Question

The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are (3, –5, 7) and (–1, 7, –6) respectively, find the coordinates of the point C.

Answer 1

Let G be the centroid of $∆$ABC.
Given: G$\equiv \left(1,1,1\right)$
Let C$\equiv \left(x,y,z\right)$
$$\begin{gathered} \mathrm{Then},1=\frac{3-1+x}{3} \\ \Rightarrow 3=3-1+x \\ \Rightarrow 3=2+x\Rightarrow x=1 \\ \mathrm{and}1=\frac{-5+7+y}{3} \\ \Rightarrow 3=2+y \\ \Rightarrow y=1 \\ \mathrm{and}1=\frac{7-6+z}{3} \\ \Rightarrow 3=1+z \\ \Rightarrow z=2 \end{gathered}$$

$\therefore C\equiv \left(1,1,2\right)$