NCERT Solutions
Class 11 Maths
Introduction to Three Dimensional Geometry

Misc. Q.1
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.
The three vertices of a parallelogram ABCD are given as A (3, –1, 2), B (1, 2, –4), and C (–1, 1, 2).
Let the coordinates of the fourth vertex be D (x, y, z).
We know that the diagonals of a parallelogram bisect each other.
Therefore, in parallelogram ABCD, AC and BD bisect each other.
So, Mid-point of AC = Mid-point of BD
{(3 - 1) ÷ 2, (-1 + 1) ÷ 2, (2 + 2) ÷ 2} = {(x + 1) ÷ 2, (y + 2) ÷ 2, (z - 4) ÷ 2}
=> (1, 0, 2) = {(x + 1) ÷ 2, (y + 2) ÷ 2, (z - 4) ÷ 2}
=> (x + 1) ÷ 2 = 1, (y + 2) ÷2 = 0, (z - 4) ÷ 2 = 2
=> x = 1, y = –2, and z = 8
Thus, the coordinates of the fourth vertex are (1, –2, 8).