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).


equation-altitudes-triangle

First we find the slope of side AB is

\frac{{4 - 2}}{{5 - \left( { - 3} \right)}} = \frac{2}{{5 + 3}} = \frac{1}{4}


Since the altitude CD is perpendicular of side AB.
So 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), so using point-slope form of equation of 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 altitude from C to AB.
Slope of side BC is

\frac{{ - 8 - 4}}{{3 - 5}} = \frac{{ - 12}}{{ - 2}} = 6


Since the altitude AE is perpendicular of side BC.
So 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), so using point-slope form of equation of 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 altitude from A to BC.
Slope of side AC is

\frac{{ - 8 - 2}}{{3 - \left( { - 3} \right)}} =  - \frac{{10}}{6} =  - \frac{5}{3}


Since the altitude BF is perpendicular of side AC.
So 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), so using point-slope form of equation of 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 altitude from B to AC.

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