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?

Question

Use a decimal response if necessary.

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 can follow these steps:

1. Calculate the area of the original square:

   • The side length of the original square is \(5 , \text{cm}\).
   • The area of a square is given by the formula \( \text{Area} = \text{side}^2 \). \[ \text{Area}_{\text{original}} = 5^2 = 25 , \text{cm}^2 \]

2. Dilate the square:

   • The scale factor is \(4\). When dilating, the new side length will be: \[ \text{New side} = 4 \times 5 = 20 , \text{cm} \]

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

4. Find the ratio of the areas: \[ \text{Ratio} = \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{400}{25} = 16 \]

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