Question

1. the mid - point of $overline{fg}$ is $m(2, - 1)$. one endpoint is f$(-5,4)$. find endpoint g. what is the distance of $overline{fg}$

Explanation:

Step1: Use mid - point formula to find coordinates of G

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Let $F(-5,4)$ be $(x_1,y_1)$ and $G(x,y)$ be $(x_2,y_2)$, and $M(2,-1)$.
For the x - coordinate: $\frac{-5 + x}{2}=2$. Multiply both sides by 2: $-5 + x = 4$, then $x=9$.
For the y - coordinate: $\frac{4 + y}{2}=-1$. Multiply both sides by 2: $4 + y=-2$, then $y=-6$. So $G(9,-6)$.

Step2: Use distance formula to find distance of $\overline{FG}$

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here $x_1=-5,y_1 = 4,x_2=9,y_2=-6$.
$d=\sqrt{(9-(-5))^2+(-6 - 4)^2}=\sqrt{(9 + 5)^2+(-10)^2}=\sqrt{14^2+100}=\sqrt{196 + 100}=\sqrt{296}=2\sqrt{74}$.

Answer:

Endpoint $G$ is $(9,-6)$ and the distance of $\overline{FG}$ is $2\sqrt{74}$