Question

the mid - point of $overline{ab}$ is $m=(2, - 1)$. one endpoint is $a=(4, - 4)$. 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 $A=(x_1,y_1)=(4, - 4)$ and $M=(x_m,y_m)=(2,-1)$ and $B=(x_2,y_2)$.
We know that $x_m=\frac{x_1 + x_2}{2}$ and $y_m=\frac{y_1 + y_2}{2}$.

Step2: Solve for $x_2$

Given $x_m = 2$ and $x_1=4$, substitute into $x_m=\frac{x_1 + x_2}{2}$.
$2=\frac{4 + x_2}{2}$
Multiply both sides by 2: $4=4 + x_2$
Subtract 4 from both sides: $x_2=0$.

Step3: Solve for $y_2$

Given $y_m=-1$ and $y_1 = - 4$, substitute into $y_m=\frac{y_1 + y_2}{2}$.
$-1=\frac{-4 + y_2}{2}$
Multiply both sides by 2: $-2=-4 + y_2$
Add 4 to both sides: $y_2 = 2$.

Answer:

$(0,2)$