Question

If the mid-points of the sides of a triangle $AB;BC;CA$ are $D(1,2,–3),E(3,0,1)$ and $F(–1,1,–4),$ then the centroid of the triangle ABC is .

Answer 1

Given that mid-point of a triangle are $D(1,2,–3),E(3,0,1)$ and $F(–1,1,–4)$.

From Geometry of centroid, centroid of triangle is same as the centroid of the triangle formed by mid-points of the sides.

The coordinates of the centroid of the triangle, whose vertices are $(x_1,y_1,z_1),(x_2,y_2,z_2)$ and $(x_3,y_3,z_3)$ are $(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_3}{3})$

$\therefore$ Centroid of triangle

$=G(\frac{1+3-1}{3},\frac{2+0+1}{3},\frac{-3+1-4}{3})$

$=G(1,1,–2)$

Hence, $(1,1,–2)$ are the coordinates of the centroid.