Question

trapezoid wxyz is the image of trapezoid wxyz under a dilation through point c.
what scale factor was used in the dilation?

○ $-\frac{6}{5}$

○ $-\frac{5}{6}$

○ $\frac{5}{6}$

○ $\frac{6}{5}$

Explanation:

Step1: Recall dilation scale factor formula

The scale factor \( k \) of a dilation is the ratio of the length of a side in the image (\( W'X' \)) to the length of the corresponding side in the original figure (\( WX \)). Also, since the dilation is through point \( C \) and the image is on the opposite side of \( C \) from the original (as seen from the diagram with dashed lines), the scale factor should be negative (indicating a reflection - like direction change in dilation).

Step2: Identify corresponding side lengths

From the diagram, the length of \( WX \) (original side) is \( 18 \) mm and the length of \( W'X' \) (image side) is \( 15 \) mm.

Step3: Calculate the scale factor

Using the formula for scale factor \( k=\frac{\text{Length of image side}}{\text{Length of original side}} \), we substitute the values: \( k = \frac{15}{18}\). Simplifying \( \frac{15}{18}=\frac{5}{6} \). But since the dilation is a reduction and the image is on the opposite side of the center of dilation \( C \) from the original trapezoid, the scale factor is negative. So \( k=-\frac{5}{6} \).

Answer:

\( \boldsymbol{-\frac{5}{6}} \) (corresponding to the option " \( -\frac{5}{6} \)")