NCERT Solutions
Class 11 Maths
Introduction to Three Dimensional Geometry

Ex.Misc. Q.3
If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (–4, 3b, –10) and R (8, 14, 2c), then find the values of a, b and c.
It is known that the coordinates of the centroid of the triangle, whose vertices are (x1, y1, z1),
(x2, y2, z2) and (x3, y3, z3), are {(x1 + x2 + x3) ÷ 3, (y1 + y2 + y3) ÷ 3, (z1 + z2 + z3) ÷ 3}
Therefore, coordinates of the centroid of ∆PQR
= {(2a - 4 + 8) ÷ 3, (2 + 3b + 14) ÷ 3, (6 - 10 + 2c) ÷ 3}
= {(2a + 4) ÷ 3, (3b + 16) ÷ 3, (2c - 4) ÷ 3}
It is given that origin is the centroid of ∆PQR.
So, (0, 0, 0) = {(2a + 4) ÷ 3, (3b + 16) ÷ 3, (2c - 4) ÷ 3}
=> (2a + 4) ÷ 3 = 0, (3b + 16) ÷ 3 = 0, (2c - 4) ÷ 3 = 0
=> a = -2, b = - , c = 2
Thus, the respective values of a, b, and c are -2, - and 2.