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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#### Solution

Let G be the centroid of\[∆\]ABC.

Given: G\[\equiv \left( 1, 1, 1 \right)\]

Let C\[\equiv \left( x, y, z \right)\]

\[\text{ Then }, 1 = \frac{3 - 1 + x}{3}\]

\[ \Rightarrow 3 = 3 - 1 + x \]

\[ \Rightarrow 3 = 2 + x \Rightarrow x = 1\]

\[\text{ and } 1 = \frac{- 5 + 7 + y}{3} \]

\[ \Rightarrow 3 = 2 + y \]

\[ \Rightarrow y = 1\]

\[\text{ and } 1 = \frac{7 - 6 + z}{3}\]

\[ \Rightarrow 3 = 1 + z\]

\[ \Rightarrow z = 2\]

\[\therefore C \equiv \left( 1, 1, 2 \right)\]

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