Question

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are \\( \triangle fgh \\) and \\( \triangle rst \\) congruent?

Explanation:

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

Using the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $FG$: $F(9,10)$ and $G(9,0)$, so $FG=\sqrt{(9 - 9)^2+(10 - 0)^2}=10$.
For $GH$: $G(9,0)$ and $H(0,10)$, so $GH=\sqrt{(0 - 9)^2+(10 - 0)^2}=\sqrt{81 + 100}=\sqrt{181}$.
For $FH$: $F(9,10)$ and $H(0,10)$, so $FH=\sqrt{(0 - 9)^2+(10 - 10)^2}=9$.

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

For $RS$: $R(-9,-10)$ and $S(-9,1)$, so $RS=\sqrt{(-9+9)^2+(1 + 10)^2}=11$.
For $ST$: $S(-9,1)$ and $T(0,-10)$, so $ST=\sqrt{(0 + 9)^2+(-10 - 1)^2}=\sqrt{81+121}=\sqrt{202}$.
For $RT$: $R(-9,-10)$ and $T(0,-10)$, so $RT=\sqrt{(0 + 9)^2+(-10 + 10)^2}=9$.

Step3: Compare the side lengths

Since $FG
eq RS$, $GH
eq ST$, and although $FH = RT = 9$, the other corresponding sides are not equal.

Answer:

No, $\triangle FGH$ and $\triangle RST$ are not congruent.