Question

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

Explanation:

Step1: Calculate side lengths of $\triangle EFG$

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
Coordinates: $E(- 10,-9)$, $F(-1,-9)$, $G(-1,1)$.
Length of $EF$: $\sqrt{(-1+10)^2+(-9 + 9)^2}=\sqrt{81}=9$.
Length of $FG$: $\sqrt{(-1+1)^2+(1 + 9)^2}=\sqrt{100}=10$.
Length of $EG$: $\sqrt{(-1 + 10)^2+(1+9)^2}=\sqrt{81 + 100}=\sqrt{181}$.

Step2: Calculate side lengths of $\triangle PQR$

Coordinates: $P(9,1)$, $Q(9,10)$, $R(0,10)$.
Length of $PQ$: $\sqrt{(9 - 9)^2+(10 - 1)^2}=\sqrt{81}=9$.
Length of $QR$: $\sqrt{(0 - 9)^2+(10 - 10)^2}=\sqrt{81}=9$.
Length of $PR$: $\sqrt{(0 - 9)^2+(10 - 1)^2}=\sqrt{81+81}=\sqrt{162}$.
Since the side - lengths of $\triangle EFG$ ($9,10,\sqrt{181}$) and $\triangle PQR$ ($9,9,\sqrt{162}$) are not equal.

Answer:

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