# 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 $AB$:
$\frac{{4 – 2}}{{5 – \left( { – 3} \right)}} = \frac{2}{{5 + 3}} = \frac{1}{4}$

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

The slope of $CD = – \frac{1}{{{\text{slope}}\,{\text{of}}\,AB}} = – 4$

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
$\begin{gathered} y – \left( { – 8} \right) = – 4\left( {x – 3} \right) \\ \Rightarrow y + 8 = – 4x + 12 \\ \Rightarrow 4x + y – 4 = 0 \\ \end{gathered}$

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

The slope of side $BC$ is
$\frac{{ – 8 – 4}}{{3 – 5}} = \frac{{ – 12}}{{ – 2}} = 6$

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

The slope of $AE = – \frac{1}{{{\text{slope}}\,{\text{of}}\,BC}} = – \frac{1}{6}$

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
$\begin{gathered} y – 2 = – \frac{1}{6}\left( {x – \left( { – 3} \right)} \right) \\ \Rightarrow – 6\left( {y – 2} \right) = x + 3 \\ \Rightarrow x + 6y – 9 = 0 \\ \end{gathered}$

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

The slope of side $AC$ is
$\frac{{ – 8 – 2}}{{3 – \left( { – 3} \right)}} = – \frac{{10}}{6} = – \frac{5}{3}$

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

The slope of $BF = – \frac{1}{{{\text{slope}}\,{\text{of}}\,AC}} = \frac{3}{5}$

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
$\begin{gathered} y – 4 = \frac{3}{5}\left( {x – 5} \right) \\ \Rightarrow 5\left( {y – 4} \right) = 3\left( {x – 5} \right) \\ \Rightarrow 3x – 5y + 5 = 0 \\ \end{gathered}$

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