Question

for items 4 and 5, use the information to answer each question. 4. triangle pqr has vertices p(1, -1), q(-2, 0), and r(2, 3). triangle tuv has vertices t(2, 2), u(-4, 0) and v(4, -6). what sequences of transformations can be used to show △pqr is similar to △tuv?

Explanation:

Step1: Calculate side - lengths of triangles

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ to find the side - lengths of $\triangle PQR$ and $\triangle TUV$.
For $\triangle PQR$:
$PQ=\sqrt{(-2 - 1)^2+(0 + 1)^2}=\sqrt{(-3)^2+1^2}=\sqrt{9 + 1}=\sqrt{10}$
$QR=\sqrt{(2 + 2)^2+(3 - 0)^2}=\sqrt{4^2+3^2}=\sqrt{16 + 9}=\sqrt{25}=5$
$PR=\sqrt{(2 - 1)^2+(3 + 1)^2}=\sqrt{1^2+4^2}=\sqrt{1 + 16}=\sqrt{17}$
For $\triangle TUV$:
$TU=\sqrt{(-4 - 2)^2+(0 - 2)^2}=\sqrt{(-6)^2+(-2)^2}=\sqrt{36+4}=\sqrt{40}=2\sqrt{10}$
$UV=\sqrt{(4 + 4)^2+(-6 - 0)^2}=\sqrt{8^2+(-6)^2}=\sqrt{64 + 36}=\sqrt{100}=10$
$TV=\sqrt{(4 - 2)^2+(-6 - 2)^2}=\sqrt{2^2+(-8)^2}=\sqrt{4 + 64}=\sqrt{68}=2\sqrt{17}$

Step2: Check for similarity ratio

We can see that $\frac{TU}{PQ}=\frac{2\sqrt{10}}{\sqrt{10}} = 2$, $\frac{UV}{QR}=\frac{10}{5}=2$, $\frac{TV}{PR}=\frac{2\sqrt{17}}{\sqrt{17}}=2$.
The ratio of corresponding side - lengths is 2.
We can first dilate $\triangle PQR$ by a scale factor of 2 centered at the origin (since the ratio of side - lengths of $\triangle TUV$ to $\triangle PQR$ is 2). Then, we may need to translate the dilated $\triangle PQR$ to match the position of $\triangle TUV$.

Answer:

First, dilate $\triangle PQR$ by a scale factor of 2 centered at the origin, and then translate the dilated triangle to match the position of $\triangle TUV$.