Question

1. a circle with an area of $8\pi$ square centimeters is dilated so that its image has an area of $32\pi$ square centimeters. what is the scale factor of the dilation? \underline{\quad\quad\quad\quad\quad\quad\quad}

Explanation:

Step1: Recall the relationship between areas and scale factors

For similar figures, if the scale factor of the linear dimensions (like radius for a circle) is \( k \), the ratio of the areas is \( k^{2} \). Let the area of the original circle be \( A_1 = 8\pi \) and the area of the dilated circle be \( A_2=32\pi \). The ratio of the areas is \( \frac{A_2}{A_1} \).

Step2: Calculate the ratio of the areas

Calculate \( \frac{A_2}{A_1}=\frac{32\pi}{8\pi} = 4 \).

Step3: Find the scale factor

Since the ratio of the areas is \( k^{2} \) and \( k^{2}=4 \), we solve for \( k \) by taking the square root of both sides. So \( k = \sqrt{4}=2 \) (we take the positive value since scale factor is a positive quantity representing a dilation).

Answer:

2