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Area and perimeter of similar figures

# Area and perimeter of similar figures

The figures below are similar. The labeled sides are corresponding.

What is the perimeter of the smaller pentagon?

Remember that $\frac{a}{b}=\frac{P_1}{P_2}$ where $\frac{a}{b}$ is the ratio of the corresponding side lengths, and $\frac{P_1}{P_2}$ is the ratio of the perimeters.

Find the ratio of the corresponding side lengths.

$\frac{a}{b}=\frac{3}{9}=\frac{1}{3}$

Find the ratio of the perimeters.

$\frac{P_1}{P_2}=\frac{P_1}{45}$

Use these two ratios to set up a proportion and solve for P1.

$\frac{1}{3}$ = $\frac{P_1}{45}$

3
 P1
=
 1 × 45
Find the cross products

3
 P1
= 45 Simplify

3
 P1
÷ 3
=
 45 ÷ 3
Divide both sides by 3

 P1
= 15

The perimeter of the smaller pentagon is 15 centimeters.

The figures below are similar. The labeled sides are corresponding.

What is the area of the larger square?

Remember that $(\frac{a}{b})^2=\frac{A_1}{A_2}$ where $\frac{a}{b}$ is the ratio of the corresponding side lengths, and $\frac{A_1}{A_2}$ is the ratio of the areas.

Find the square of the ratio of the corresponding side lengths.

$(\frac{a}{b})^2=(\frac{7}{4})^2=\frac{49}{16}$

Find the ratio of the areas.

$\frac{A_1}{A_2}=\frac{A_1}{16}$

Use these two ratios to set up a proportion and solve for A1.

 $\frac{49}{16}$
=
 $\frac{A_1}{16}$

16
 A1
=
 49 × 16
Find the cross products

16
 A1
= 784 Simplify

16
 A1
÷ 16
=
 784 ÷ 16
Divide both sides by 16

 A1
= 49

The area of the larger square is 49 square feet.

The figures below are similar. The labeled sides are corresponding.
What is the perimeter of the smaller rectangle?

 27 ft 9 ft 84 ft

P2 = ft