Question

The centroid of a triangle $ABC$ is at the point $(2,3,-1)$.If the coordinates of $A$ and $B$ are $(1,-5,5)$ and $(0,5,-4)$ respectively.Find the coordinates of the point $C$

Answer 1

Two vertices of a triangle are $A(1,-5,5)$ and $B(0,5,-4)$

Let the co-ordinates of the third vertex be $C(a,b,c)$, then the co-ordinates of the centroid of a triangle $ABC$ are

$(\frac{1+0+a}{3},\frac{-5+5+b}{3},\frac{5-4+c}{3})$

$=(\frac{1+a}{3},\frac{b}{3},\frac{1+c}{3})$

But the centroid of the triangle $ABC$ is $(1,2,1)$

$\therefore \frac{a+1}{3}=2,\frac{b}{3}=3,\frac{c+1}{3}=-1$

$\Rightarrow a+1=6,b=9,c+1=-3$

$\Rightarrow a=6-1=5,b=9,c=-3-1=-4$

Hence, the co-ordinate of $C$ are $(5,9,-4)$