# How do I find the equation of a perpendicular bisector of a line segment with the endpoints (-2, -4) and (6, 4)?

Jul 16, 2016

$x + y - 2 = 0$

#### Explanation:

Let $\left(x , y\right)$ be any point on the perpendicular bisector. From elementary geometry, we can easily see that this point must be equidistant from the two points $\left(- 2 , - 4\right) \mathmr{and} \left(6 , 4\right)$. Using the Euclidean distance formula gives us the equation

${\left(x + 2\right)}^{2} + {\left(y + 4\right)}^{2} = {\left(x - 6\right)}^{2} + {\left(y - 4\right)}^{2}$

This can be rewritten as

${\left(x + 2\right)}^{2} - {\left(x - 6\right)}^{2} = {\left(y - 4\right)}^{2} - {\left(y + 4\right)}^{2}$

Using ${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$, this simplifies to

$8 \left(2 x - 4\right) = - 8 \cdot 2 y$ which simplifies to

$x + y - 2 = 0$