Question

the coordinates of the mid - point of $overline{gh}$ are $m(-\frac{13}{2},-6)$ and the coordinates of one endpoint are $g(-4,1)$.

Explanation:

Step1: Recall mid - point formula

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 $G(x_1,y_1)=(-4,1)$ and $H(x_2,y_2)$. The mid - point $M(\frac{-13}{2},-6)$.

Step2: Find the x - coordinate of H

We know that $\frac{x_1 + x_2}{2}=\frac{-13}{2}$, substituting $x_1=-4$ into the equation:
$\frac{-4 + x_2}{2}=\frac{-13}{2}$
Multiply both sides by 2: $-4+x_2=-13$
Add 4 to both sides: $x_2=-13 + 4=-9$

Step3: Find the y - coordinate of H

We know that $\frac{y_1 + y_2}{2}=-6$, substituting $y_1 = 1$ into the equation:
$\frac{1 + y_2}{2}=-6$
Multiply both sides by 2: $1+y_2=-12$
Subtract 1 from both sides: $y_2=-12 - 1=-13$

Answer:

The coordinates of point $H$ are $(-9,-13)$