Question

If the orthocenter and the circumcenter of a triangle are (-3, 5, 2), (6, 2, 5) then its centroid is

Answer 1

(3, 3, 4)

Answer 2

The orthocenter (O), centroid (G), and circumcenter (C) of a triangle are collinear. The centroid G divides the line segment joining the orthocenter O and the circumcenter C in the ratio 2:1, i.e., OG : GC = 2 : 1. Given: Orthocenter $O = (-3, 5, 2)$ Circumcenter $C = (6, 2, 5)$ Let the centroid be $G = (x, y, z)$.

Using the section formula for internal division, the coordinates of G are given by: $G = \left( \frac{1 \cdot O_x + 2 \cdot C_x}{1+2}, \frac{1 \cdot O_y + 2 \cdot C_y}{1+2}, \frac{1 \cdot O_z + 2 \cdot C_z}{1+2} \right)$

Substituting the given coordinates: $x = \frac{1 \cdot (-3) + 2 \cdot 6}{3} = \frac{-3 + 12}{3} = \frac{9}{3} = 3$ $y = \frac{1 \cdot 5 + 2 \cdot 2}{3} = \frac{5 + 4}{3} = \frac{9}{3} = 3$ $z = \frac{1 \cdot 2 + 2 \cdot 5}{3} = \frac{2 + 10}{3} = \frac{12}{3} = 4$

Therefore, the centroid is $(3, 3, 4)$.