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).


equation-altitudes-triangle
 

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.