Question

Find the possible pairs of co-ordinates of the fourth vertex D of the parallelogram, if three of its vertices are A (5, 6), B (1, -2) and C (3, -2).

Answer 1

Let the points A (5, 6), B (1, -2) and C (3, -2) be the three vertices of a parallelogram.

The fourth vertex can be point D or point Di or point D₂ as shown in the figure.

Let D(x₁, y₁), D, (x₂, y₂) and D (x₃, y₃).

Consider the parallelogram ACBD.

The diagonals of a parallelogram bisect each other. 

∴ midpoint of DC = midpoint of AB

Co-ordinates of point D(x₁, y₁) are (3, 6). 

Consider the parallelogram ABD₁C.

The diagonals of a parallelogram bisect each other. 

∴ midpoint of AD₁ = midpoint of BC

∴ Co-ordinates of D₁(x₂, y₂) are (-1,-10). 

Consider the parallelogram ABCD₂. 

The diagonals of a parallelogram bisect each other.

 ∴ midpoint of BD₂ = midpoint of AC

∴ co-ordinates of point D₂(x₃, y₃) are (7, 6). 

∴ The possible pairs of co-ordinates of the fourth vertex D of the parallelogram are (3, 6), (-1,-10) and (7, 6).