# Equation of the Altitudes of a Triangle

To find the equation of the altitude of a triangle, we examine the following example: Consider the triangle having vertices $A\left( { - 3,2} \right)$, $B\left( {5,4} \right)$ and $C\left( {3, - 8} \right)$.

First we find the slope of side $AB$:

The altitude $CD$ is perpendicular to side $AB$.

The slope of

Since the altitude $CD$ passes through the point $C\left( {3, - 8} \right)$, using the point-slope form of the equation of a line, the equation of $CD$ is

This is the required equation of the altitude from $C$ to $AB$.

The slope of side $BC$ is

The altitude $AE$ is perpendicular to side $BC$.

The slope of

Since the altitude $AE$ passes through the point $A\left( { - 3,2} \right)$, using the point-slope form of the equation of a line, the equation of $AE$ is

This is the required equation of the altitude from $A$ to $BC$.

The slope of side $AC$ is

The altitude $BF$ is perpendicular to side $AC$.

The slope of

Since the altitude $BF$ passes through the point $B\left( {5,4} \right)$, using the point-slope form of the equation of a line, the equation of $BF$ is

This is the required equation of the altitude from $B$ to $AC$.