Question

the mid - point of $overline{ab}$ is $m=(-2,-1)$. one endpoint is $a = (-7,1)$. 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 $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. We know $A(-7,1)$, $M(-2,-1)$, and we want to find $B(x_2,y_2)$.

Step2: Solve for $x$ - coordinate of $B$

Set up the equation for the $x$ - coordinate of the mid - point: $\frac{-7+x_2}{2}=-2$. Multiply both sides by 2: $-7 + x_2=-4$. Then add 7 to both sides: $x_2=-4 + 7=3$.

Step3: Solve for $y$ - coordinate of $B$

Set up the equation for the $y$ - coordinate of the mid - point: $\frac{1+y_2}{2}=-1$. Multiply both sides by 2: $1 + y_2=-2$. Then subtract 1 from both sides: $y_2=-2 - 1=-3$.

Answer:

$(3,-3)$