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 $\boxed{{\displaystyle 16}}$.