Question

the figures below are similar. the labeled sides are corresponding.
4 cm 2 cm
a₁ = ? a₂= 9 cm²
what is the area of the larger rectangle?
a₁ = square centimeters

Explanation:

Step1: Find the scale factor

The corresponding sides of the similar rectangles are 4 cm (larger) and 2 cm (smaller). The scale factor \( k \) is the ratio of the larger side to the smaller side: \( k=\frac{4}{2} = 2 \).

Step2: Relate areas of similar figures

For similar figures, the ratio of their areas is the square of the scale factor. Let \( A_1 \) be the area of the larger rectangle and \( A_2 = 9 \, \text{cm}^2 \) be the area of the smaller one. The ratio of areas is \( \frac{A_1}{A_2}=k^2 \). Substituting \( k = 2 \) and \( A_2=9 \): \( \frac{A_1}{9}=2^2=4 \).

Step3: Solve for \( A_1 \)

Multiply both sides by 9: \( A_1=4\times9 = 36 \).

Answer:

36