Q. In a three dimensional co-ordinate system, $P,\ Q$ and $R$ are images of a point $A(a,\ b,\ c)$ in the $x$-$y$, $y$-$\mathcal{z}$ and $\mathcal{z}$-$x$ 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) $\frac{2}{3}(a^{2} + b^{2} + c^{2})$
D) None of these
Explanation
(a)
Point $A$ is $(a,\ b,\ c) \Rightarrow$ Points $P,\ Q,\ R$ are $(a,\ b,\ - c),\ ( - a,\ b,\ c)$ and$\ (a,\ - b,\ c)$, respectively
$\Rightarrow$ Centroid of triangle $PQR$ is $\left( \frac{a}{3},\frac{b}{3},\frac{c}{3} \right) \Rightarrow G \equiv \left( \frac{a}{3},\frac{b}{3},\frac{c}{3} \right)$
$\Rightarrow A,\ O,\ G$ are collinear $\Rightarrow$ area of triangle $AOG$ is zero
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.