Question

the midpoint of $overline{ab}$ is $m(-1,6)$. if the coordinates of $a$ are $(5,4)$, 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 $x_1 = 5$, $y_1 = 4$, and $M(-1,6)$.

Step2: Find the x - coordinate of B

Set up the equation for the x - coordinate of the mid - point: $\frac{x_1 + x_2}{2}=-1$. Substitute $x_1 = 5$ into it: $\frac{5 + x_2}{2}=-1$. Multiply both sides by 2: $5 + x_2=-2$. Then solve for $x_2$: $x_2=-2 - 5=-7$.

Step3: Find the y - coordinate of B

Set up the equation for the y - coordinate of the mid - point: $\frac{y_1 + y_2}{2}=6$. Substitute $y_1 = 4$ into it: $\frac{4 + y_2}{2}=6$. Multiply both sides by 2: $4 + y_2 = 12$. Then solve for $y_2$: $y_2=12 - 4 = 8$.

Answer:

$(-7,8)$