Question

the mid - point of $overline{ab}$ is $m(1,2)$. if the coordinates of $a$ are $(8, - 4)$, 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 $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

Step2: Set up equations for x - coordinate

Let the coordinates of $B$ be $(x,y)$. We know $x_1 = 8$, $y_1=-4$, and the mid - point $M(1,2)$. For the x - coordinate of the mid - point: $\frac{8 + x}{2}=1$. Cross - multiply: $8 + x=2$. Solve for $x$: $x=2 - 8=-6$.

Step3: Set up equations for y - coordinate

For the y - coordinate of the mid - point: $\frac{-4 + y}{2}=2$. Cross - multiply: $-4 + y = 4$. Solve for $y$: $y=4 + 4 = 8$.

Answer:

$(-6,8)$