To find the area of the original rectangle in terms of \( x \) and \( y \), we use the formula for the area of a rectangle, which is:
\[ \text{Area} = \text{length} \times \text{width} \]
For DeeDee's original rectangle, the dimensions are \( x \) inches and \( y \) inches. Thus, the area \( A_{\text{original}} \) is:
\[ A_{\text{original}} = x \times y \]
Now, DeeDee dilates the rectangle using a scale factor of \( \frac{1}{4} \). When we dilate a shape, each dimension is multiplied by the scale factor. Therefore, the dimensions of the dilated rectangle are:
\[ \text{New length} = x \times \frac{1}{4} = \frac{x}{4} \] \[ \text{New width} = y \times \frac{1}{4} = \frac{y}{4} \]
Next, we can find the area of the dilated rectangle \( A_{\text{dilated}} \):
\[ A_{\text{dilated}} = \left(\frac{x}{4}\right) \times \left(\frac{y}{4}\right) \]
Calculating this gives:
\[ A_{\text{dilated}} = \frac{x}{4} \times \frac{y}{4} = \frac{xy}{16} \]
Now we will compare the area of the original rectangle to the area of the dilated rectangle. We can do this by calculating the ratio of the areas:
\[ \text{Ratio of areas} = \frac{A_{\text{original}}}{A_{\text{dilated}}} = \frac{xy}{\frac{xy}{16}} \]
This simplifies to:
\[ \text{Ratio of areas} = \frac{xy \cdot 16}{xy} = 16 \]
Thus, the area of the original rectangle is 16 times greater than the area of the dilated rectangle.
Summary:
- Area of the original rectangle: \( A_{\text{original}} = xy \)
- Area of the dilated rectangle: \( A_{\text{dilated}} = \frac{xy}{16} \)
- The area of the original rectangle is 16 times greater than the area of the dilated rectangle.