Course 3

5-7 Indirect Measurement

Warm UpSolve each proportion.

x75

35

=1. 2.48

6x

=2.

x6

9 27

=3. 87

x3.5

=4.

x = 45 x = 20

x = 2 x = 4

Course 3

5-7 Indirect Measurement

Essential Question: How do you find an unknown value by using indirect measurement?

Spi: 706.4.1

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Course 3

5-7 Indirect Measurement

Vocabulary

indirect measurement

Sometimes, distances cannot be measured directly. One way to find such a distance is to use indirect measurement, a way of using similar figures and proportions to find a measure.

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Course 3

5-7 Indirect Measurement

Additional Example 1: Geography Application

Triangles ABC and EFG are similar.

Triangles ABC and EFG are similar. Find the length of side EG.

B

A C

3 ft

4 ft

F

E G

9 ft

x

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Course 3

5-7 Indirect Measurement

Additional Example 1 Continued

ABAC = EF

EG Set up a proportion.

Substitute 3 for AB, 4 for AC, and 9 for EF.

3x = 36 Find the cross products.

The length of side EG is 12 ft.x = 12

Triangles ABC and EFG are similar. Find the length of side EG.

34 = 9

x

3x3 = 36

3 Divide both sides by 3.

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Course 3

5-7 Indirect Measurement

Check It Out: Example 1

Triangles DEF and GHI are similar.

Triangles DEF and GHI are similar. Find the length of side HI.

2 in

E

D F

7 in

H

G I

8 in x

Course 3

5-7 Indirect Measurement

Check It Out: Example 1 Continued

DEEF = GH

HI Set up a proportion.

Substitute 2 for DE, 7 for EF, and 8 for GH.

2x = 56 Find the cross products.

The length of side HI is 28 in.

x = 28

27 = 8

x

2x2 = 56

2 Divide both sides by 2.

Triangles DEF and GHI are similar. Find the length of side HI.

Course 3

5-7 Indirect Measurement

• Indirect Measurement video

Course 3

5-7 Indirect Measurement

A 30-ft building casts a shadow that is 75 ft long. A nearby tree casts a shadow that is 35 ft long. How tall is the tree?

Additional Example 2: Problem Solving Application

11 Understand the Problem

The answer is the height of the tree.

List the important information:

• The length of the building’s shadow is 75 ft.

• The height of the building is 30 ft.

• The length of the tree’s shadow is 35 ft.

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Course 3

5-7 Indirect Measurement

Additional Example 2 Continued

Use the information to draw a diagram.22 Make a Plan

h

Draw dashed lines to form triangles. The building with its shadow and the tree with its shadow form similar right triangles.

Solve3375 feet

30 feet

35 feet

Course 3

5-7 Indirect Measurement

30 75

= h35

Solve33

Corresponding sides of similar figures are proportional.

75h = 1050 Find the cross products.

The height of the tree is 14 feet.

h = 14

75h75 = 1050

75 Divide both sides by 75.

Additional Example 2 Continued

Course 3

5-7 Indirect Measurement

Since = 2.5, the building’s shadow is

2.5 times its height. So, the tree’s shadow should also be 2.5 times its height and 2.5 of 14 is 35 feet.

44 Look Back

75 30

Additional Example 2 Continued

Course 3

5-7 Indirect Measurement

A 24-ft building casts a shadow that is 8 ft long. A nearby tree casts a shadow that is 3 ft long. How tall is the tree?

Check It Out: Example 2

11 Understand the Problem

The answer is the height of the tree.

List the important information:

• The length of the building’s shadow is 8 ft.

• The height of the building is 24 ft.

• The length of the tree’s shadow is 3 ft.

Course 3

5-7 Indirect Measurement

Use the information to draw a diagram.22 Make a Plan

h

Draw dashed lines to form triangles. The building with its shadow and the tree with its shadow form similar right triangles.

Solve33

3 feet

8 feet

24 feet

Check It Out: Example 2 Continued

Course 3

5-7 Indirect Measurement

24 8

= h3

Solve33

Corresponding sides of similar figures are proportional.

72 = 8h Find the cross products.

The height of the tree is 9 feet.

9 = h

728 = 8h

8 Divide both sides by 8.

Check It Out: Example 2 Continued

Course 3

5-7 Indirect Measurement

Since = , the building’s shadow is

times its height. So, the tree’s

shadow should also be times its

height and of 9 is 3 feet.

44 Look Back

8 24

1 3

1 3

1 3

1 3

Check It Out: Example 2 Continued

Course 3

5-7 Indirect Measurement

1. Vilma wants to know how wide the river near her house is. She drew a diagram and labeled it with her measurements. How wide is the river?

2. A yardstick casts a 2-ft shadow. At the same

time, a tree casts a shadow that is 6 ft long. How

tall is the tree?

Lesson Quiz

7.98 m

9 ft

w

7 m

5 m

5.7 m

Course 3

5-7 Indirect Measurement

Course 3

5-7 Indirect Measurement

Tree = 3.3 m

Park to House = 15 m. You need to add the missing value (10) and the distance from red-light to park (5).

Course 3

5-7 Indirect Measurement

200 ft 21 ft

37.5 m 13.5 m

4.2ft

103.3 ft

Course 3

5-7 Indirect Measurement

• There are 2 ways to solve #15. Either turn everything into inches first, then go back to feet or plug in the answers in the proportion and see which fractions are closest to each other.

Course 3

5-7 Indirect Measurement

• Use Indirect Measurement

• MAPS In the figure, ΔGHK ΔJIK. Find ∼the distance across the forest.

48 mi

Course 3

5-7 Indirect Measurement

Homework: • Worksheet practice 11-6