Question

1. the midpoint of $overline{ab}$ is $m(-1,3)$. if the coordinates of $a$ are $(3, - 2)$, what is the length of $overline{ab}$?

Explanation:

Step1: Find coordinates of point B

Let the coordinates of B be $(x,y)$. The mid - point formula is $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Given $A(3,-2)$ and $M(-1,3)$, we have $\frac{3 + x}{2}=-1$ and $\frac{-2 + y}{2}=3$.
For $\frac{3 + x}{2}=-1$, we get $3+x=-2$, so $x=-5$.
For $\frac{-2 + y}{2}=3$, we get $-2 + y = 6$, so $y = 8$. Thus, $B(-5,8)$.

Step2: Calculate the length of AB

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here $x_1 = 3,y_1=-2,x_2=-5,y_2 = 8$.
$d=\sqrt{(-5 - 3)^2+(8+2)^2}=\sqrt{(-8)^2+10^2}=\sqrt{64 + 100}=\sqrt{164}=2\sqrt{41}$.

Answer:

$2\sqrt{41}$