Question

If the centroid of a tetrahedron \(OABC\) is \((1,2,-1)\), where O is the origin, \(A(a,2,3),B(1,b,2),C(2,1,c)\) are the other vertices, then the distance of the point \(P(a,b,c)\) from the origin is...

• A. \(42\) units
• B. \(\sqrt{107}\) units
• C. \(25\) units
• D. \(15\) units

Hint:

The centroid is the mean of the four vertices; solve for a, b, c.

Answer 1

The Correct Option is B

Step 1: Centroid formula:
For a tetrahedron with vertices \(O(0,0,0)\), \(A(a,2,3)\), \(B(1,b,2)\), \(C(2,1,c)\), the centroid is \(\left(\dfrac{a+1+2}{4},\dfrac{2+b+1}{4},\dfrac{3+2+c}{4}\right)\).

Step 2: Equate with (1, 2, -1):
\(\dfrac{a+3}{4}=1\) gives \(a=1\). \(\dfrac{b+3}{4}=2\) gives \(b=5\). \(\dfrac{c+5}{4}=-1\) gives \(c=-9\).

Step 3: Distance of P(1, 5, -9) from the origin:
\(OP=\sqrt{1+25+81}=\sqrt{107}\). Option B.

Step 4: Why the other options are wrong.
42, 25 and 15 are not square roots of 107. They arise from adding squares without taking the square root or from wrong values of c.

Final Answer:
The distance is sqrt 107 units. \[ \boxed{\text{(B) }\sqrt{107}\ \text{units}} \]