Question

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

Answer 1

Two vertices of a triangle are $A(1,-3,3)$ and $B(2,4,-2)$

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

$(\frac{1+2+a}{3},\frac{-3+4+b}{3},\frac{3-2+c}{3})$

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

But the centroid of the triangle $ABC$ is $(1,3,-1)$

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

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

$\Rightarrow a=3-3=0,b=9-1=8,c=-3-1=-4$

Hence, the co-ordinate of $C$ are $(0,8,-4)$