1. 4.2
Using Similar Shapes
How can you use similar shapes to find unknown measures?

2. Texas Essential Knowledge and Skills
The student is expected to:
Proportionality—7.5.A
Generalize the critical attributes of similarity, including ratios within and between similar shapes.
Mathematical Processes
7.1.E
Create and use representations to organize, record, and communicate mathematical ideas.

3. 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

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

5. Additional Example 1: Finding Unknown Lengths in Similar Figures
Find the unknown length in similar figures. AC QS = AB QR
Write a proportion using corresponding sides. 12 48 14 w =
Substitute lengths of the sides.
12 · w = 48 · 14
Find the cross product.
12w = 672
Multiply.
12w 12
672 12 =
Divide each side by 12 to isolate the variable.
w = 56
QR is 56 centimeters.

6. Find the unknown length in similar figures.
Check It Out: Example 1
Find the unknown length in similar figures. x
10 cm Q R A B
12 cm
24 cm D C S T AC QS AB QR =
Write a proportion using corresponding sides. 12 24 10 x =
Substitute lengths of the sides.
12 · x = 24 · 10
Find the cross product.
12x = 240
Multiply.
12x 12
240 12 =
Divide each side by 12 to isolate the variable.
x = 20
QR is 20 centimeters.

7. 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.

8. 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.

9. 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.

10. 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.

11. ADDITIONAL EXAMPLE 1 ∆QRS ~ ∆BCD. Find the unknown measurements. 1.
Find the unknown side, x.
7.5 m 2.
Find m y.
30°

12. ADDITIONAL EXAMPLE 2
The rectangles shown are similar. Find the length of the larger rectangle.
120 m

13. ADDITIONAL EXAMPLE 3
A billboard that is 60 feet tall is near a flagpole. The flagpole has a shadow that is 14 feet long at the same time that the shadow of the billboard is 24 feet long. Find the height of the flagpole.
35 ft

14. 4.2 LESSON QUIZ
7.5.A
The shapes in each pair are similar. Find the unknown measures. 1.
t = 32 cm, d = 34°

15. The shapes in each pair are similar. Find the unknown measures.2.
x = 16.5 ft, S = 29° 3.
x = 110 cm

16. 4.
A cell phone tower and a fence are on the same property. The fence is 8 feet tall and has a shadow that measures 14.5 feet. The shadow of the cell phone tower measures 72.5 feet. How tall is the cell phone tower?
40 feet

17. Will is standing in the shadow of a 40 foot tall telephone pole
Will is standing in the shadow of a 40 foot tall telephone pole. He is standing so that his shadow and the telephone pole’s shadow end at the same place. The shadow of the telephone pole measures 64 feet. If Will is 5 feet 6 inches tall, how far away from the telephone pole is he? Explain.

18. 55.2 feet; Will is 5 feet 6 inches tall, or 5.5 feet tall.
Write a proportion using corresponding sides of the
similar triangles Solve for the
length of Will’s shadow, which is 8.8 feet. Since the
telephone pole’s shadow is 64 feet long, subtract to
find how far from the telephone pole Will is. 64 – 8.8 = 55.2

19. How can you use similar shapes to find unknown measures?
Sample answer:
To find an unknown length, write a proportion using corresponding sides. To find an unknown angle, compare it to the corresponding angle on the similar shape.