Multiple choice

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

  1. $\sqrt{195}$
  2. $\sqrt{14}$
  3. $\sqrt{\dfrac{107}{14}}$
  4. $\sqrt{13}$
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A Correct answer
Explanation

The centroid of a tetrahedron with vertices (x1,y1,z1), (x2,y2,z2), (x3,y3,z3), (x4,y4,z4) is the average of the coordinates. Given O(0,0,0), A(a,2,3), B(1,b,2), C(2,1,c), the centroid is ((a+1+2)/4, (2+b+1)/4, (3+2+c)/4) = (1,2,-2). Solving gives a=1, b=5, c=-13. The distance of P(1,5,-13) from origin is sqrt(1^2 + 5^2 + (-13)^2) = sqrt(1+25+169) = sqrt(195).

AI explanation

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 given by ((a plus 1 plus 2) divided by 4, (2 plus b plus 1) divided by 4, (3 plus 2 plus c) divided by 4). Equating this to (1, 2, minus 2) yields a equals 1, b equals 5, and c equals minus 13. The distance of the point P(a, b, c) from the origin is the square root of (1 squared plus 5 squared plus (minus 13) squared). Calculating this gives the square root of (1 plus 25 plus 169), which is the square root of 195.

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