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.

Given: ABCD is a parallelogram, with vertices A (3, -1, 2), B (1, 2, -4), C (-1, 1, 2).

⇒ x1 = 3, y1 = -1, z1 = 2; x2 = 1, y2 = 2, z2 = -4; x3 = -1, y3 = 1, z3 = 2



Let the coordinates of the fourth vertex be D (x, y, z).


We know that the diagonals of a parallelogram bisect each other, so the mid points of AC and BD are equal, i.e. Midpoint of AC = Midpoint of BD ……….(i)


Now, by Midpoint Formula, we know that the coordinates of the mid-point of the line segment joining two points P (x1, y1, z1) and Q (x2, y2, z2) are .


So, we have


Coordinates of the midpoint of AC


Coordinates of the midpoint of BD


So, using (i), we have




⇒ 1 + x = 2, 2 + y = 0, -4 + z = 4


⇒ x = 1, y = -2, z = 8


Hence, the coordinates of the fourth vertex is D (1, -2, 8).


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