Q.If the origin is the centroid of the triangle whose vertices are $A(2, p, -3)$, $B(q, -2, 5)$ and $R(-5, 1, r)$, then find the values of $p, q, r$.

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. …