Question

If \(G(2,-1,2)\) is the centroid of tetrahedron \(OABC\) where \(O = (0,0,0)\) and \(G_1\) is the centroid of \(\Delta ABC\), then \(|OG_1| = ?\)

• A. 1
• B. \(\frac{3}{2}\)
• C. 4
• D. \(\frac{9}{2}\)

Hint:

For a tetrahedron, relate the centroid of the tetrahedron to the centroid of a face to find distances from the origin or vertices.

Answer 1

The Correct Option is C

Step 1: Use the fact that O is the origin.
For a tetrahedron \(OABC\), the centroid is \(G = \frac{O+A+B+C}{4}\). Since \(O=(0,0,0)\), this simplifies directly to
\[ G = \frac{A+B+C}{4} \implies A+B+C = 4G \]

Step 2: Write the centroid of triangle ABC.
\[ G_1 = \frac{A+B+C}{3} = \frac{4G}{3} \]

Step 3: Substitute the given point G.
\[ G_1 = \frac{4}{3}(2,-1,2) = \left(\frac{8}{3},-\frac{4}{3},\frac{8}{3}\right) \]

Step 4: Find the distance from the origin.
\[ |OG_1| = \sqrt{\left(\frac{8}{3}\right)^2+\left(\frac{4}{3}\right)^2+\left(\frac{8}{3}\right)^2} = \sqrt{\frac{64+16+64}{9}} = \sqrt{\frac{144}{9}} = \sqrt{16} \]

Step 5: State the final value.
\[ \boxed{4} \]