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Using Similar Figures

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Lesson PresentationLesson Presentation

Problem of the DayProblem of the Day

Lesson QuizzesLesson Quizzes


Warm UpSolve each proportion.

1. k4

= 7525

2. 619

= 24x

3.

Triangles JNZ and KOA are similar. Identify the side that corresponds to the given side of the similar triangles.

J

N Z K O

A

JN KO

k = 12 x = 76


Problem of the Day

Harvey starts to fill a 3-gallon tank from a hose that delivers 1 gallon per minute. However, the tanks leaks 1 quart per minute. In how many minutes will the tank begin to overflow?4


Learn to use similar figures to find unknown lengths.


Vocabulary

indirect measurement


Indirect measurement is a method of using proportions to find an unknown length or distance in similar figures.


Find the unknown measures in the similar figures.

Additional Example 1: Finding Unknown Lengths in Similar Figures

ABJG

= BCHG Write a proportion using corresponding sides.

105

= 6x

Substitute lengths of the sides.

10 · x = 5 · 6 Find the cross product.10x = 30 Multiply.

10x10

= 3010

x = 3

HG is 3 centimeters.

Divide each side by 12 to isolate the variable.

HB A

CGJ

10 cm

6 cm 116 cm

5.8 cmx

5 cm

31°

59°

y



Additional Example 1 Continued

Step 2 Find y.

Corresponding angles of similar triangles have equal angle measures.

H corresponds to C

y = 59

HB A

CGJ

10 cm

6 cm 116 cm

5.8 cmx

5 cm

31°

59°

y



Check It Out: Example 1

ABFE

= BCDE Write a proportion using corresponding sides.

147

= 9x

Substitute lengths of the sides.

14 · x = 9 · 7 Find the cross product.14x = 63 Multiply.

14x14

= 6314

x = 4.5

HG is 4.5 centimeters.


DB A

CEF

14 cm

9 cm 116 cm

5.8 cmx

7 cm

27°

63°

y



Check It Out: Example 1 Continued

Step 2 Find y.

Corresponding angles of similar triangles have equal angle measures.

D corresponds to C

y = 63

DB A

CEF

10 cm

6 cm 116 cm

5.8 cmx

5 cm

27°

63°

y


The inside triangle is similar in shape to the outside triangle. Find the length of the base of the inside triangle.

Let x = the base of the inside triangle.

82

=12x

8 · x = 2 · 128x = 24

8x8

= 248

x = 3The base of the inside triangle is 3 inches.

Write a proportion using corresponding sidelengths.

Find the cross products.Multiply.


Additional Example 2: Measurement Application



The rectangle on the left is similar in shape to the rectangle on the right. Find the width of the right rectangle.

3 cm

6 cm12 cm

Let w = the width of the right rectangle.

612

= 3w

6 ·w = 12 · 3

6w = 36

6w6

= 366

w = 6

The right rectangle is 6 cm wide.

Write a proportion using correspondingside lengths.

Find the cross products.Multiply.


?


Additional Example 3: Estimating with Indirect Measurement

City officials want to know the height of a traffic light. Estimate the height of the traffic light.

27.2515

= 48.75h

Write a proportion.

Use compatible numbers to estimate.

53

≈ 50h

Simplify.

5h ≈ 150

The traffic light is about 30 feet tall.

27.25 ft

48.75 ft

h ft 2515

≈ 50h

Cross multiply.

h ≈ 30 Divide each side by 5 to isolate the variable.



The inside triangle is similar in shape to the outside triangle. Find the height of the outside triangle.

514.75

= h30.25

Write a proportion.

Use compatible numbers to estimate.

13

≈ h30

Simplify.

1 • 30 ≈ 3 • h

The outside triangle is about 10 feet tall.

14.75 ft

30.25 ft

h ft

515

≈ h30

30 ≈ 3h Divide each side by 3 to isolate the variable.

5 ft

Cross multiply.

10 ≈ h


Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems


Lesson Quiz: Part I

Find the unknown length in each pair of similar figures.

1.

2.

x = 120 cmy = 98◦

t = 150 cms = 120◦

33°49°

98° y

s

19°

120° 41°


Lesson Quiz: Part II

Find the unknown length in each pair of similar figures.

3. The width of the smaller rectangular cake is 5.75 in. The width of a larger rectangular cake is 9.25 in. Estimate the length of the larger rectangular cake.

about 15 inches


1. Identify the unknown length in the given pair of similar figures.

A. x = 174 cm

B. x = 182 cm

C. x = 196 cm

D. x = 204 cm



2. Identify the unknown length in the given pair of similar figures.

A. t = 166 cm

B. t = 198 cm

C. t = 206 cm

D. t = 248 cm



3. The width of the smaller rectangular box is 6.25 inches. The width of a similar larger rectangular box is 8.75 inches. Estimate the length of the larger rectangular box.

A. about 16 in.

B. about 17 in.

C. about 18 in.

D. about 19 in.


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