Question

The sides of a triangle are $y=x,y=2x$ and $y=3x+4$. Find the equation of the medians and hence the co-ordinates of its centroid.

Answer 1

Solving the given equations in pairs the coordinates of the vertices of the triangle are
$A(-4,-8),B(-2,-2),C(0,0)$
The midpoints of the sides of the triangle are
$D(-1,-1),E(-2,-4),F(-3,-5)$
Equation to median $AD$ joining $(-4,-8)$ and $(-1,-1)$ is
$y+8=\frac{-1-(-8)}{-1-(-4)}(x+4)$
or $3(y+8)=7(x+4)$
or $7x-3y-4=0$
Similarly the equations to medians $BE$ and $CF$ are respectively.
$x+2=\mathrm{0......}\left(2\right)$
and $5x-3y=\mathrm{0......}\left(3\right)$
Solving any two of (1), (2) and (3) we get the point of intersection as $(-2,-10/3)$ which clearly satisfies the third. Hence the three medians are concurrent and the point of concurrency is called centoid of the triangle.
We can, however, find the coordinates of the centroid by using the formula
$(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$ ie., $(-2,\frac{-10}{3})$