Question

the midpoint of $overline{ab}$ is $m=(3, - 6)$. one endpoint is $a=(8, - 4)$. 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 $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Let $A=(x_1,y_1)=(8,-4)$ and $B=(x_2,y_2)$, and $M=(3,-6)$.

Step2: Solve for $x_2$

We know that $\frac{x_1 + x_2}{2}=x_M$. Substituting $x_1 = 8$ and $x_M=3$ into the formula:
\[

$$\begin{align*} \frac{8 + x_2}{2}&=3\\ 8+x_2&=6\\ x_2&=6 - 8\\ x_2&=-2 \end{align*}$$

\]

Step3: Solve for $y_2$

We know that $\frac{y_1 + y_2}{2}=y_M$. Substituting $y_1=-4$ and $y_M = - 6$ into the formula:
\[

$$\begin{align*} \frac{-4 + y_2}{2}&=-6\\ -4+y_2&=-12\\ y_2&=-12 + 4\\ y_2&=-8 \end{align*}$$

\]

Answer:

$(-2,-8)$