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Question: Answered & Verified by Expert
Let $\mathrm{ABCD}$ be a tetrahedron in which the coordinates of each of its vertices are in arithmetic progression with same common difference. If the centroid $\mathrm{G}$ of the tetrahedron is $(2,3, \mathrm{k})$, then the distance of $\mathrm{G}$ from the origin is
MathematicsThree Dimensional GeometryAP EAMCETAP EAMCET 2023 (15 May Shift 2)
Options:
  • A $\sqrt{38}$
  • B $7$
  • C $\sqrt{22}$
  • D $\sqrt{29}$
Solution:
1868 Upvotes Verified Answer
The correct answer is: $\sqrt{29}$
Let coordinates of vertices of tetrahedron are
$\begin{aligned}
& A\left(a_1-d, a_1, a_1+d\right) ; B\left(a_2-d, a_2, a_2+d\right) ; \\
& C\left(a_3-d, a_3, a_3+d\right) \text { and } D\left(a_4-d, a_4, a_4+d\right)
\end{aligned}$
Centroid $=(2,3, k)$
$\begin{aligned}
\therefore \quad & a_1+a_2+a_3+a_4-4 d=4...(i) \\
& a_1+a_2+a_3+a_4=6 ...(ii)\\
& a_1+a_2+a_3+a_4+4 d=2 k...(iii)
\end{aligned}$
From (i), (ii) and (iii) we get $d=\frac{1}{2}$ and $k=4$
Now distance of $G$ from origin $=\sqrt{4+9+16}=\sqrt{29}$.

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