# Question #d902d

Nov 18, 2016

$3 x + y = 14$

#### Explanation:

The line segment through the points $\left(2 , 3\right)$ and $\left(5 , 4\right)$:

[1]$\textcolor{w h i t e}{\text{XXX}}$has a mid-point at $\left(\frac{2 + 5}{2} , \frac{3 + 4}{2}\right) = \left(3.5 , 3.5\right)$

[2]$\textcolor{w h i t e}{\text{XXX}}$has a slope of $m = \frac{\Delta y}{\Delta x} = \frac{4 - 3}{5 - 2} = \frac{1}{3}$

The perpendicular bisector of this line segment:

[1]$\textcolor{w h i t e}{\text{XXX}}$passes through the midpoint $\left(3.5 , 3.5\right)$

[2]$\textcolor{w h i t e}{\text{XXX}}$has a slope of $- \frac{1}{m} = - 3$

Using the slope-point form for the perpendicular
$\textcolor{w h i t e}{\text{XXX}} y - 3.5 = \left(- 3\right) \left(x - 3.5\right)$
or
$\textcolor{w h i t e}{\text{XXX}} 3 x + y = 14$

Here is a picture to help verify the results: