Question

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

Answer 1

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

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

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

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

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

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

$\Rightarrow a+3=3,b=12,c+1=-6$

$\Rightarrow a=3-3=0,b=12,c=-6-1=-7$

Hence, the co-ordinate of $C$ are $(0,12,-7)$