Question

DeeDee draws a rectangle with sides measuring x inches and y inches. She dilates the rectangle using a scale factor of 1/4. 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

To find the area of the original rectangle and the area of the dilated rectangle, we will follow these steps:

### Step 1: Calculate the Area of the Original Rectangle
The area \( A \) of a rectangle is calculated using the formula:
\[
A = \text{length} \times \text{width}
\]
For the original rectangle with sides measuring \( x \) inches and \( y \) inches:
\[
A_{\text{original}} = x \times y = xy \quad \text{(area in square inches)}
\]

### Step 2: Determine the Dimensions of the Dilated Rectangle
When DeeDee dilates the rectangle using a scale factor of \( \frac{1}{4} \), the new dimensions (length and width) can be found by multiplying the original dimensions by the scale factor. Thus, the dimensions of the dilated rectangle are:
\[
\text{New Length} = \frac{1}{4}x
\]
\[
\text{New Width} = \frac{1}{4}y
\]

### Step 3: Calculate the Area of the Dilated Rectangle
Now, we apply the formula for the area again:
\[
A_{\text{dilated}} = \text{New Length} \times \text{New Width}
\]
Substituting the new dimensions:
\[
A_{\text{dilated}} = \left(\frac{1}{4}x\right) \times \left(\frac{1}{4}y\right)
\]
Calculating this gives:
\[
A_{\text{dilated}} = \frac{1}{4}x \cdot \frac{1}{4}y = \frac{1}{16}xy
\]

### Step 4: Compare the Areas
We need to find how many times greater the area of the original rectangle is compared to the area of the dilated rectangle. To do this, we can set up a ratio of the two areas:
\[
\text{Ratio} = \frac{A_{\text{original}}}{A_{\text{dilated}}} = \frac{xy}{\frac{1}{16}xy}
\]
Now simplifying this ratio:
\[
\text{Ratio} = \frac{xy \times 16}{xy} = 16
\]

### Conclusion
- The area of the original rectangle in terms of \( x \) and \( y \) is \( xy \).
- The area of the dilated rectangle is \( \frac{1}{16}xy \).
- The area of the original rectangle is **16 times greater** than the area of the dilated rectangle.

This concludes the explanation and calculations, demonstrating how dilation affects the area of geometric shapes.