Question

Find the equation of the medians of $△ABC$ whose vertices are $A(-1,2),B(2,1)$ and $C(0,4)$,Hence find the co-ordinates of the centroids of $△ABC$ ?

Answer 1

since $AD,BE,CF$ are the medians of $\Delta ABC$

therefore $D,E,F$ are the midpoints of $BC,CA,AB$ respectively

Hence coordinates of$D$$=(\frac{2+0}{2},\frac{1+4}{2})=(1,\frac{5}{2})$

coordinate of $E$$=(\frac{0+(-1)}{2},\frac{4+2}{2})=(\frac{-1}{2},3)$

coordinate of $F$$=(\frac{-1+2}{2},\frac{2+1}{2})=(\frac{1}{2},\frac{3}{2})$

Equation of median $AD$ is given as ;$\frac{y-\frac{5}{2}}{2-\frac{5}{2}}=\frac{x-1}{-1-1}\Rightarrow x-4y+9=0⟶\left(1\right)$

Equation of median $BE$ is given as;$\frac{y-3}{1-3}=\frac{x-\left(\frac{-1}{2}\right)}{2-\left(\frac{-1}{2}\right)}\Rightarrow 4x+5y-13=0⟶\left(2\right)$

Equation of median $CF$ is given as;$\frac{y-\frac{3}{2}}{4-\frac{3}{2}}=\frac{x-\frac{1}{2}}{0-\frac{1}{2}}\Rightarrow 10x+2y-8=0⟶\left(3\right)$

Also the coordinate of the centroid $O$ of $\Delta ABC$ is given as ;

$O_x=\frac{A_x+B_x+C_x}{3}=\frac{-1+2+0}{3}=\frac{1}{3}{O}_y=\frac{A_y+B_y+C_y}{3}=\frac{2+1+4}{3}=\frac{7}{3}\therefore (O_x,O_y)=(\frac{1}{3},\frac{7}{3})$