Let's break down the question step by step:
-
Calculate the area of the original rectangle:
- The area \( A \) of a rectangle is given by the formula: \[ A = \text{length} \times \text{width} \]
- In terms of \( x \) and \( y \), the area of the original rectangle is: \[ A = xy \]
-
Dilate the rectangle:
- The problem states that DeeDee dilates the rectangle with a scale factor of \( \frac{1}{4} \) (note: the scale factor given is \( 1414 \) which appears to be written incorrectly, but we'll assume it should indicate a scale factor of \( \frac{1}{4} \)).
- After dilation, the new dimensions would be: \[ \text{New length} = \frac{x}{4}, \quad \text{New width} = \frac{y}{4} \]
-
Calculate the area of the dilated rectangle:
- The area of the dilated rectangle is: \[ A_{\text{dilated}} = \left(\frac{x}{4}\right) \times \left(\frac{y}{4}\right) = \frac{xy}{16} \]
-
Compare the areas:
- Now, we can compare the area of the original rectangle to the area of the dilated rectangle: \[ \text{Ratio} = \frac{A_{\text{original}}}{A_{\text{dilated}}} = \frac{xy}{\frac{xy}{16}} = 16 \]
- Therefore, the area of the original rectangle is 16 times greater than the area of the dilated rectangle.
Summary:
- The area of the original rectangle is \( xy \).
- The area of the dilated rectangle is \( \frac{xy}{16} \).
- The area of the original rectangle is 16 times greater than the area of the dilated rectangle.
Final Answers:
- Area of original rectangle: \( xy \)
- Area of dilated rectangle: \( \frac{xy}{16} \)
- Area comparison: The area of the original rectangle is 16 times the area of the dilated rectangle.