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...

Step-by-step solution

The coordinates of the centroid of a tetrahedron with vertices (x₁, y₁, z₁) , (x₂, y₂, z₂) , (x₃, y₃, z₃) , and (x₄, y₄, z₄) are given by ( x₁ + x₂ + x₃ + x₄ 4 , y₁ + y₂ + y₃ + y₄ 4 , z₁ + z₂ + z₃ + z₄ 4 ) . Given the vertices O(0, 0, 0) , A(a, 2, 3) , B(1, b, 2) , and C(2, 1, c) , and the centroid (1, 2, -1) , equating the coordinates gives: 0 + a + 1 + 2 4 = 1 a + 3 = 4 a = 1 0 + 2 + b + 1 4 = 2 b + 3 = 8 b = 5 0 + 3 + 2 + c 4 = -1 c + 5 = -4 c = -9 The point P(a, b, c) is (1, 5, -9) . The distance of P from the