Question

the triangle uvw is a dilation of the triangle uvw. what is the scale factor of the dilation? simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

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 side $UV$

Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, for points $U(-5,0)$ and $V(0,-10)$, we have $UV=\sqrt{(0 + 5)^2+(-10 - 0)^2}=\sqrt{25 + 100}=\sqrt{125}=5\sqrt{5}$.

Step3: Calculate the length of side $U'V'$

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

Step4: Find the scale - factor

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

Answer:

$\frac{1}{5}$