Q.Two vertices of a triangle are $A(3, -2)$ and $B(-1, 4)$. If the centroid of the triangle is $G(2, 3)$, find the third vertex $C$.

Centroid of a Triangle

The centroid of a triangle is the point where its three medians (each joining a vertex to the midpoint of the opposite side) meet. It is the triangle's balance point, and its coordinates are simply the average of the three vertices. For a triangle with vertices $A$, $B$ and $C$, the centroid is

In the plane, if $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$, this reads $G = \left(\dfrac{x_1+x_2+x_3}{3},\ \dfrac{y_1+y_2+y_3}{3}\right)$, and the same averaging works coordinate-by-coordinate in three-dimensional space by including the $z$-values. A useful geometric fact is that the centroid divides each median in the ratio $2:1$, measured from the vertex. …