Question

the triangle uvw is a dilation of the triangle uvw. what is the scale factor of the dilation?

Explanation:

Step1: Select a corresponding side

Let's consider side $UV$ and $U'V'$. The coordinates of $U(- 5,0)$, $V(0,-10)$, $U'(-1,0)$ and $V'(0, - 2)$.

Step2: Calculate the length of the original - side

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $U(-5,0)$ and $V(0,-10)$, we have $d_{UV}=\sqrt{(0 + 5)^2+(-10 - 0)^2}=\sqrt{25 + 100}=\sqrt{125}=5\sqrt{5}$.

Step3: Calculate the length of the dilated - side

For points $U'(-1,0)$ and $V'(0,-2)$, we have $d_{U'V'}=\sqrt{(0 + 1)^2+(-2 - 0)^2}=\sqrt{1 + 4}=\sqrt{5}$.

Step4: Find the scale factor

The scale factor $k$ of a dilation is given by the ratio of the length of a side of the dilated figure to the length of the corresponding side of the original figure. So $k=\frac{d_{U'V'}}{d_{UV}}=\frac{\sqrt{5}}{5\sqrt{5}}=\frac{1}{5}$.

Answer:

$\frac{1}{5}$