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: Use sum of coordinates:
The centroid coordinates times 4 give the sum of vertex coordinates: $4(1,2,-1)=(4,8,-4)$.

Step 2: Add the vertices:
x: $0+a+1+2=4$ so $a=1$. y: $0+2+b+1=8$ so $b=5$. z: $0+3+2+c=-4$ so $c=-9$.

Step 3: Compute the distance:
$|OP|^2=1^2+5^2+(-9)^2=107$, so the distance is $\sqrt{107}$ units. Option B.

Final Answer:
The vertices sum to (4, 8, -4), giving P(1, 5, -9) and distance sqrt 107. \[ \boxed{\text{(B) }\sqrt{107}\ \text{units}} \]