1. Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes

2. Warm Up Solve each proportion. 1. 2.k 4 75 25 6 19 24 x 1. =
k = 12 2.
x = 76 =
Triangles JNZ and KOA are similar. Identify the side that corresponds to the given side of the similar triangles. 3. J A N Z K O JN KO

3. 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

4. Learn to use similar figures to find unknown lengths.

5. Vocabulary
indirect measurement

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

7. Additional Example 1: Finding Unknown Lengths in Similar Figures
Find the unknown measures in the similar figures. H B
10 cm
31° A y
5.8 cm x
6 cm
116 cm
59° J G C
5 cm AB JG = BC HG
Write a proportion using corresponding sides. 10 5 6 x =
Substitute lengths of the sides.
10 · x = 5 · 6
Find the cross product.
10x = 30
Multiply.
10x 10 30 10 =
Divide each side by 12 to isolate the variable.
x = 3
HG is 3 centimeters.

8. Additional Example 1 Continued
Find the unknown measures in the similar figures. H B
10 cm
31° A y
5.8 cm x
6 cm
116 cm
59° J G C
5 cm
Step 2 Find y.
Corresponding angles of similar
triangles have equal angle
measures.
H corresponds to C
y = 59

9. Find the unknown measures in the similar figures.
Check It Out: Example 1
Find the unknown measures in the similar figures. D B
14 cm
27° A y
5.8 cm x
9 cm
116 cm
63° F E C
7 cm AB FE = BC DE
Write a proportion using corresponding sides. 14 7 9 x =
Substitute lengths of the sides.
14 · x = 9 · 7
Find the cross product.
14x = 63
Multiply.
14x 14 63 14 =
Divide each side by 12 to isolate the variable.
x = 4.5
HG is 4.5 centimeters.

10. Check It Out: Example 1 Continued
Find the unknown measures in the similar figures. D B
10 cm
27° A y
5.8 cm x
6 cm
116 cm
63° F E C
5 cm
Step 2 Find y.
Corresponding angles of similar
triangles have equal angle
measures.
D corresponds to C
y = 63

11. Additional Example 2: Measurement Application
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. 8 2 12 x
Write a proportion using corresponding side
lengths. =
8 · x = 2 · 12
Find the cross products.
8x = 24
Multiply. 8x 8 24 8 =
Divide each side by 8 to isolate the variable.
x = 3
The base of the inside triangle is 3 inches.

12. Check It Out: Example 2
The rectangle on the left is similar in shape to the rectangle on the right. Find the width of the right rectangle.
12 cm
6 cm
3 cm ?
Let w = the width of the right rectangle. 6 12 3 w
Write a proportion using corresponding
side lengths. =
6 ·w = 12 · 3
Find the cross products.
6w = 36
Multiply. 6w 6 = 36 6
Divide each side by 6 to isolate the
variable.
w = 6
The right rectangle is 6 cm wide.

13. 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.
27.25 15
48.75 h =
Write a proportion. 25 15 50 h
Use compatible numbers to estimate.
h ft ≈ 5 3 50 h ≈
Simplify.
27.25 ft
5h ≈ 150
Cross multiply.
48.75 ft
h ≈ 30
Divide each side by 5 to isolate the variable.
The traffic light is about 30 feet tall.

14. Check It Out: Example 3
The inside triangle is similar in shape to the outside triangle. Find the height of the outside triangle. 5
14.75 h
30.25 =
Write a proportion. 5 15 h 30
Use compatible numbers to estimate. ≈
h ft
5 ft 13 h 30 ≈
Simplify.
1 • 30 ≈ 3 • h
Cross multiply.
14.75 ft
30 ≈ 3h
Divide each side by 3 to isolate the variable.
30.25 ft
10 ≈ h
The outside triangle is about 10 feet tall.

15. Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems

16. Find the unknown length in each pair of similar figures.
Lesson Quiz: Part I
Find the unknown length in each pair of similar figures.
x = 120 cm
y = 98◦ 1.
33°
49°
98° y 2.
19°
t = 150 cm
s = 120◦ s
120°
41°

17. 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

18. Lesson Quiz for Student Response Systems
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

19. Lesson Quiz for Student Response Systems
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

20. Lesson Quiz for Student Response Systems
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.