Question

If the origin is the centroid of the triangle with vertices $A(3a, 4, -5)$, $B(-2, 4b, 6)$ and $C(6, 10, c)$, then find the value of $a$, $b$, $c$.

• A. $-\frac{4}{3}, -\frac{7}{2}, -1$
• B. $\frac{4}{3}, \frac{7}{2}, 1$
• C. $-\frac{4}{3}, -\frac{7}{2}, 1$
• D. None of these

Answer 1

Answer: (A)

Here $A(3a, 4, -5)$, $B(-2, 4b, 6)$ and $C(6,10, c)$ be the three vertices of $\Delta ABC$, then coordinates of centroid are
$\left[\frac{3a-2+6}{3}, \frac{4+4b+10}{3}, \frac{-5+6+c}{3}\right]$.
But it is given that coordinates of centroid are $\left(0,0,0\right)$.
$\therefore \frac{3a-2+6}{3} = 0$
$\Rightarrow 3a=-4$
$\Rightarrow a = -\frac{4}{3}$
$\frac{4+4b+10}{3}=0$
$\Rightarrow 4b =-14$
$\Rightarrow b=-\frac{7}{2}$
$\frac{-5+6+c}{3}=0$
$\Rightarrow c=-1$