Question

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

Answer 1

Two vertices of a triangle are $(3,1,5)$ and $(4,-5,-1)$

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

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

$=(\frac{7+a}{3},\frac{-4+b}{3},\frac{4+c}{3})$

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

$\therefore \frac{a+7}{3}=2,\frac{b-4}{3}=2,\frac{c+4}{3}=-1$

$\Rightarrow a+7=6,b-4=6,c+4=-3$

$\Rightarrow a=7-6=1,b=6+4=10,c=-3-4=-7$

Hence, the co-ordinate of $C$ are $(1,10,-7)$