Question

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are △ghi and △vwx congruent?

Explanation:

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

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $GH$: $G(4,6)$, $H(- 2,0)$.
$GH=\sqrt{(4 + 2)^2+(6 - 0)^2}=\sqrt{36 + 36}=\sqrt{72}=6\sqrt{2}$.
For $HI$: $H(-2,0)$, $I(-2,9)$.
$HI=\sqrt{(-2+2)^2+(9 - 0)^2}=9$.
For $GI$: $G(4,6)$, $I(-2,9)$.
$GI=\sqrt{(4 + 2)^2+(6 - 9)^2}=\sqrt{36+9}=\sqrt{45}=3\sqrt{5}$.

Step2: Calculate the side lengths of $\triangle VWX$

For $VX$: $V(2,-3)$, $X(1,-9)$.
$VX=\sqrt{(2 - 1)^2+(-3 + 9)^2}=\sqrt{1 + 36}=\sqrt{37}$.
For $XW$: $X(1,-9)$, $W(9,-9)$.
$XW=\sqrt{(9 - 1)^2+(-9 + 9)^2}=8$.
For $VW$: $V(2,-3)$, $W(9,-9)$.
$VW=\sqrt{(9 - 2)^2+(-9 + 3)^2}=\sqrt{49 + 36}=\sqrt{85}$.

Answer:

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