Question

if a dilation with center o and scale factor r takes a quadrilateral abcd to abcd, what dilation will take abcd back to abcd?

choose the correct answer below.

a. a dilation with center o - r and scale factor r
b. a dilation with center o and scale factor r
c. a dilation with center o and scale factor \\(\frac{1}{r}\\)
d. a dilation with center o - r and scale factor \\(\frac{1}{r}\\)

Explanation:

Step1: Recall Dilation Properties

Dilation with center \( O \) and scale factor \( r \) maps a point \( P \) to \( P' \) such that \( \overrightarrow{OP'} = r\overrightarrow{OP} \). To reverse this, we need a scale factor \( s \) where \( s \cdot r = 1 \), so \( s=\frac{1}{r} \). The center remains \( O \) (dilation center doesn't change for inverse dilation here).

Step2: Evaluate Options

• Option A: Center \( O - r \) is invalid (center is a point, not subtracted by \( r \)), and scale factor \( r \) won't reverse. Eliminate.
• Option B: Scale factor \( r \) would repeat the original dilation, not reverse. Eliminate.
• Option C: Center \( O \) (correct) and scale factor \( \frac{1}{r} \) (since \( r \cdot \frac{1}{r}=1 \), reversing the dilation). Valid.
• Option D: Center \( O - r \) is invalid. Eliminate.

Answer:

C. A dilation with center \( O \) and scale factor \( \frac{1}{r} \)