Question

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are \\( \triangle uvw \\) and \\( \triangle qrs \\) congruent?

Explanation:

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

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $UV$: $U(- 3,10)$ and $V(8,10)$. $UV=\sqrt{(8 + 3)^2+(10 - 10)^2}=\sqrt{121}=11$.
For $VW$: $V(8,10)$ and $W(8,1)$. $VW=\sqrt{(8 - 8)^2+(1 - 10)^2}=\sqrt{81}=9$.
For $UW$: $U(-3,10)$ and $W(8,1)$. $UW=\sqrt{(8 + 3)^2+(1 - 10)^2}=\sqrt{121 + 81}=\sqrt{202}$.

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

For $QR$: $Q(-2,4)$ and $R(-2,-8)$. $QR=\sqrt{(-2+2)^2+(-8 - 4)^2}=\sqrt{144}=12$.
For $RS$: $R(-2,-8)$ and $S(-9,-8)$. $RS=\sqrt{(-9 + 2)^2+(-8+8)^2}=\sqrt{49}=7$.
For $QS$: $Q(-2,4)$ and $S(-9,-8)$. $QS=\sqrt{(-9 + 2)^2+(-8 - 4)^2}=\sqrt{49+144}=\sqrt{193}$.

Since the side - lengths of $\triangle UVW$ ($11,9,\sqrt{202}$) and $\triangle QRS$ ($12,7,\sqrt{193}$) are not equal.

Answer:

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