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),
find the values of a, b and c.
The centroid of a triangle with vertices
(x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃) is given by:
\( \left( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3} \right) \)
Since the centroid is the origin (0, 0, 0), we equate each coordinate to zero.
x-coordinate:
\( \frac{2a + (-4) + 8}{3} = 0 \)
\( 2a + 4 = 0 \Rightarrow a = -2 \)
y-coordinate:
\( \frac{2 + 3b + 14}{3} = 0 \)
\( 3b + 16 = 0 \Rightarrow b = -\frac{16}{3} \)
z-coordinate:
\( \frac{6 + (-10) + 2c}{3} = 0 \)
\( 2c - 4 = 0 \Rightarrow c = 2 \)
Final Values:
a = –2,
b = –\( \frac{16}{3} \),
c = 2