a length of 7 feet and a height of 11 feet. The length of the threedimensional diagonal of the box is 20 feet. What is the width of the box? Chapter 6 Skills Practice 607
76 Lesson 6.6 Skills Practice page The length of the diagonal across the side of a rectangular box is 16 centimeters, and the length of the diagonal across the top of the box is 18 centimeters. The length of the threedimensional diagonal is 20 centimeters. What is the length of the diagonal across the front of the box? 29. A rectangular box has a height of 3 feet and a width of 4 feet. The length of the threedimensional diagonal of the box is 13 feet. What is the length of the box? 608 Chapter 6 Skills Practice
77 Lesson 6.6 Skills Practice page 21 Name Date 30. The length of the diagonal across the front of a rectangular box is 30 inches, and the length of the diagonal across the side of the box is 30 inches. The length of the threedimensional diagonal is 40 inches. What is the length of the diagonal across the top of the box? Chapter 6 Skills Practice 609
78 610 Chapter 6 Skills Practice
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