Question

Centroid of the tetrahedron OABC, where coordinates of A, B, C are (a, 2, 3), (1, b, 3) and (2, 1, c) respectively, is (1, 2, 3). Find the distance of the point (a, b, c) from the origin O

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

Centroid of the tetrahedron with vertices $A(x_1,y_1,z_1),$ $B(x_2,y_2,z_2),$ $(x_3,y_2,z_3)$ and $D(x_4,y_4,z_4)$ is given by
$G\equiv (\frac{x_1+x_2+x_3+x_4}{4},\frac{y_1+y_2+y_3+y_4}{4},\frac{z_1+z_2+z_3+z_4}{4})$
We are given (1, 2, 3 ) is the centroid of the tetrahedron with vertices O(0,0,0), A(a, 2, 3), B(1, b, 3) and C(2, 1, c)
Using the relation (1), we get
$G\equiv (\frac{0+a+1+2}{4},\frac{0+2+b+1}{4},\frac{0+3+3+c}{4})$
This is given as (1, 2, 3 )
$\Rightarrow$ $\frac{0+a+1+2}{4}=1,\frac{0+2+b+1}{4}=2$ and $\frac{0+3+3+c}{4}=3$
$\Rightarrow$ $a=2,b=5$ and $c=3$
$\Rightarrow$ Distance of (a, b, c ) from origin = $\sqrt{a^2+b^2+c^2}$
=$\sqrt{38}$