Question

the midpoint of $overline{ab}$ is $m(0, -2)$. if the coordinates of $a$ are $(5, -1)$, what are the coordinates of $b$?

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 $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Here $x_1 = 5$, $y_1=-1$, and the mid - point $M(0,-2)$.

Step2: Find the x - coordinate of B

Set up the equation for the x - coordinate of the mid - point: $\frac{x_1 + x_2}{2}=0$. Substitute $x_1 = 5$ into it: $\frac{5 + x_2}{2}=0$. Multiply both sides by 2: $5 + x_2=0$. Then solve for $x_2$: $x_2=-5$.

Step3: Find the y - coordinate of B

Set up the equation for the y - coordinate of the mid - point: $\frac{y_1 + y_2}{2}=-2$. Substitute $y_1=-1$ into it: $\frac{-1 + y_2}{2}=-2$. Multiply both sides by 2: $-1 + y_2=-4$. Then solve for $y_2$: $y_2=-3$.

Answer:

$(-5,-3)$