Question

Three vertices of a parallelogram ABCD are A(3, -1, 2), B(1, 2, -4) and C(-1, 1, 2). Find the coordinates of the fourth vertex D.

Answer 1

Let D(x, y, z) be the required point. Then, the midpoint of diagonal BD is

$(\frac{x+1}{2},\frac{y+2}{2},\frac{z-4}{2})$.

And, the midpoint of diagonal AC is

$(\frac{3-1}{2},\frac{-1+1}{2},\frac{2+2}{2})$, i.e., (1, 0, 2).

But, the midpoints of the diagonals of a parallelogram always coincide

$\therefore \frac{x+1}{2}=1,\frac{y+2}{2}=0$ and $\frac{z-4}{2}=2$.

So, x=1, y=-2 and z=8.

Hence, the required point is D(1, -2, 8).