Question

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are △efg and △ijk congruent?

Explanation:

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

Using the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ (or counting units on the grid for right - angled sides).
For $\triangle EFG$:

• $FG$: Counting units, $FG=\sqrt{(- 10+8)^2+(0 + 8)^2}=\sqrt{(-2)^2+8^2}=\sqrt{4 + 64}=\sqrt{68}$.
• $EG$: Counting units, $EG = 8$ (horizontal distance from $E(-1, - 8)$ to $G(-9,-8)$).
• $EF$: Counting units, $EF=\sqrt{(-10 + 1)^2+(0 + 8)^2}=\sqrt{(-9)^2+8^2}=\sqrt{81+64}=\sqrt{145}$.

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

For $\triangle IJK$:

• $IJ$: Counting units, $IJ=\sqrt{(8 - 0)^2+(10 - 1)^2}=\sqrt{8^2+9^2}=\sqrt{64 + 81}=\sqrt{145}$.
• $JK$: Counting units, $JK = 8$ (horizontal distance from $J(0,1)$ to $K(8,1)$).
• $IK$: Counting units, $IK=\sqrt{(8 - 0)^2+(10 - 1)^2}=\sqrt{68}$.

Step3: Compare the side lengths

By the SSS (Side - Side - Side) congruence criterion, since $FG = IK=\sqrt{68}$, $EG=JK = 8$, and $EF=IJ=\sqrt{145}$.

Answer:

Yes, $\triangle EFG$ and $\triangle IJK$ are congruent.