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 the 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: Identify corresponding sides

Let's take side \( A'C' \) and \( AC \). From the diagram, \( AC = 4 \) and \( A'C' = 6.4 \). Or we can take \( A'B' \) and \( AB \), where \( AB = 3 \) and \( A'B' = 4.8 \), or \( B'C' \) and \( BC \), where \( BC = 5 \) and \( B'C' = 8 \).

Step3: Calculate the scale factor

Using \( AC \) and \( A'C' \): \( k=\frac{A'C'}{AC}=\frac{6.4}{4} = 1.6 \). Let's check with another pair, say \( AB \) and \( A'B' \): \( k=\frac{A'B'}{AB}=\frac{4.8}{3}=1.6 \). Or with \( BC \) and \( B'C' \): \( k = \frac{B'C'}{BC}=\frac{8}{5}=1.6 \). All give the same result.

Answer:

The scale factor of the dilation is \( 1.6 \) (or \(\frac{8}{5}\) or \(\frac{16}{10}\) simplified, but \( 1.6 \) is a common decimal representation).