Question

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.

Answer 1

It is known that the coordinates of the centroid of the triangle, whose vertices are (x₁, y₁, z₁), (x₂, y₂, z₂) and (x, y, z), are (\(\frac{x_1+x_2+x_3}{3}\), \(\frac{y_1+y_2+y_3}{3}\), \(\frac{z_1+z_2+z_3}{3}\)). 
Therefore, the coordinates of the centroid of
PQR = (\(\frac{2a-4+8}{3}\), \(\frac{2+3b+14}{3}\), \(\frac{6-10+2c}{3}\)) = (\(\frac{2a+4}{3}\), \(\frac{3b+16}{3}\), \(\frac{2c-4}{3}\))
It is given that the origin is the centroid of PQR.
∴ (0,0,0)= (\(\frac{2a+4}{3},\frac{3b+16}{3},\frac{2c-4}{3}\))
⇒ \(\frac{2a+4}{3}=0,\) and \(\frac{2c-4}{3}=0\)
a=-2, b= \(-\frac{16}{3}\) and c = 2
Thus, the respective values of a, b, and c are -2, \(-\frac{16}{3}\), and 2.