Point Q is the center of dilation. Rectangle H I J K is dilated to create rectangle H prime I prime J prime K prime. The length of H prime I prime is 32 inches. The length of I prime J prime is 24 inches.

Question

Sylvia enlarged a photo to make a 24 x 32 inch poster using the dilation DQ,4. What are the dimensions, in inches, of the original photo?

3 × 8
6 × 8
12 × 16
18 × 24

Answer 1

To find the dimensions of the original photo before dilation, we will use the scale factor of the dilation. In this case, the dilation is represented by \( D_{Q, 4} \), which means we are dilating by a scale factor of 4.

Given the dimensions of the enlarged poster (the dilated rectangle):

• Length of \( H' I' \) (which corresponds to the original width) = 32 inches
• Length of \( I' J' \) (which corresponds to the original height) = 24 inches

To find the original dimensions, we need to divide the dimensions of the enlarged poster by the scale factor:

1. For the length: \[ \text{Original length} = \frac{\text{Dilation length}}{\text{scale factor}} = \frac{32}{4} = 8 \text{ inches} \]

2. For the height: \[ \text{Original height} = \frac{\text{Dilation height}}{\text{scale factor}} = \frac{24}{4} = 6 \text{ inches} \]

So, the dimensions of the original photo are \( 8 \text{ inches} \times 6 \text{ inches} \).

From the options given:

• 3 × 8
• 6 × 8
• 12 × 16
• 18 × 24

The closest matching answer to the original dimensions is 6 × 8.