Find the third vertex of a triangle, if two of its vertices are at $(-3, 1)$ and $(0, -2)$ and the centroid is at the origin.

Given:

Two vertices of a triangle are $(-3, 1), (0, -2)$ and its centroid is at the origin.

To do:

We have to find the third vertex.

Solution:

Let the coordinates of the third vertex be $(x,y)$.

We know that,

Coordinates of the centroid of a triangle are $(\frac{Sum\ of\ abscissa}{3}, \frac{Sum\ of\ ordinates}{3})$

Therefore,

The coordinates of the centroid of the given triangle are,

$(0,0)=(\frac{-3+0+x}{3}, \frac{1+(-2)+y}{3})$

$(0,0)=(\frac{x-3}{3}, \frac{y-1}{3})$

On comparing, we get,

$0=\frac{x-3}{3}$

$0(3)=x-3$

$x=0+3$

$x=3$

$0=\frac{y-1}{3}$

$0(3)=y-1$

$y=0+1$

$y=1$

The third vertex of the triangle is $(3, 1)$.