Q: In a three dimensional co-ordinate system P, Q and R are images of a point A(a, b, c) in the x-y, the y-z and the z-x planes respectively. If G is the centroid of triangle PQR then area of triangle AOG is (O is the origin)

(A) 0

(B) a^{2} + b^{2} + c^{2}

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

(D) none of these

Sol. Point A is (a, b, c)

⇒ Points P, Q, R are (a, b, –c), (–a, b, c) and (a, –b, c) respectively.

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

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

⇒ A, O, G are collinear

⇒ area of triangle AOG is zero.