Question:
Find the third vertex of triangle whose centroid is origin and two vertices are (2, 4, 6) and (0, –2, –5).
Solution:
Given the centroid is origin and two vertices are $(2,4,6)$ and $(0,-2,-5)$. Let the third vertex be $(x, y, z)$ For a triangle the coordinates of the centroid is given by the average of the coordinates of its vertices.
$\Rightarrow(0,0,0)=\left(\frac{2+0+x}{3}, \frac{4+(-2)+y}{3}, \frac{6+(-5)+z}{3}\right)$
$\Rightarrow \frac{2+\mathrm{x}}{3}=0, \therefore \mathrm{x}=-2$
$\Rightarrow \frac{2+y}{3}=0, \therefore y=-2$
$\Rightarrow \frac{1+\mathrm{X}}{3}=0, \therefore \mathrm{X}=-1$
Therefore the third vertex is $(-2,-2,-1)$.
$\Rightarrow \frac{1+x}{3}=0, \therefore x=-1$
Therefore the third vertex is $(-2,-2,-1)$.