Question

If the centroid of the tetrahedron $OABC$, where $A, B, C$ are given by $(\alpha, 5, 6), (1, \beta, 4), (3, 2, \gamma)$ respectively be $(1, -1 ,2)$, then value of $\alpha^{2}+\beta^{2}+\gamma^{2}$ equals

• A. $\alpha^{2}+\beta^{2}$
• B. $\gamma^{2}+\beta^{2}$
• C. $\alpha^{2}+\gamma^{2}$
• D. $0$

Answer 1

Answer: (B)

Centroid of tetrahedron, $x = \frac{x_{1}+x_{2}+x_{3}+x_{4}}{4}$
or $1 \times 4 =\alpha+1+3+0$
or $\alpha=0$
$\therefore \alpha^{2}+\beta^{2}+\gamma^{2} = \beta^{2}+\gamma^{2}$ (as $\alpha=0)$