Question

line segment $overline{gh}$ has endpoints at $(-7, -8)$ and $(-9, 13)$. if $overline{qr}$ is congruent to $overline{gh}$ and has one endpoint at $(8, 5)$, which of the following points could be the second endpoint? (10, 24) (-8, -16) (10, -16)

Explanation:

Step1: Calculate length of $\overline{GH}$ using distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Let $(x_1,y_1)=(-7,-8)$ and $(x_2,y_2)=(-9,13)$. Then $d_{GH}=\sqrt{(-9+7)^2+(13 + 8)^2}=\sqrt{(-2)^2+(21)^2}=\sqrt{4 + 441}=\sqrt{445}$.

Step2: Check distance from $(8,5)$ to each option using distance formula.

For option $(10,24)$: $d=\sqrt{(10 - 8)^2+(24 - 5)^2}=\sqrt{2^2+19^2}=\sqrt{4+361}=\sqrt{365}$.
For option $(-8,-16)$: $d=\sqrt{(-8 - 8)^2+(-16 - 5)^2}=\sqrt{(-16)^2+(-21)^2}=\sqrt{256+441}=\sqrt{697}$.
For option $(10,-16)$: $d=\sqrt{(10 - 8)^2+(-16 - 5)^2}=\sqrt{2^2+(-21)^2}=\sqrt{4 + 441}=\sqrt{445}$.

Answer:

$(10, -16)$