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  4. 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.

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Q. 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 $C$.

Introduction to Three Dimensional Geometry

Solution:

Let the coordinates of $C$ be $(x,y, z)$. Coordinates of the centroid $G = (1,1,1)$. Then
$\frac{x+3-1}{3} = 1$,
$\frac{y-5+7}{3}=1$,
$\frac{z+7-6}{3}=1$
$\Rightarrow x = 1$, $y = 1$, $z = 2$.
Hence, coordinates of $C$ are $(1,1,2)$.