Question

1. identify the scale factor for the dilation given the graph of each preimage △efg and image △efg.

Explanation:

Step1: Select a corresponding side

Let's consider side $EG$ and $E'G'$. The coordinates of $E(6,0)$, $G(0,6)$, $E'(2,0)$ and $G'(0,2)$.

Step2: Calculate the lengths of the sides

The length of a line - segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $EG$, $x_1 = 6,y_1 = 0,x_2 = 0,y_2 = 6$. Then $EG=\sqrt{(0 - 6)^2+(6 - 0)^2}=\sqrt{(-6)^2+6^2}=\sqrt{36 + 36}=\sqrt{72}=6\sqrt{2}$.
For $E'G'$, $x_1 = 2,y_1 = 0,x_2 = 0,y_2 = 2$. Then $E'G'=\sqrt{(0 - 2)^2+(2 - 0)^2}=\sqrt{(-2)^2+2^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}$.

Step3: Find the scale factor

The scale factor $k$ of a dilation is the ratio of the length of a side of the image to the length of the corresponding side of the pre - image. So $k=\frac{E'G'}{EG}=\frac{2\sqrt{2}}{6\sqrt{2}}=\frac{1}{3}$.

Answer:

$\frac{1}{3}$