What is the fourth vertex D?

For the following two (02) items: Let A(1,-1,0), B(-2,1,8) and C(-1,2,7) are three consecutive vertices of a parallelogram ABCD.

  1. A. (0,-2,1)
  2. B. (2,0,-1) ✓
  3. C. (1, 0, 1)
  4. D. (1, 2, 0)

Correct Answer: B. (2,0,-1)

Explanation

In a parallelogram, the diagonals bisect each other, so the midpoint of AC is the same as the midpoint of BD. Midpoint of AC = (\frac{1-1}{2}, \frac{-1+2}{2}, \frac{0+7}{2}) = (0, \frac{1}{2}, \frac{7}{2}). Let D be (x, y, z). The midpoint of BD is (\frac{x-2}{2}, \frac{y+1}{2}, \frac{z+8}{2}). Equating them: \frac{x-2}{2} = 0 \implies x=2, \frac{y+1}{2} = \frac{1}{2} \implies y=0, \frac{z+8}{2} = \frac{7}{2} \implies z=-1. Thus, D is (2, 0, -1).

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