Question

Let $ABCD$ be a tetrahedron in which the coordinates of each of its vertices are in arithmetic progression. If the centroid $G$ of the tetrahedron is $(2,3,k)$ then the distance of $G$ from the origin is

• A. $\sqrt{38}$
• B. $7$
• C. $\sqrt{22}$
• D. $\sqrt{29}$

Answer 1

Answer: (D)

Coordinates of verticies of tetrahedron are in AP. Let
$A\left(x_1,y_1,z_1\right),B\left(x_2,y_2,z_2\right),C\left(x_3,y_3,z_3\right)$
and $D\left(x_4,y_4,z_4\right)$
Now, each coordinates are in $AP$
$\therefore y_1=x_1+d,z_1=x_1+2d$
$y_2=x_2+d_1{z}_2=x_2+2d$
$y_3=x_3+d,z_3=x_3+2d$
$\therefore$ centroid of tetrahedron
$=\frac{x_1+x_2+x_3+x_4}{4},\frac{y_1+y_2+y_3+y_4}{4}$
$\frac{z_1+z_2+z_3+z_4}{4}$
$\therefore 2=\frac{x_1+x_2+x_3+x_4}{4}$,
$3=\frac{x_1+x_2+x_3+x_4+4d}{4}$
and $k=\frac{x_1+x_2+x_3+x_4+8d}{4}$
$\therefore x_1+x_2+x_3+x_4=8,d=1$
$\Rightarrow k=\frac{8+8}{4}=4$
$\therefore$ Distance of $G$ from the origion
$=\sqrt{2^2+3^2+4^2}=\sqrt{29}$