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.

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 each of the areas.

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

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

3. Area of the dilated square: Using the same area formula: \[ A_{\text{dilated}} = (s_{\text{dilated}})^2 = 20^2 = 400 , \text{cm}^2 \]

4. Finding the ratio of the areas: The ratio of the area of the dilated square to the area of the original square is: \[ \text{Ratio} = \frac{A_{\text{dilated}}}{A_{\text{original}}} = \frac{400}{25} \] Simplifying the ratio: \[ \text{Ratio} = 16 \]

Therefore, the ratio of the area of the dilated square to the area of the original square is \(\boxed{16}\).