# Given point A (-2,1) and point B (1,3), how do you find the equation of the line perpendicular to the line AB at its midpoint?

Jan 7, 2016

Find the midpoint and slope of the Line AB and make the slope a negative reciprocal then to find the y axis plug in the midpoint coordinate. Your answer will be $y = - \frac{2}{3} x + 2 \frac{2}{6}$

#### Explanation:

If point A is (-2, 1) and point B is (1, 3) and you need to find the line perpendicular to that line and passes through the midpoint you first need to find the midpoint of AB. To do this you plug it into the equation $\left(\frac{x 1 + x 2}{2} , \frac{y 1 + y 2}{2}\right)$ (Note: The numbers after the variables are subscripts) so plug the cordinates into the equation...

$\left(\frac{- 2 + 1}{2} , 1 + \frac{3}{2}\right)$
$\left(\frac{- 1}{2} , \frac{4}{2}\right)$
$\left(- .5 , 2\right)$

So for our midpoint of AB we get (-.5, 2). Now we need to find the slope of AB. to do this we use $\frac{y 1 - y 2}{x 1 - x 2}$ Now we plug A and B into the equation...

$\frac{- 2 - 1}{1 - 3}$
$\frac{- 3}{-} 2$
$\frac{3}{2}$

So our slope of line AB is 3/2. Now we take the opposite reciprocal* of the slope to make a new line equation. Which is $y = m x + b$ and plug in the slope for $y = - \frac{2}{3} x + b$. Now we put in the cordinates of the midpoint to get...

$2 = - \frac{2}{3} \cdot - .5 + b$
$2 = - \frac{2}{6} + b$
$2 \frac{2}{6} = b$

So put b back in the get $y = - \frac{2}{3} x + 2 \frac{2}{6}$as your final answer.

*opposite reciprocal is a fraction with the top and bottom numbers switched then multiplied by -1