Prove that when the point s A (7, 3), B (9, 6), C (10, 12) and D (8, 9) are joined in order, then they will form a parallelogram.

We know that a quadrilateral is a parallelogram if the co-ordinates of mid-point s of its both the diagonals are same.

Therefore, we’ll find the mid-point s of diagonal AC and BD.

Let the co-ordinates of mid-point of AC be (x₃, y3).

And, we know mid-point formula, i.e. the coordinates of mid-point of line joining (x₁, y₁) and (x₂, y₂) is

(Where, (x₁, y₁) and (x₂, y₂) are the coordinates of A and C

⇒ Co – ordinates of mid-point of AC =

Now, Let the co-ordinates of mid-point of BD be (x₄, y₄).

And, since it is a mid-point –

(Where, (x₅, y₅) and (x₆, y₆) are the coordinates of B and D.

⇒ Co – ordinates of mid-point of BD =

And, since these are equal –

⇒ ABCD is a parallelogram.