Question

The centroid of a triangle with vertices \( (a, 1, 3), (-2, b, -5), (4, 7, c) \) lies at the origin. Find \( a^2 + b^2 + c^2 \).

• A. 68
• B. 64
• C. 72
• D. 54

Hint:

The centroid of a triangle with vertices \( A, B, C \) is given by the average of coordinates: \( G = \frac{A + B + C}{3} \).

Answer 1

The Correct Option is C

Centroid of triangle: \[ \left( \frac{a - 2 + 4}{3}, \frac{1 + b + 7}{3}, \frac{3 - 5 + c}{3} \right) = (0, 0, 0) \] Equating: \[ \begin{align} \frac{a + 2}{3} = 0 \Rightarrow a = -2
\frac{b + 8}{3} = 0 \Rightarrow b = -8
\frac{-2 + c}{3} = 0 \Rightarrow c = 2 \] Now compute: \[ a^2 + b^2 + c^2 = (-2)^2 + (-8)^2 + (2)^2 = 4 + 64 + 4 = 72 \]