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 D 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 D (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)