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ln a three-dimensional coordinate system, P, Q and R are images of a point A (a, b, c) in the x-y, y-z and z-x planes, respectively. If G is the centroid of triangle PQR, thin area of triangle AOC is (O is the origin)

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$a^{2} + b^{2} + c^{2}$

$\frac{2}{3} (a^{2} + b^{2} + c^{2})$

none of these

answer is A.

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Detailed Solution

Point A is (a, b, c) $\Rightarrow$ Points P, Q, R are (a, b, -c),
(-a, b, c) and (a, -b, c), respectively

Centroid of triangle PQR is $(\frac{a}{3}, \frac{b}{2}, \frac{c}{3})$

$\Rightarrow G \equiv (\frac{a}{3}, \frac{b}{3}, \frac{c}{3})$

A, O, G are collinear

Area of triangle AOG is zero.

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