Question

DeeDee draws a rectangle with sides measuring x inches and y inches. She dilates the rectangle using a scale factor of 14 . What is the area of the original rectangle in terms of x and y? What is the area of the dilated rectangle? How many times greater is the area of the original rectangle compared to the area of the dilated rectangle? Explain your answers and show all your work.

Answer 1

the scale factor is 1/4 or one-fourth.

Answer 2

To find the areas of the original and dilated rectangles, let’s start with the area of the original rectangle.

1. **Area of the Original Rectangle:**
The area \( A \) of a rectangle is given by the formula:
\[
A = \text{length} \times \text{width}
\]
In this case, the sides of the rectangle measure \( x \) inches and \( y \) inches. Therefore, the area of the original rectangle is:
\[
A = x \times y
\]

2. **Dilating the Rectangle:**
When DeeDee dilates the rectangle using a scale factor of \( 14 \), each dimension of the rectangle (the sides) will be multiplied by \( 14 \). Hence, the new dimensions of the dilated rectangle are:
\[
\text{Length} = 14x \quad \text{and} \quad \text{Width} = 14y
\]

3. **Area of the Dilated Rectangle:**
To find the area of the dilated rectangle, we use the same formula for the area:
\[
A_{\text{dilated}} = \text{Length} \times \text{Width} = (14x) \times (14y)
\]
Simplifying this expression yields:
\[
A_{\text{dilated}} = 14x \times 14y = 196xy
\]

4. **Comparison of Areas:**
Now, we want to compare the area of the original rectangle with the area of the dilated rectangle. Recall that the area of the original rectangle is \( xy \) and the area of the dilated rectangle is \( 196xy \). We want to determine how many times greater the area of the dilated rectangle is compared to the area of the original rectangle:
\[
\text{Ratio} = \frac{\text{Area of the dilated rectangle}}{\text{Area of the original rectangle}} = \frac{196xy}{xy}
\]
Simplifying this ratio, we find:
\[
\text{Ratio} = 196
\]

Thus, the area of the dilated rectangle is 196 times greater than the area of the original rectangle.

In summary:
- The area of the original rectangle is \( xy \).
- The area of the dilated rectangle is \( 196xy \).
- The area of the dilated rectangle is 196 times greater than the area of the original rectangle.