Question

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are \\( \triangle pqr \\) and \\( \triangle vwx \\) congruent?

Explanation:

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

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $PQ$: $P(-10,0)$ and $Q(-10,-9)$, so $PQ=\sqrt{(-10+10)^2+(-9 - 0)^2}=\sqrt{0 + 81}=9$.
For $QR$: $Q(-10,-9)$ and $R(1,-9)$, so $QR=\sqrt{(1 + 10)^2+(-9+9)^2}=\sqrt{121+0}=11$.
For $PR$: $P(-10,0)$ and $R(1,-9)$, so $PR=\sqrt{(1 + 10)^2+(-9 - 0)^2}=\sqrt{121 + 81}=\sqrt{202}$.

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

For $WX$: $W(1,10)$ and $X(1,0)$, so $WX=\sqrt{(1 - 1)^2+(0 - 10)^2}=\sqrt{0+100}=10$.
For $VX$: $V(9,10)$ and $X(1,0)$, so $VX=\sqrt{(1 - 9)^2+(0 - 10)^2}=\sqrt{64 + 100}=\sqrt{164}$.
For $VW$: $V(9,10)$ and $W(1,10)$, so $VW=\sqrt{(1 - 9)^2+(10 - 10)^2}=\sqrt{64+0}=8$.

Step3: Compare the side lengths

Since the corresponding side lengths of $\triangle PQR$ and $\triangle VWX$ are not equal ($PQ
eq WX$, $QR
eq VW$, $PR
eq VX$), the triangles are not congruent.

Answer:

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