Name Date - rcsdk8.org Practice... · Chapter 6 Skills Practice • 533 ... theorem a. diagonal of...

Chapter 6 Skills Practice • 533

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Lesson 6.1 Skills Practice

Name ________________________________________________________ Date _________________________

Soon You Will Determine the Right Triangle ConnectionThe Pythagorean Theorem

VocabularyMatch 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

e.theorem

2. Either of the two shorter sides of a right triangle. b. right triangle

f. leg

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

d. right angle

4. A series of steps used to prove the validity of an if-then d. right angle

statement.

i. proof

5. A line segment connecting opposite vertices of a square. e. theorem

a. diagonal of a square

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 a2 1 b2 5 c2.

c. Pythagorean Theorem

7. A mathematical statement that cannot be proven but is g. postulate

considered to be true.

g. postulate

8. A triangle with a right angle. h. hypotenuse

b. right triangle

534 • Chapter 6 Skills Practice

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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.

h. hypotenuse

Problem SetThe 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 15.

9 2 1 12 2 5 15 2

81 1 144 5 225

225 5 225

2. 10, 26, 24


10 2 1 24 2 5 26 2

100 1 576 5 676

676 5 676

3. 20, 12, 16


12 2 1 16 2 5 20 2

144 1 256 5 400

400 5 400

4. 6, 8, 10


6 2 1 8 2 5 10 2

36 1 64 5 100

100 5 100


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Name ________________________________________________________ Date _________________________

7.

18

24

c 2 5 a 2 1 b 2

c 2 5 24 2 1 18 2

c 2 5 576 1 324

c 2 5 900

c 5 √____

900

c 5 30

8.

48

14

c2 5 a2 1 b2

c2 5 142 1 482

c2 5 196 1 2304

c2 5 2500

c 5 √_____

2500

c 5 50

5. 25, 15, 20


15 2 1 20 2 5 25 2

225 1 400 5 625

625 5 625

6. 15, 36, 39


15 2 1 36 2 5 39 2

225 1 1296 5 1521

1521 5 1521

Calculate the length of the hypotenuse of each given triangle.


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11.

18

6

c 2 5 a 2 1 b 2

c 2 5 6 2 1 18 2

c 2 5 36 1 324

c 2 5 360

c 5 √____

360

12.

5

10

c 2 5 a 2 1 b 2

c 2 5 5 2 1 10 2

c 2 5 25 1 100

c 2 5 125

c 5 √____

125

9.

2

1.5

c 2 5 a 2 1 b 2

c 2 5 1.5 2 1 2 2

c 2 5 2.25 1 4

c 2 5 6.25

c 5 √_____

6.25

c 5 2.5

10.

12

9

c 2 5 a 2 1 b 2

c 2 5 9 2 1 12 2

c 2 5 81 1 144

c 2 5 225

c 5 √____

225

c 5 15


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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 2 5 4 2 1 12 2

c 2 5 16 1 144

c 2 5 160

c < 12.65

Clayton must extend his ladder about 12.65 feet.


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14. 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

c2 5 a2 1 b2

c 2 5 3 2 1 8 2

c 2 5 9 1 64

c 2 5 73

c < 8.5

Each support beam should be about 8.5 feet long.


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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

c2 5 a2 1 b2

c 2 5 4 2 1 10 2

c 2 5 16 1 100

c 2 5 116

c < 10.77

The length of Perry’s ladder must be a minimum of about 10.77 feet.


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16. 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

c 2 5 a 2 1 b 2

c 2 5 12 2 1 16 2

c 2 5 144 1 256

c 2 5 400

c 5 20

The path is 20 feet long. Ling will need 10 stepping stones to create the path.


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Name ________________________________________________________ Date _________________________

19.4 8

c 2 5 a 2 1 b 2

8 2 5 4 2 1 b 2

64 2 16 5 b 2

48 5 b 2

6.93 < b

20.5

7

c 2 5 a 2 1 b 2

c 2 5 5 2 1 7 2

c 2 5 25 1 49

c 2 5 74

c < 8.60

Calculate the length of the missing side of each given triangle.

17.6

15

c 2 5 a 2 1 b 2

15 2 5 6 2 1 b 2

225 2 36 5 b 2

189 5 b 2

13.75 < b

18. 22

24

c 2 5 a 2 1 b 2

24 2 5 22 2 1 b 2

576 2 484 5 b 2

92 5 b 2

9.59 < b


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23.

6

3

c 2 5 a 2 1 b 2

6 2 5 3 2 1 b 2

36 2 9 5 b 2

27 5 b 2

5.20 < b

24. 8

10

c 2 5 a 2 1 b 2

c 2 5 8 2 1 10 2

c 2 5 64 1 100

c 2 5 164

c < 12.81

21. 12

18

c 2 5 a 2 1 b 2

18 2 5 12 2 1 b 2

324 2 144 5 b 2

180 5 b 2

13.42 < b

22.8

14

c 2 5 a 2 1 b 2

14 2 5 8 2 1 b 2

196 2 64 5 b 2

132 5 b 2

11.49 < b


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Can That Be Right?The Converse of the Pythagorean Theorem

VocabularyWrite the term that best completes the statement.

1. The Converse of the Pythagorean Theorem states: If a 2 1 b 2 5 c 2 , then the triangle is a right

triangle.

2. The converse of a theorem is created when the if-then 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) Pythagorean triple .

Problem SetDetermine whether each triangle with the given side lengths is a right triangle.


Name ________________________________________________________ Date _________________________

1. 8, 15, 17

c 2 5 a 2 1 b 2

17 2 5 15 2 1 8 2

289 5 225 1 64

289 5 289

This is a right triangle.

2. 6, 9, 14

c 2 5 a 2 1 b 2

14 2 5 9 2 1 6 2

196 5 81 1 36

196 fi 117

This is not a right triangle.


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3. 12, 15, 18

c 2 5 a 2 1 b 2

18 2 5 15 2 1 12 2

324 5 225 1 144

324 fi 369


4. 5, 12, 13

c 2 5 a 2 1 b 2

13 2 5 12 2 1 5 2

169 5 144 1 25

169 5 169


5. 6, 8, 10

c 2 5 a 2 1 b 2

10 2 5 8 2 1 6 2

100 5 64 1 36

100 5 100


6. 9, 12, 16

c 2 5 a 2 1 b 2

16 2 5 12 2 1 9 2

256 5 144 1 81

256 fi 225



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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 15-inch

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 2 1 13 2 5 15 2

a 2 1 169 5 225

a 2 5 56

a 5 √___

56

a < 7.5

The height of the computer monitor screen is about 7.5 inches.


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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?

a 2 1 b 2 5 c 2

3 2 1 4 2 5 c 2

9 1 16 5 c 2

25 5 c 2

√___

25 5 c

5 5 c

Luisa can measure the diagonal. For the corners to be right angles, the diagonal must be

5 feet long.


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Name ________________________________________________________ Date _________________________

9. Firefighters need to cross from the roof of a 25-feet-tall building to the roof of a 35-feet-tall 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

25 ft

20 ft

35 ft

The difference in the heights of the two buildings is 35 2 25 5 10 feet.

c2 5 a2 1 b2

c2 5 102 1 202

c2 5 100 1 400

c2 5 500

c 5 √____

500

c < 22.36

The ladder must be a minimum of approximately 22.36 feet long.


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10. 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

4 feet 5 48 inches

a 2 1 b 2 5 c 2

48 2 1 18 2 5 c 2

2304 1 324 5 c 2

2628 5 c 2

√_____

2628 5 c

51.3 < c

The piece of wood for the ramp should be about 51.3 inches long.


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Name ________________________________________________________ Date _________________________

11. Perry wants to use a 12-foot 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.

a 2 1 b 2 5 c 2

a 2 1 11 2 5 12 2

a 2 1 121 5 144

a 2 5 23

a 5 √___

23

a < 4.8

Perry should place the base of the ladder about 4.8 feet from the wall.

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.

a 2 1 b 2 5 c 2

6 2 1 4 2 5 c 2

36 1 16 5 c 2

52 5 c 2

√___

52 5 c

7 < c

The soccer field is about 7 blocks from Lea’s home.


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Calculate the length of the segment that connects the points in each. Write your answer as a radical

if necessary.

13.

3

4

a 2 1 b 2 5 c 2

3 2 1 4 2 5 c 2

9 1 16 5 c 2

25 5 c 2

√___

25 5 c

5 5 c

14.

12

5

a 2 1 b 2 5 c 2

5 2 1 12 2 5 c 2

25 1 144 5 c 2

169 5 c 2

√____

169 5 c

13 5 c


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Name ________________________________________________________ Date _________________________

15.

4

8

a 2 1 b 2 5 c 2

4 2 1 8 2 5 c 2

16 1 64 5 c 2

80 5 c 2

√___

80 5 c

16.

10

6

a 2 1 b 2 5 c 2

6 2 1 10 2 5 c 2

36 1 100 5 c 2

136 5 c 2

√____

136 5 c


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17.

10

5

a 2 1 b 2 5 c 2

5 2 1 10 2 5 c 2

25 1 100 5 c 2

125 5 c 2

√____

125 5 c

18.

3

6

a 2 1 b 2 5 c 2

3 2 1 6 2 5 c 2

9 1 36 5 c 2

45 5 c 2

√___

45 5 c



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Pythagoras to the RescueSolving for Unknown Lengths

Problem SetDetermine the length of the hypotenuse of each given triangle.


Name ________________________________________________________ Date _________________________

1.

10

24

c

c2 5 102 1 242

c2 5 100 1 576

c2 5 676

c 5 √____

676

c 5 26

The length of the hypotenuse is 26 units.

2. 6

6c

c2 5 62 1 62

c2 5 36 1 36

c2 5 72

c 5 √___

72

c < 8.49

The length of the hypotenuse

is approximately 8.49 units.


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3.

4

7.5

c

c2 5 42 1 7.52

c2 5 16 1 56.25

c2 5 72.25

c 5 √______

72.25

c 5 8.50

The length of the hypotenuse is 8.50 units.

4.

4

7

c

c2 5 42 1 72

c2 5 16 1 49

c2 5 65

c 5 √___

65

c < 8.06

The length of the hypotenuse



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Name ________________________________________________________ Date _________________________

5. 20

4.5c

c2 5 4.52 1 202

c2 5 20.25 1 400

c2 5 420.25

c 5 √_______

420.25

c 5 20.50

The length of the hypotenuse is 20.50 units.

6.

20

20

c

c2 5 202 1 202

c2 5 400 1 400

c2 5 800

c 5 √____

800

c < 28.28

The length of the hypotenuse is

approximately 28.28 units.


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7.

1220

b

12 2 1 b2 5 202

144 1 b2 5 400

b2 5 400 2 144

b2 5 256

b 5 √____

256

b 5 16

The length of the leg is 16 units.

8.

11

13a

a2 1 112 5 132

a2 1 121 5 169

a2 5 169 2 121

a2 5 48

a 5 √___

48

a < 6.93

The length of the leg is approximately

6.93 units.

Determine each unknown leg length.


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Name ________________________________________________________ Date _________________________

9.

17

12

b

122 1 b2 5 172

144 1 b2 5 289

b2 5 289 2 144

b2 5 145

b 5 √____

145

b < 12.04


12.04 units.

10. 9

12a

a2 1 92 5 122

a2 1 81 5 144

a2 5 144 2 81

a2 5 63

a 5 √___

63

a < 7.94


7.94 units.


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11.41

9

b

92 1 b2 5 412

81 1 b2 5 1681

b2 5 1681 2 81

b2 5 1600

b 5 √_____

1600

b 5 40


12.

3355

a

a2 1 332 5 552

a2 1 1089 5 3025

a2 5 3025 2 1089

a2 5 1936

a 5 √_____

1936

a 5 44



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Name ________________________________________________________ Date _________________________

13.

17

15

6

62 1 152 0 172

36 1 225 0 289

261 fi 289

The triangle is not a right triangle.

15.

119

8

82 1 92 0 112

64 1 81 0 121

145 fi 121


14. 24

257

72 1 242 0 252

49 1 576 0 625

625 5 625

The triangle is a right triangle.

16.

3.75

4.25

2

22 + 3.752 0 4.252

4 1 14.0625 0 18.0625

18.0625 5 18.0625


Use the Pythagorean Theorem to determine whether each given triangle is a right triangle.


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17.

35

28

21

212 1 282 0 352

441 1 784 0 1225

1225 5 1225


18.

28

26

5

52 1 262 0 282

25 1 676 0 784

701 fi 784



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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

182 1 242 5 x2

324 1 576 5 x2

900 5 x2

√____

900 5 x

30 5 x

Each diagonal brace is 30 feet long.


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20. 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.

502 1 942 5 c2

2500 1 8836 5 c2

11,336 5 c2

√_______

11,336 5 c

106.47 < c

The distance of each practice shot is approximately 106.47 feet.


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Name ________________________________________________________ Date _________________________

21. The water company installed a 40-yard 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

122 1 b2 5 402

144 1 b2 5 1600

b2 5 1600 2 144

b2 5 1456

b 5 √_____

1456

b < 38.16

The height of each vertical beam is approximately 38.16 yards.


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22. The lengths of the legs of a right triangle are 15 meters each. Determine the length of the

hypotenuse.

c2 5 152 1 152

c2 5 225 1 225

c2 5 450

c 5 √____

450

c < 21.21

The length of the hypotenuse is approximately 21.21 meters.

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.

a2 1 a2 5 502

2a2 5 2500

a2 5 1250

a 5 √_____

1250

a < 35.36

The length of each leg is approximately 35.36 inches.


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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.

182 + 802 5 c2

324 1 6400 5 c2

6724 5 c2

√_____

6724 5 c

82 5 c

The direct distance from Walker Dock to Blue Haven Dock is 82 miles.


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Meeting FriendsThe Distance Between Two Points in a Coordinate System

Problem SetDetermine 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)

x

86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y c2 5 a2 1 b2

c2 5 62 1 32

c2 5 36 1 9

c2 5 45

c 5 √___

45

c < 6.71

The distance between (2, 2) and (8, 5)



Name ________________________________________________________ Date _________________________


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2. (3, 7) and (7, 3)

x

86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

c2 5 a2 1 b2

c2 5 42 1 42

c2 5 16 1 16

c2 5 32

c 5 √___

32

c < 5.66

The distance between (3, 7) and (7, 3)


3. (26, 8) and (6, 3)

x

86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

c2 5 a2 1 b2

c2 5 52 1 122

c2 5 25 1 144

c2 5 169

c 5 √____

169

c 5 13

The distance between (26, 8) and

(6, 3) is 13 units.


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Name ________________________________________________________ Date _________________________

4. (7, 5) and (3, 23)

x

86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

c2 5 a2 1 b2

c2 5 42 1 82

c2 5 16 1 64

c2 5 80

c 5 √___

80

c < 8.94


(3, 23) is approximately 8.94 units.

5. (24, 24) and (5, 8)

x

86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

c2 5 a2 1 b2

c2 5 92 1 122

c2 5 811 144

c2 5 225

c 5 √____

225

c 5 15


(5, 8) is 15 units.


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6. (29, 3) and (7, 5)

x

86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

c2 5 a2 1 b2

c2 5 162 1 22

c2 5 256 1 4

c2 5 260

c 5 √____

260

c < 16.12



7. (27, 3) and (8, 25)

x

86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

c2 5 a2 1 b2

c2 5 82 1 152

c2 5 64 1 225

c2 5 289

c 5 √____

289

c 5 17

The distance between (27, 3)

and (8, 25) is 17 units.


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Name ________________________________________________________ Date _________________________

8. (29, 6) and (8, 1)

x

y

–2

–4

–6

–8

86–2 42–4–6–8

2

4

6

8

c2 5 a2 1 b2

c2 5 52 1 172

c2 5 25 1 289

c2 5 314

c 5 √____

314

c < 17.72



Archaeologists map each item they find at a dig on a 1-foot by 1-foot 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.

x

1 2 3 4 5 6 7 8 900

1

2

3

4

5

6

7

8

9

y

spindle

beads

c2 5 a2 1 b2

c2 5 42 1 32

c2 5 16 1 9

c2 5 25

c 5 √___

25

c 5 5

The distance between the spindle and

the beads is 5 feet.

7721B_C3_Skills_CH06_533-610.indd 571 6/14/11 11:12 AM


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10. Determine the distance between the pottery shard and the axe head.

x

1 2 3 4 5 6 7 8 900

1

2

3

4

5

6

7

8

9

y

axe head

pottery shard

c2 5 a2 1 b2

c2 5 62 1 62

c2 5 36 1 36

c2 5 72

c 5 √___

72

c < 8.49

The distance between the pottery

shard and the axe head is

approximately 8.49 feet.

11. Determine the distance between the coins and the beads.

x

1 2 3 4 5 6 7 8 900

1

2

3

4

5

6

7

8

9

y

coins

beads

c2 5 a2 1 b2

c2 5 12 1 42

c2 5 1 1 16

c2 5 17

c 5 √___

17

c < 4.12

The distance between the coins

and the beads is approximately

4.12 feet.


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Name ________________________________________________________ Date _________________________

12. Determine the distance between the coins and the axe head.

x

1 2 3 4 5 6 7 8 900

1

2

3

4

5

6

7

8

9

y

coins

axe head

c2 5 a2 1 b2

c2 5 22 1 32

c2 5 4 1 9

c2 5 13

c 5 √___

13

c < 3.61

The distance between the coins

and the axe head is approximately

3.61 feet.

13. Determine the distance between the mask and the beads.

x

1 2 3 4 5 6 7 8 900

1

2

3

4

5

6

7

8

9

y

beads

mask

c2 5 a2 1 b2

c2 5 82 1 42

c2 5 64 1 16

c2 5 80

c 5 √___

80

c < 8.94

The distance between the mask and

the beads is approximately 8.94 feet.


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14. Determine the distance between the pottery shard and the beads.

x

1 2 3 4 5 6 7 8 900

1

2

3

4

5

6

7

8

9

y

beads

pottery shard

c2 5 a2 1 b2

c2 5 12 1 32

c2 5 1 1 9

c2 5 10

c 5 √___

10

c < 3.16

The distance between the pottery

shard and the beads is approximately

3.16 feet.

15. Determine the distance between the spindle and the axe head.

x

1 2 3 4 5 6 7 8 900

1

2

3

4

5

6

7

8

9

y

spindle

axe head

c2 5 a2 1 b2

c2 5 62 1 32

c2 5 36 1 9

c2 5 45

c 5 √___

45

c < 6.71

The distance between the spindle and

the axe head is approximately

6.71 feet.


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Name ________________________________________________________ Date _________________________

16. Determine the distance between the mask and the coins.

x

1 2 3 4 5 6 7 8 900

1

2

3

4

5

6

7

8

9

y

coins

mask

c2 5 a2 1 b2

c2 5 42 1 52

c2 5 16 1 25

c2 5 41

c 5 √___

41

c < 6.40

The distance between the mask and

the coins is approximately 6.40 feet.


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DiagonallyDiagonals in Two Dimensions

Problem SetDetermine the length of the diagonals in each given quadrilateral.

1. The quadrilateral is a square.

A B

CD

15 ft

c2 5 a2 1 b2

c2 5 152 1 152

c2 5 225 1 225

c2 5 450

c 5 √____

450

c < 21.21

The length of diagonal AC is approximately 21.21 feet. The length of diagonal

BD is approximately 21.21 feet.


Name ________________________________________________________ Date _________________________


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2. The quadrilateral is a rectangle.

E F

GH

10 in.

18 in.

c2 5 a2 1 b2

c2 5 102 1 182

c2 5 100 1 324

c2 5 424

c 5 √____

424

c < 20.59

The length of diagonal EG is approximately 20.59 inches. The length of diagonal

FH is approximately 20.59 inches.


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Name ________________________________________________________ Date _________________________

3. The quadrilateral is a parallelogram.

11 m

8 m

6 m

K

MN

J

c2 5 a2 1 b2

c2 5 62 1 82

c2 5 36 1 64

c2 5 100

c 5 √____

100

c 5 10

The length of diagonal JM is 10 meters.

c2 5 a2 1 b2

c2 5 112 1 62

c2 5 1211 36

c2 5 157

c 5 √____

157

c < 12.53

The length of diagonal KN is approximately 12.53 meters.


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4. The quadrilateral is a trapezoid.

x

1 2 3 4 5 6 7 8 900

1

2

3

4

5

6

7

8

9

y

P Q

RS

c2 5 a2 1 b2

c2 5 42 1 62

c2 5 16 1 36

c2 5 52

c 5 √___

52

c < 7.21

The length of diagonal PR


c2 5 a2 1 b2

c2 5 42 1 42

c2 5 16 1 16

c2 5 32

c 5 √___

32

c < 5.66

The length of diagonal QS is approximately 5.66 units.


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Name ________________________________________________________ Date _________________________

5. The quadrilateral is an isosceles trapezoid.

x

1 2 3 4 5 6 7 8 900

1

2

3

4

5

6

7

8

9

y

W X

YZ

c2 5 a2 1 b2

c2 5 52 1 52

c2 5 25 1 25

c2 5 50

c 5 √___

50

c < 7.07

The length of diagonal WY is

approximately 7.07 units. The length of

diagonal XZ is approximately

7.07 units.


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6. The quadrilateral is a rhombus.

x

1 2 3 4 5 6 7 8 900

1

2

3

4

5

6

7

8

9

y

B G

KM

c2 5 a2 1 b2

c2 5 22 1 42

c2 5 4 1 16

c2 5 20

c 5 √___

20

c < 4.47

The length of diagonal BK


c2 5 a2 1 b2

c2 5 82 1 42

c2 5 64 1 16

c2 5 80

c 5 √___

80

c < 8.94

The length of diagonal GM is approximately 8.94 units.


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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.

9 in.

The area of the rectangle is:

A 5 bh

A 5 (4)(9)

A 5 36 in.2

The length of the rectangle’s diagonal is:

c2 5 a2 1 b2

c2 5 42 1 92

c2 5 16 1 81

c2 5 97

c 5 √___

97

c < 9.85 in.

The area of the circle is:

A 5 πr2

A < (3.14)(4.93)2

A < 76.32 in.2

The area of the shaded region

is approximately 76.32 2 36 5 40.32 in.2.


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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

The area of the smaller square is:

A 5 s2

A 5 (10)2

A 5 100 ft2

The length of the smaller square’s diagonal is:

c2 5 a2 1 b2

c2 5102 1 102

c2 5 100 1 100

c2 5 200

c 5 √____

200

c < 14.14 ft

The area of the larger square is:

A 5 s2

A 5 (14.14)2

A 5 200 ft2


is 200 2 100 5 100 ft2.


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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

The area of the triangle is:

A 5 1 __ 2

bh

A 5 1 __ 2

(12)(5)

A 5 30 m2

The length of the triangle’s hypotenuse is:

c2 5 a2 1 b2

c2 5 52 1 122

c2 5 25 1 144

c2 5 169

c 5 √____

169

c 5 13 m

The area of the circle is:

A 5 πr2

A < (3.14)(6.5)2

A < 132.67 m2

The area of the shaded region is

approximately 132.67 2 30 5 102.67 m2.


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10. 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


A 5 1 __ 2

bh

A 5 1 __ 2

(20)(15)

A 5 150 yd2


c2 5 a2 1 b2

c2 5 152 1 202

c2 5 225 1 400

c2 5 625

c 5 √____

625

c 5 25 yd

The area of the square is:

A 5 s2

A 5 (25)2

A 5 625 yd2

The area of the shaded region is

625 1 150 5 775 yd2.


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Name ________________________________________________________ Date _________________________

11. The figure is composed of a right triangle and a semi-circle. The hypotenuse of the right triangle is

the same length as the diameter of the semi-circle.

5 ft 5 ft


A 5 1 __ 2

bh

A 5 1 __ 2

(5)(5)

A 5 12.50 ft2


c2 5 a2 1 b2

c2 5 52 1 52

c2 5 25 1 25

c2 5 50

c 5 √___

50

c < 7.07 ft

The area of the semi-circle is:

A 5 1 __ 2

πr2

A < 1 __ 2

(3.14)(3.54)2

A < 19.67 ft2


is approximately 19.67 1 12.50 5 32.17 ft2.


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12. 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

The area of the bottom triangle is:

A 5 1 __ 2

bh

A 5 1 __ 2

(4)(3)

A 5 6 cm2

The length of the bottom triangle’s hypotenuse is:

c2 5 a2 1 b2

c2 5 32 1 42

c2 5 9 1 16

c2 5 25

c 5 √___

25

c 5 5 cm

The area of the top triangle is:

A 5 1 __ 2

bh

A 5 1 __ 2

(5)(3)

A 5 7.50 cm2


is 7.50 1 6 5 13.50 cm2.


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Two Dimensions Meet Three DimensionsDiagonals in Three Dimensions

Problem SetDraw all of the edges you cannot see in each rectangular solid using dotted lines. Then draw a

three-dimensional diagonal using a solid line.


Name ________________________________________________________ Date _________________________

1. 2.

3. 4.


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Name ________________________________________________________ Date _________________________

Determine the length of the three-dimensional diagonal in the given rectangular solid using each

Pythagorean Theorem.

7.

5 m

8 m8 m

Length of second leg:

c2 5 82 1 82

c2 5 64 1 64

c2 5 128

c 5 √____

128

c < 11.31

Length of diagonal:

d2 < 11.312 1 52

d2 < 127.92 1 25

d2 < 152.92

d < √_______

152.92

d < 12.37

The length of the three-dimensional diagonal in the rectangular solid is approximately

12.37 meters.


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8.

10 in.

4 in.1 in.


c2 5 42 1 12

c2 5 16 1 1

c2 5 17

c 5 √___

17

c < 4.12

Length of diagonal:

d2 < 4.122 1 102

d2 < 16.97 1 100

d2 < 116.97

d < √_______

116.97

d < 10.82


10.82 inches.


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Name ________________________________________________________ Date _________________________

9.

11 cm

3 cm

6 cm


c2 5 62 1 32

c2 5 36 1 9

c2 5 45

c 5 √___

45

c < 6.71

Length of diagonal:

d2 < 6.712 1 112

d2 < 45.02 1 121

d2 < 166.02

d < √_______

166.02

d < 12.88


12.88 centimeters.


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10.

15 m

3 m4 m


c2 5 42 1 32

c2 5 16 1 9

c2 5 25

c 5 √___

25

c 5 5

Length of diagonal:

d2 5 52 1 152

d2 5 25 1 225

d2 5 250

d 5 √____

250

d < 15.81


15.81 meters.


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Name ________________________________________________________ Date _________________________

11.

9 ft

5 ft

12 ft


c2 5 52 1 122

c2 5 25 1 144

c2 5 169

c 5 √____

169

c 5 13

Length of diagonal:

d2 5 132 1 92

d2 5 169 1 81

d2 5 250

d 5  250

d < 15.81


15.81 feet.


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12.

14 in.

13 in.

7 in.


c2 5 72 1 132

c2 5 49 1 169

c2 5 218

c 5 √____

218

c < 14.76

Length of diagonal:

d2 < 14.762 1 142

d2 < 217.86 1 196

d2 < 413.86

d < √_______

413.86

d < 20.34


20.34 inches.


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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 three-dimensional diagonal. Use a formula.

13.

8"

3"

6"

d2 5 1 __ 2 (32 1 62 1 82)

d2 5 1 __ 2 (9 1 36 1 64)

d2 5 1 __ 2 (109)

d2 5 54.50

d 5 √______

54.50

d < 7.38

The length of the three-dimensional diagonal is approximately 7.38 inches.


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14.

3 m10 m

9 m

d2 5 1 __ 2 (32 1 92 1 102)

d2 5 1 __ 2 (9 1 81 1 100)

d2 5 1 __ 2 (190)

d2 5 95

d 5 √___

95

d < 9.75

The length of the three-dimensional diagonal is approximately 9.75 meters.


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Name ________________________________________________________ Date _________________________

15.

12 ft

8 ft

10 ft

d2 5 1 __ 2 (82 1 102 1 122)

d2 5 1 __ 2 (64 1 100 1 144)

d2 5 1 __ 2 (308)

d2 5 154

d 5  154

d < 12.41

The length of the three-dimensional diagonal is approximately 12.41 feet.


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16.6 m

6 m5 m

d2 5 1 __ 2 (52 1 62 1 62)

d2 5 1 __ 2 (25 1 36 1 36)

d2 5 1 __ 2 (97)

d2 5 48.50

d 5 √______

48.50

d < 6.96

The length of the three-dimensional diagonal is approximately 6.96 meters.


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Name ________________________________________________________ Date _________________________

17.

4 yd10 yd

8 yd

d2 5 1 __ 2 (42 1 82 1 102)

d2 5 1 __ 2 (16 1 64 1 100)

d2 5 1 __ 2 (180)

d2 5 90

d 5 √___

90

d < 9.49

The length of the three-dimensional diagonal is approximately 9.49 yards.


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18.

15"

3"

13"

d2 5 1 __ 2 (32 1 132 1 152)

d2 5 1 __ 2 (9 1 169 1 225)

d2 5 1 __ 2 (403)

d2 5 201.50

d 5 √_______

201.50

d < 14.20

The length of the three-dimensional diagonal is approximately 14.20 inches.


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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 three-dimensional

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?

d2 5 52 1 62 1 82

d2 5 25 1 36 1 64

d2 5 125

d 5 √____

125

d < 11.18

The diagonal support will need to be approximately 11.18 feet.

20. A plumber needs to transport a 12-foot 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?

d 2 5 902 1 402 1 402

d 2 5 8100 1 1600 1 1600

d 2 5 11,300

d 5 √_______

11,300

d < 106.30

The pipe will not fit in the car. The three-dimensional diagonal length of the van is

approximately 106.30 inches. The length of the pipe is 12 feet or 144 inches.


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21. 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?

d 2 5 602 1 452 1 402

d 2 5 3600 1 2025 1 1600

d 2 5 7225

d 5 √_____

7225

d 5 85

The tree will fit in the car. The three-dimensional diagonal length of the car’s interior is

85 inches. The height of the tree is 5 feet or 60 inches.

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 three-dimensional 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?

d 2 5 1 __ 2

(22 1 32 1 72)

d 2 5 1 __ 2

(4 1 9 1 49)

d 2 5 1 __ 2

(62)

d 2 5 31

d 5 √___

31

d < 5.57

The three-dimensional diagonal through the center of the box is approximately 5.57 feet long.


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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 10-inch stick into the box and be able to

close the lid. Will the stick fit inside the box?

d 2 5 1 __ 2

(92 1 52 1 72)

d 2 5 1 __ 2

(81 1 25 1 49)

d 2 5 1 __ 2

(155)

d 2 5 77.50

d 5 √______

77.50

d < 8.80

A 10-inch stick will not fit inside the box. The three-dimensional diagonal through the center of

the box is approximately 8.80 inches long.


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24. 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 three-dimensional diagonal

of the television is 24 inches. Will the television fit in the box?

d 2 5 1 __ 2

(172 1 192 1 202)

d 2 5 1 __ 2

(289 1 361 1 400)

d 2 5 1 __ 2

(1050)

d 2 5 525

d 5 √____

525

d < 22.91

A television with a three-dimensional diagonal of 24 inches will not fit in the box.

The three-dimensional diagonal through the center of the box is approximately

22.91 inches long.

Determine each unknown measurement.

25. A rectangular box has a length of 8 inches and a width of 5 inches. The length of the three-

dimensional diagonal of the box is 12 inches. What is the height of the box?

d2 5 l2 1 w2 1 h2

122 5 82 1 52 1 h2

144 5 64 1 25 1 h2

55 5 h2

√___

55 5 h

7.42 < h

The height of the box is approximately 7.42 inches.


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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 three-dimensional diagonal is

14 feet. What is the length of the diagonal across the side of the box?

d2 5 1 __ 2 (d1

2 1 d22 1 d3

2)

142 5 1 __ 2

(62 1 92 1 d32)

196 5 1 __ 2

(36 1 81 1 d32)

392 5 117 1 d32

275 5 d32

√____

275 5 d3

16.58 < d3

The length of the diagonal across the side of the box is approximately 16.58 feet.

27. A rectangular box has a length of 7 feet and a height of 11 feet. The length of the

three-dimensional diagonal of the box is 20 feet. What is the width of the box?

d2 5 l2 1 w2 1 h2

202 5 72 1 w2 1 112

400 5 49 1 w2 1 121

230 5 w2

√____

230 5 w

15.17 < w

The width of the box is approximately 15.17 feet.


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28. 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 three-dimensional

diagonal is 20 centimeters. What is the length of the diagonal across the front of the box?

d2 5 1 __ 2 (d1

2 1 d22 1 d3

2)

202 5 1 __ 2

(162 1 182 1 d32)

400 5 1 __ 2 (256 1 324 1 d3

2)

800 5 580 1 d32

220 5 d32

√____

220 5 d3

14.83 < d3

The length of the diagonal across the front of the box is approximately 14.83 centimeters.

29. A rectangular box has a height of 3 feet and a width of 4 feet. The length of the three-dimensional

diagonal of the box is 13 feet. What is the length of the box?

d2 5 l2 1 w2 1 h2

132 5 l2 1 42 1 32

169 5 l2 1 16 1 9

144 5 l2

√____

144 5 l

12 5 l

The length of the box is 12 feet.


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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 three-dimensional diagonal is

40 inches. What is the length of the diagonal across the top of the box?

d2 5 1 __ 2

(d12 1 d2

2 1 d32)

402 5 1 __ 2 (302 1 302 1 d3

2)

1600 5 1 __ 2

(900 1 900 1 d32)

3200 5 1800 1 d32

1400 5 d32

√_____

1400 5 d3

37.42 < d3

The length of the diagonal across the top of the box is approximately 37.42 inches.

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