If the origin is the centroid of a triangle ABC having vertices A (a, 1, 3), B (– 2, b, – 5) and C (4, 7, c), find the values of a, b, c.
If the origin is the centroid of a triangle ABC having vertices A (a, 1, 3), B (– 2, b, – 5) and C (4, 7, c), find the values of a, b, c.

Solution:

It is given that the triangle ABC having vertices A(a, 1,3), B(-2, b,-5) and C(4,7, c) and origin is the centroid.
The coordinates of the centroid for a triangle is given by the average of the coordinates of its vertices.
So,
\Rightarrow(0,0,0)=\left(\frac{\mathrm{a}+(-2)+4}{3}, \frac{1+\mathrm{b}+7}{3}, \frac{3+(-5)+\mathrm{c}}{3}\right)
By comparing the each point we obtain
\begin{array}{l} \Rightarrow \frac{a+2}{3}=0, \therefore \mathrm{a}=-2 \\ \Rightarrow \frac{b+8}{3}=0, \therefore b=-8 \\ \Rightarrow \frac{c-2}{3}=0, \therefore c=2 \end{array}