Question

Alcivious draws a square with a side that measures 5 cm. He dilates the square using a scale factor of 4. What is the ratio of the area of the dilated square to the area of the original square?

Answer 1

To solve the problem, we start by finding the area of the original square.

The side length of the original square is given as 5 cm. The area \( A \) of a square is calculated using the formula:

\[
A = \text{side length}^2
\]

Thus, the area of the original square is:

\[
A_{\text{original}} = 5^2 = 25 \text{ cm}^2
\]

Next, we apply the dilation. The square is dilated using a scale factor of 4. This means that each side length of the square will be multiplied by 4. Therefore, the new side length \( s_{\text{dilated}} \) of the dilated square is:

\[
s_{\text{dilated}} = 5 \times 4 = 20 \text{ cm}
\]

We then calculate the area of the dilated square using the same area formula:

\[
A_{\text{dilated}} = (s_{\text{dilated}})^2 = 20^2 = 400 \text{ cm}^2
\]

Now, we want to find the ratio of the area of the dilated square to the area of the original square. This ratio \( R \) is given by:

\[
R = \frac{A_{\text{dilated}}}{A_{\text{original}}} = \frac{400}{25}
\]

Calculating this ratio gives:

\[
R = \frac{400}{25} = 16
\]

Thus, the ratio of the area of the dilated square to the area of the original square is:

\[
\boxed{16}
\]