# Equation of the Medians of a Triangle

To find the equation of the 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 $C\left( {3, - 8} \right)$.

If $G$ is the midpoint of 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 $G$, using the two-point form of the equation of a straight line, the equation of median $CG$ can be found as

If $H$ is the midpoint of side$BC$ 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 $H$, using the two-point form of the equation of a straight line, the equation of median $AH$ can be found as

If $I$ is the midpoint of side$AC$ 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 $I$, using the two-point form of the equation of a straight line, the equation of median $BI$ can be found as