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

Solution:

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, diagonals AC and BD bisect each other.

i.e., 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]

Hence,

(x + 1)/2 = 1, (y + 2)/2 = 0 and (z - 4)/2 = 2

⇒ x = 1, y = - 2 and z = 8

Thus, the coordinates of the fourth vertex D are (1, - 2, 8)

NCERT Solutions Class 11 Maths Chapter 12 Exercise ME Question 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

Summary:

The coordinates of the fourth vertex of a parallelogram ABCD whose three vertices are A(3, - 1, 2), B (1, 2, - 4) and C (- 1, 1, 2) are (1, - 2, 8)