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? Use a decimal response if necessary.

Question

The ratio of the areas is ____:1

Answer 1

To find the ratio of the area of the dilated square to the area of the original square, we first need to calculate the areas of both squares.

1. Calculate the area of the original square:
   The formula for the area of a square is given by \( A = s^2 \), where \( s \) is the length of a side of the square. \[ A_{\text{original}} = 5 , \text{cm} \times 5 , \text{cm} = 25 , \text{cm}^2 \]

2. Determine the side length of the dilated square:
   The scale factor for dilation is 4, so the new side length will be: \[ s_{\text{dilated}} = 4 \times 5 , \text{cm} = 20 , \text{cm} \]

3. Calculate the area of the dilated square: \[ A_{\text{dilated}} = 20 , \text{cm} \times 20 , \text{cm} = 400 , \text{cm}^2 \]

4. Find the ratio of the area of the dilated square to the area of the original square: \[ \text{Ratio} = \frac{A_{\text{dilated}}}{A_{\text{original}}} = \frac{400 , \text{cm}^2}{25 , \text{cm}^2} = 16 \]

This can be expressed as a ratio of 16:1. Therefore, the final answer is:

The ratio of the areas is 16:1.