Question

1. the midpoint of $overline{ab}$ is $m(2,6)$. if the coordinates of $a$ are $(-5,4)$, what is the length of $overline{ab}$?

Explanation:

Step1: Find coordinates of 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 $M(2,6)$ and $A(-5,4)$, we have $\frac{-5 + x}{2}=2$ and $\frac{4 + y}{2}=6$.
From $\frac{-5 + x}{2}=2$, we get $-5+x = 4$, so $x=9$. From $\frac{4 + y}{2}=6$, we get $4 + y=12$, so $y = 8$. So B is $(9,8)$.

Step2: Use distance formula

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, $x_1=-5,y_1 = 4,x_2=9,y_2 = 8$.
$d=\sqrt{(9-(-5))^2+(8 - 4)^2}=\sqrt{(9 + 5)^2+4^2}=\sqrt{14^2+4^2}=\sqrt{196 + 16}=\sqrt{212}=2\sqrt{53}$.

Answer:

$2\sqrt{53}$