# Equation of Altitudes of Triangle

To find the equation of 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 $AB$ is

Since the altitude $CD$ is perpendicular of side $AB$.
So the slope of

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

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

Since the altitude $AE$ is perpendicular of side $BC$.
So the slope of

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

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

Since the altitude $BF$ is perpendicular of side $AC$.
So the slope of

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

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