Lesson 24 Solve Problems Using the Pythagorean...

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©Curriculum Associates, LLC Copying is not permitted. 243Lesson 24 Solve Problems Using the Pythagorean Theorem

Name: Solve Problems Using the Pythagorean Theorem

Lesson 24

Vocabularysquare root a factor

of a number that when

multiplied by itself

results in the number

For example, 9 is a

square root of 81

because 9 3 9 5 81

Prerequisite: Finding Square Roots

Study the example showing how to solve a problem that involves square roots. Then solve problems 1–7.

1 Suppose you know the area of the yard is 7,500 square feet How can you fi nd the value of a?

2 Use your answer to problem 1 to fi nd the value of a

Show your work.

Solution:

3 What is the perimeter of the backyard?

Example

Leroy’s backyard is a rectangle that is three times as long as it is wide What is the perimeter and the area of the yard?

First draw a diagram to help you solve the problem Let a be the width of Leroy’s yard

a

3a

Leroy’s Backyard

The perimeter is a 1 3a 1 a 1 3a, or 8a The area of a rectangle is , • w, so the area of the backyard is a • 3a, or 3a2

243243

The area of the backyard is 3a2, so I can solve the equation 3a2 5 7,500.

3a2 5 7,500

a2 5 2,500

a 5 Ï······ 2,500

a 5 50

The value of a is 50 feet.

8a 5 8 • 50 5 400. The perimeter is 400 feet.

B

M

M

©Curriculum Associates, LLC Copying is not permitted.244 Lesson 24 Solve Problems Using the Pythagorean Theorem

Solve.

4 The area of the fl oor of a square room is 144 square feet What is the perimeter of the room? Include a diagram in your solution

Show your work.

Solution:

5 Which two numbers in the box are perfect squares? Explain how you know they are perfect squares

6 What is the side length of a square whose area is 1 ·· 64 square mile?

Explain your answer

7 Natasha says that if the area of a square is a, then the length of a side of the square is Ï·· a Do you agree? Explain

20 125 196 500 625

244

The area of the floor is s2.

s2 5 144

s 5 Ï···· 144 5 Ï···· 122 5 12

The perimeter of the fl oor is 4 • s.

4 • s 5 4 • 12 5 48

14 • 14 5 196 and 25 • 25 5 625, so 196 and 625 are perfect squares. Possible explanation:

You cannot multiply an integer by itself to get 20, 125, or 500, so they are not perfect

squares.

1 ·· 8 mile; Possible explanation: To find the side length, I can take the square root of 1 ··· 64 ,

which is 1 ·· 8 .

Yes; Possible explanation: If Ï·· a is the length of a side of a square, then ( Ï·· a )2 is the area.

( Ï·· a )2 5 a, so if the area of a square is a, then the length of a side is Ï·· a .

Area144 ft2 s

s

M

M

M

C

The perimeter of the room is 48 feet.

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Unit 4 G

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Unit 4


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Name: Lesson 24

Solve Problems Using Right Triangles

Study the example showing how to solve a problem that involves distance. Then solve problems 1–5.

1 Find the distance if Mr Nichols had been able to drive straight from the entrance to his campsite

Show your work.

Solution:

2 How much shorter would Mr Nichols’ drive have been if he had been able to drive straight from the entrance to his campsite? Explain

Example

Mr Nichols followed the road from the entrance of a park to his campsite He drove 6 miles south, then 5 miles east, 3 miles south, and finally 7 miles east to the campsite In all, how far did Mr Nichols drive? Suppose Mr Nichols had been able to drive straight from the entrance of the park to the campsite How could you find that distance?

You can use the diagram to help you solve this problem Add the individual distances to find that Mr Nichols drove a total of 6 1 5 1 3 1 7 5 21 miles

You can use the Pythagorean Theorem to find the distance that Mr Nichols would have driven if he could have driven straight from the entrance to the campsite The distance is the hypotenuse of a right triangle with leg lengths of 9 miles and 12 miles, so you can write and solve the equation 92 1 122 5 c2 and solve for c

Campsite

Entrance

245

a2 1 b2 5 c2 92 1 122 5 c2 81 1 144 5 c2

225 5 c2

15 5 c

Mr. Nichols would have driven 15 miles if he had been able to drive straight from

the entrance to his campsite.

6 miles shorter; 21 2 15 5 6

M

B


Solve.

3 You tie a spherical balloon that is 2 feet in diameter to a stake in the ground The string is 15 feet long The wind blows and you observe that the top of the balloon is 8 feet over from the stake, as shown in the diagram What is the height, b, of the balloon?

Show your work.

Solution:

4 In problem 5, the wind blows harder and the top of the balloon is now 15 feet over from the stake What is the height of the balloon now? Draw a diagram to help you solve the problem and explain your answer

5 The perimeter of an equilateral triangle is 48 cm Use estimation to fi nd the height of the triangle to the nearest whole number Include a sketch in your answer

Show your work.

Solution:

8 ft

b15 ft

2 ft

246

8 ft; Possible explanation: The base of the triangle is 15 feet and the length of the hypotenuse has not changed, so the other leg of the triangle will be 8 feet because 82 1 152 5 172.

I can use the Pythagorean Theorem to find b.

a2 1 b2 5 c2

82 1 b2 5 172

64 1 b2 5 289 b2 5 225 b 5 Ï···· 225 5 15

The height, b, of the balloon is 15 feet.

The height of the triangle is about 14 cm.

The sides of the triangle are all the same length: 48 4 3 5 16.

82 1 h2 5 162

64 1 h2 5 256

h2 5 192

h 5 Ï···· 192 Ï···· 192 is between 13 and 14; it is closer to 14.

a

15 ft

17 ft

8

16h

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Unit 4 G

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nit 4Practice Lesson 24 Solve Problem

s Using the Pythagorean Theorem


Name: Lesson 24

The Pythagorean Theorem and Three-Dimensional Figures

Study the example showing how to apply the Pythagorean Theorem. Then solve problems 1–6.

1 How do you know that the triangle with side lengths 6, 8, and c is a right triangle?

2 How do you know that the triangle with side lengths c, 24, and d is a right triangle?

3 Use the Pythgorean Theorem to write equations to fi nd the values of c and d Find the value of c and then the value of d

Example

The diagram shows a diagonal drawn from point A to point B in a rectangular prism How can you find the length of this diagonal?

You can use what you know about right triangles that are related to the diagonal

The triangle on the base of the prism with side lengths 6, 8, and c is a right triangle The triangle that includes the diagonal from point A to point B with side lengths c, 24, and d is a right triangle

Use the Pythagorean Theorem to solve for c Then use c to solve for d

8

24

6

A

B

c

d

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62 1 82 5 c2 c2 1 242 5 d2

36 1 64 5 c2 100 1 576 5 d2

100 5 c2 676 5 d2

Ï···· 100 5 Ï·· c2 Ï···· 676 5 Ï·· d2

10 5 c 26 5 d

The three-dimensional figure is a rectangular prism, so all of the faces are rectangles.

Rectangles have square corners. The triangle with side lengths 6, 8, and c is on the base of

the prism, so the angle between the sides labeled 6 and 8 is a right angle.

The three-dimensional figure is a rectangular prism, so all of the faces are perpendicular.

In the triangle with side lengths c, 24, and d, lengths c and 24 are on adjacent faces, so

they are perpendicular and the angle between them is a right angle.

B

M

M


Solve.

4 Between what two whole numbers is the length of the diagonal from M to N in the rectangular prism?

Show your work.

Solution:

5 Tyler wants to ship a fi shing pole that is 7 feet long He will use a box that is a rectangular prism with a base that is 4 feet by 3 feet What is the shortest whole-number height that will hold the fi shing pole? Explain

Show your work.

Solution:

6 Write a real-world problem about a rectangular prism that can be solved by using the Pythagorean Theorem Then solve your problem

8 cm

6 cm 12 cm

M

N

4 ft3 ftA

B

d ft

248

Possible work: diagonal of base, x:x2 5 82 1 122 5 208x 5 Ï···· 208 diagonal from M to N:MN2 5 62 1 ( Ï···· 208 )2 5 36 1 208 5 244MN 5 Ï···· 244 152 5 225 and 162 5 256, so MN is between 15 and 16.

Possible explanation: The hypotenuse of the base is:x2 5 42 1 32 5 16 1 9 5 25 x 5 Ï··· 25 5 5I can use this length and the length of diagonal AB, which is 7 feet, to find d: 52 1 d2 5 72

25 1 d2 5 49 d2 5 24 d 5 Ï··· 24 The least whole-number value that makes this true is 5 because Ï··· 25 5 5 and Ï··· 25 Ï··· 24 .

The diagonal from M to N is between 15 and 16 centimeters long.

The height of the box must be at least 5 feet for the fishing pole to fit.

Problems will vary. Possible answer: Will a 12-inch ruler fit in a box that is a rectangular prism with a width of 3 inches, a length of 5 inches, and a height of 10 inches? Answer: No; The diagonal of the base x would be x2 5 32 1 52 5 34, so x 5 Ï··· 34 . The diagonal of the box d would be d2 5 ( Ï··· 34 )2 1 102 5 134, so d 5 Ï···· 134 . Ï···· 134 , 12, so the diagonal in the box is not long enough for the ruler.

M

C

C

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Solving

Unit 4 G

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Unit 4


sing the Pythagorean Theorem


Name: Lesson 24

1 Elsa is fl ying a kite She let out 25 meters of string The kite is directly above a spot that is 7 meters away How high above the ground is the kite?

Show your work.

Solution:

Solve Problems Using the Pythagorean Theorem

Solve the problems.

3 Margo drew a diagram of a square wooden deck, with side length c feet, that she wants to build in her square yard Explain how to fi nd the perimeter of the deck Then fi nd the perimeter

Show your work.

Solution:

2 The dimensions of a box that is a rectangular prism are 3 inches, 4 inches, and 6 inches What is the length of the diagonal from point R to point S, to the nearest tenth of an inch?

A 5 inches C 7 8 inches

B 6 7 inches D 61 inches

Rodney chose B as the correct answer How did he get that answer?

What lengths of the right triangle are you given: both legs or one leg and the hypotenuse?

What is the length of the hypotenuse of the base? How can this length help you solve the problem?

How is the length c related to the lengths 9 and 12?

4 in.6 in.

R

S3 in.

c9 ft

12 ft

249

24 meters

60 feet

He thought the diagonal of the base was the same as the diagonal of the prism.

Possible explanation: The length of one side of the deck is the hypotenuse of a right triangle with legs that are 9 and 12 feet long.c2 5 92 1 122 5 81 1 144 5 225, so c 5 Ï···· 225 5 15.The perimeter of the deck is: 4c 5 4 3 15 5 60.

72 1 b2 5 252

49 1 b2 5 625 b2 5 576 b 5 Ï···· 576 5 24

B

M

M


Solve.

4 Tell whether each statement is True or False about the right triangle shown

a

bc

a. The square of b is always equal to the square of c minus the square of a u True u False

b. The square of c equals the square of b plusthe square of a u True u False

c. The value of c is equal to the square of the sum ofb and c u True u False

5 An isosceles right triangle has a hypotenuse of 12 Between which two whole numbers are the lengths of the legs of the triangle?

Show your work.

Solution:

6 The length of the base of a rectangular prism is 3 feet and its width is 2 feet The length of the diagonal of the prism is 6 feet What is the height of the prism to the nearest tenth of a foot?

A 3 3 ft C 13 ft

B 4 8 ft D 23 ft

What do you know about the sides of an isosceles triangle?

What does the Pythagorean Theorem tell you about the side lengths of a right triangle?

You can draw a diagram to help you visualize the problem.

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3

3

3

The lengths of the legs are between 8 and 9.

Let a be the length of one of the legs of the triangle.a2 1 a2 5 122

2a2 5 144a2 5 72a 5 Ï··· 72 ø 8.5

C

M

C

Lesson 24 Solve Problems Using the Pythagorean...

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