Question

If the centroid of a triangle with vertices \[ (4,p,-3),\quad (-1,-1,2),\quad (3,5,-8) \] is given by the midpoint of \[ (1,4,-2) \] and \[ (q,2,-4), \] then \(p^2+q^2=\)

• A. \(26\)
• B. \(25\)
• C. \(24\)
• D. \(34\)

Hint:

For three-dimensional coordinate geometry, use the centroid formula coordinate-wise and equate it with the given midpoint coordinate-wise.

Answer 1

The Correct Option is D

Step 1: Compute the centroid.
Centroid \(= \left(\dfrac{4-1+3}{3},\dfrac{p-1+5}{3},\dfrac{-3+2-8}{3}\right) = \left(2,\dfrac{p+4}{3},-3\right)\).

Step 2: Compute the midpoint.
Midpoint of \((1,4,-2)\) and \((q,2,-4)\) \(= \left(\dfrac{1+q}{2}, 3, -3\right)\).

Step 3: Solve for p and q.
\(\dfrac{p+4}{3}=3 \Rightarrow p=5\). \(\dfrac{1+q}{2}=2 \Rightarrow q=3\). \[p^2+q^2 = 25+9 = 34\]
\[ \boxed{34} \]