Question

△abc is a dilation of △abc with a center of dilation at point d.
what is the scale factor of this dilation?

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 (\(\triangle A'B'C'\)) to the length of the corresponding side in the original figure (\(\triangle ABC\)).

Step2: Choose corresponding sides

Let's take side \( B'C' = 1.02 \) and corresponding side \( BC = 3 \), or \( A'C' = 2.04 \) and \( AC = 6 \), or \( A'B' = 2.38 \) and \( AB = 7 \). Let's use \( A'C' \) and \( AC \).

Step3: Calculate the scale factor

Using the formula \( k=\frac{\text{Length of image side}}{\text{Length of original side}} \), substitute \( A'C' = 2.04 \) and \( AC = 6 \). So \( k = \frac{2.04}{6}=0.34 \). We can check with another pair, e.g., \( B'C' = 1.02 \) and \( BC = 3 \), \( \frac{1.02}{3}=0.34 \), or \( A'B' = 2.38 \) and \( AB = 7 \), \( \frac{2.38}{7}=0.34 \). All give the same result.

Answer:

\( 0.34 \)