Question

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are \\( \triangle hij \\) and \\( \triangle bcd \\) congruent?

Explanation:

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

Using the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $HJ$: $H(- 10,1)$,$J(0,1)$, $HJ=\sqrt{(0 + 10)^2+(1 - 1)^2}=10$.
For $IJ$: $I(-1,9)$,$J(0,1)$, $IJ=\sqrt{(0 + 1)^2+(1 - 9)^2}=\sqrt{1 + 64}=\sqrt{65}$.
For $HI$: $H(-10,1)$,$I(-1,9)$, $HI=\sqrt{(-1 + 10)^2+(9 - 1)^2}=\sqrt{81 + 64}=\sqrt{145}$.

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

For $BC$: $B(9,-10)$,$C(3,-1)$, $BC=\sqrt{(3 - 9)^2+(-1+10)^2}=\sqrt{36 + 81}=\sqrt{117}$.
For $CD$: $C(3,-1)$,$D(10,0)$, $CD=\sqrt{(10 - 3)^2+(0 + 1)^2}=\sqrt{49+1}=\sqrt{50}$.
For $BD$: $B(9,-10)$,$D(10,0)$, $BD=\sqrt{(10 - 9)^2+(0 + 10)^2}=\sqrt{1 + 100}=\sqrt{101}$.
Since the side - lengths of $\triangle HIJ$ and $\triangle BCD$ are not equal.

Answer:

No, $\triangle HIJ$ and $\triangle BCD$ are not congruent.