Question

The coordinates of mid-points of the sides AB, BC & CA of a triangle are $(1,2),(0,-1)$ and $(2,-1)$ respectively. Coordinates of its vertex A are?

Answer 1

Now the line joining the mid-points of BC and CA is parallel to the side AB and if this parallel line passes through the mid point of AB then we will get the equation of AB.

The equation of the line joining the mid-points of BC and CA is

$\frac{x-0}{2-0}=\frac{y+1}{-1+1}$

or, $y=-1$.....(1)

Equation of line parallel to (1) is $y=c$ which passes through $(1,2)$, so equation of AB is $y=2$......(2)

Similarly to get the equation of AC we join the mid-points of AB and BC.

Its equation is

$\frac{x-0}{1-0}=\frac{y+1}{2+1}$

or, $y=3x-1$......(3)

The equation of line parallel to (3) is

$y=3x+c$, which passes through $(2,-1)$ then $c=-7$.

So, equation of AC is $y=3x-7$.......(4)

It is clear that the point of intersection of AB and AC is the vertex A.

Solving (2) and (4) we get the co-ordinate of vertex A and it is $(3,2)$.