Equation of Medians of Triangle

To find the equation of median 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-medians-triangle

If G is the midpoint of the side AB of the given triangle, then its coordinates are given as \left(  {\frac{{ - 3 + 5}}{2},\frac{{2 + 4}}{2}} \right) = \left(  {\frac{2}{2},\frac{6}{2}} \right) = \left( {1,3} \right).
Since the median CG passes through points C and G, so using two-point form of equation of straight line, the equation of median CG can be find as

\begin{gathered} \frac{{y - 3}}{{ - 8 - 3}} = \frac{{x -  1}}{{3 - 1}} \\ \Rightarrow \frac{{y - 3}}{{ - 11}} =  \frac{{x - 1}}{2} \\ \Rightarrow 2\left( {y - 3} \right) =  - 11\left( {x - 1} \right) \\ \Rightarrow 11x + 2y - 17 = 0 \\ \end{gathered}


If H is the midpoint of the sideBC of the given triangle, then its coordinates are given as \left(  {\frac{{3 + 5}}{2},\frac{{ - 8 + 4}}{2}} \right) = \left( {\frac{8}{2},\frac{{  - 4}}{2}} \right) = \left( {4, - 2} \right).
Since the median AH passes through points A and H, so using two-point form of equation of straight line, the equation of median AH can be find as

\begin{gathered} \frac{{y - \left( { - 2} \right)}}{{2 -  \left( { - 2} \right)}} = \frac{{x - 4}}{{ - 3 - 4}} \\ \Rightarrow \frac{{y + 2}}{4} = \frac{{x -  4}}{{ - 7}} \\ \Rightarrow   - 7\left( {y + 2} \right) = 4\left( {x - 4} \right) \\ \Rightarrow 5x + 7y - 2 = 0 \\ \end{gathered}


If I is the midpoint of the sideAC of the given triangle, then its coordinates are given as \left(  {\frac{{ - 3 + 3}}{2},\frac{{2 - 8}}{2}} \right) = \left( {0,\frac{{ - 6}}{2}}  \right) = \left( {0, - 3} \right).
Since the median BI passes through points B and I, so using two-point form of equation of straight line, the equation of median BI can be find as

\begin{gathered} \frac{{y - \left( { - 3} \right)}}{{4 -  \left( { - 3} \right)}} = \frac{{x - 0}}{{5 - 0}} \\ \Rightarrow \frac{{y + 3}}{7} = \frac{x}{5} \\ \Rightarrow 5\left( {y + 3} \right) = 7x \\ \Rightarrow 7x - 5y - 15 = 0 \\ \end{gathered}

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