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)
Step-by-Step Solution
Key Concept: Collinear points form a degenerate triangle with zero area; verify collinearity by checking if points lie on the same line through the origin.
Point $A$ is at $(a, b, c)$ and points $P$, $Q$, $R$ are at $(a, b, -c)$, $(-a, b, c)$, and $(a, -b, c)$ respectively. The centroid of triangle $PQR$ is $G = \left(\frac{a}{3}, \frac{b}{3}, \frac{c}{3}\right)$. Since $A$, $O$ (the origin), and $G$ are collinear (all lie on the line from origin through $(a, b, c)$), the area of triangle $AOG$ is zero.
Correct Answer: 4