Question

In a three dimensional co - ordinate system P, Q and R are images of a point A(a, b, c) in the xy the yz and the zx 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. $\frac{1}{2}(a^2+b^2+c^2)$

Answer 1

The correct option is A $0$

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 $(\frac{a}{3},\frac{b}{3},\frac{c}{3})$ $\Rightarrow G\equiv (\frac{a}{3}.\frac{b}{3},\frac{c}{3})$
$\Rightarrow$ A, O, G are collinear $\Rightarrow$ area of triangle AOG is zero. The equation of that line is $\frac{x}{a}=\frac{y}{b}=\frac{z}{c}$