Question

If the centroid of tetrahedron $OABC$ where $A,B,C$ are given by $(a,2,3),(1,b,2)$ and $(2,1,c)$ respectively is $(1,2,-2)$, then distance of $P(a,b,c)$ from origin is

• A. $\sqrt{195}$
• B. $\sqrt{14}$
• C. $\sqrt{\frac{107}{14}}$
• D. $\sqrt{13}$

Answer 1

The correct option is A $\sqrt{195}$

The Centroid of tetrahedron $OABC$ with vertices $O(0,0,0),A(a,2,3),B(1,b,2)$ and $C(2,1,c)$ is $G(\frac{a+3}{4},\frac{b+3}{4},\frac{5+c}{4})$.
Comparing the coordinates of the Centroid $G$ with the given coordinates of centroid $(1,2,-2)$, we get
$a=1,b=5,c=-13$
$\therefore$ distance of $P(1,5,-13)$ from origin $O(0,0,0)$ is
$D:\sqrt{1^2+5^2+(-13{)}^2}=\sqrt{195}$