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.