Question

The coordinates of the mid-points of sides AB, BC and CA of $△$ are D (1, 2, -3) E(3, 0, 1) and F(-1, 1, -4) respectively. Write the coordinates of its centroid.

Answer 1

$\text{Let the coordinates of the triangle be A}$

$(x_1,y_1,z_1),B(x_2,y_2,z_2),C(x_3,y_3,z_3)$

$\text{Now,}$ $\text{Mid point of AB is D}(1,2,-3)$

$\therefore \frac{x_2+x_3}{2}=1,\frac{y_2+y_3}{2}=2,\frac{z_2+z_3}{2}=-3$

$\Rightarrow x_1+x_2=2,y_1+y_2=4,$

$and{z}_1+z_2=-6...\left(i\right)$

$\text{Mid-point of BC is E (3, 0, 1)}$

$\therefore \frac{x_1+x_2}{2}=3,\frac{y_1+y_2}{2}=0,\frac{z_1+z_2}{2}=1$

$\Rightarrow \frac{x_1+x_2+x_3}{3}=1,\frac{y_1+y_2+y_3}{3}=1and{z}_1+z_2+z_3=-6$
$\Rightarrow \frac{x_1+x_2+x_3}{3}=1,\frac{y_1+y_2+y_3}{3}=1$
$and\frac{z_1+z_2+z_3}{3}=-2$
$\text{Hence, the centroid of the triangle ABC is (1, 1, -2)}$