Question

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are △uvw and △ghi congruent?

Explanation:

Step1: Find the side lengths of $\triangle GHI$

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $GH$: $G(-4,9)$, $H(2,2)$

$$\begin{align*} GH&=\sqrt{(2 + 4)^2+(2 - 9)^2}\\ &=\sqrt{36 + 49}\\ &=\sqrt{85} \end{align*}$$

For $HI$: $H(2,2)$, $I(-10,2)$

$$\begin{align*} HI&=\sqrt{(-10 - 2)^2+(2 - 2)^2}\\ &=\sqrt{144+0}\\ & = 12 \end{align*}$$

For $GI$: $G(-4,9)$, $I(-10,2)$

$$\begin{align*} GI&=\sqrt{(-10 + 4)^2+(2 - 9)^2}\\ &=\sqrt{36+49}\\ &=\sqrt{85} \end{align*}$$

Step2: Find the side lengths of $\triangle UVW$

For $UV$: $U(1,-5)$, $V(7,2)$

$$\begin{align*} UV&=\sqrt{(7 - 1)^2+(2 + 5)^2}\\ &=\sqrt{36 + 49}\\ &=\sqrt{85} \end{align*}$$

For $VW$: $V(7,2)$, $W(7,-10)$

$$\begin{align*} VW&=\sqrt{(7 - 7)^2+(-10 - 2)^2}\\ &=\sqrt{0 + 144}\\ &=12 \end{align*}$$

For $UW$: $U(1,-5)$, $W(7,-10)$

$$\begin{align*} UW&=\sqrt{(7 - 1)^2+(-10 + 5)^2}\\ &=\sqrt{36+25}\\ &=\sqrt{61} \end{align*}$$

Answer:

No, $\triangle UVW$ and $\triangle GHI$ are not congruent.