Question

the midpoint of $overline{ab}$ is $m=(4, -6)$. one endpoint is $a=(2, -8)$. find the coordinates of the other endpoint, $b$.

Explanation:

Step1: Recall mid - point formula

The mid - point formula for two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Given $A(2,-8)$ and $M(4,-6)$, we can set up equations for $x$ and $y$ coordinates.
For the $x$ - coordinate: $\frac{2 + x_2}{2}=4$.

Step2: Solve for $x_2$

Multiply both sides of the equation $\frac{2 + x_2}{2}=4$ by 2: $2 + x_2=8$. Then subtract 2 from both sides: $x_2=8 - 2=6$.

Step3: Solve for $y$ - coordinate

For the $y$ - coordinate, we have the equation $\frac{-8 + y_2}{2}=-6$. Multiply both sides by 2: $-8 + y_2=-12$. Then add 8 to both sides: $y_2=-12 + 8=-4$.

Answer:

$(6,-4)$