Question

the midpoint of $overline{ab}$ is $m = (-4, 1)$. one endpoint is $a = (-1, 7)$. find the coordinates of the other endpoint, $b$.

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 $A=(x_1,y_1)=(-1,7)$ and $B=(x_2,y_2)$ and $M=(-4,1)$.

Step2: Solve for $x_2$

We know that $\frac{x_1 + x_2}{2}=-4$. Substitute $x_1=-1$ into the equation: $\frac{-1 + x_2}{2}=-4$. Multiply both sides by 2: $-1+x_2=-8$. Then add 1 to both sides: $x_2=-8 + 1=-7$.

Step3: Solve for $y_2$

We know that $\frac{y_1 + y_2}{2}=1$. Substitute $y_1 = 7$ into the equation: $\frac{7 + y_2}{2}=1$. Multiply both sides by 2: $7+y_2=2$. Then subtract 7 from both sides: $y_2=2 - 7=-5$.

Answer:

$B=(-7,-5)$