Question

1. triangle def is the image that resulted from a dilation of △abc using the origin as the center of dilation. what scale factor was used to dilate △abc?

Explanation:

Step1: Select corresponding side lengths

Let's take the base - for example, the horizontal side of the triangles. The base of $\triangle ABC$ (the length between points $A$ and $C$) has endpoints $A(8,9)$ and $C(14,9)$. So the length of the base of $\triangle ABC$ is $14 - 8=6$. The base of $\triangle DEF$ (the length between points $D$ and $F$) has endpoints $D(6,4)$ and $F(8,4)$. So the length of the base of $\triangle DEF$ is $8 - 6 = 2$.

Step2: Calculate the scale factor

The scale factor $k$ of a dilation is given by the ratio of the length of a side of the image to the length of the corresponding side of the pre - image. So $k=\frac{\text{length of side in }\triangle DEF}{\text{length of corresponding side in }\triangle ABC}=\frac{2}{6}=\frac{1}{3}$.

Answer:

$\frac{1}{3}$