If the points A (6, 1), B (8, 2), C (9, 4) and D (p, 3) taken in order are the vertices of a parallelogram, find the value of p using distance formula.

Formula used:

Let A, B, C and D represent the points (6, 1), (8, 2), (9, 4) and (p, 3)

Distance of AB

⇒ AB = √ ((8 – 6))² + (2 – 1)²)

⇒ AB = √ ((2)² + (1)²)

⇒ AB = √ (4 + 1)

⇒ AB = √ 5

Distance of CD

⇒ CD = √ ((p – 9)² + (3 – 4)²)

⇒ CD = √ ((p – 9)² + (1)²)

⇒ CD = √ (p² + 81 – 18p + 1)

⇒ CD = √ p² – 18p + 82

i.e., The opposite sides are equal.

∴ AB = CD

⇒ √5 = √ p² – 18p + 82

Squaring both sides

⇒ 5 = p² – 18p + 82

⇒ p² – 18p + 82 – 5 =0

⇒ p² – 18p + 77 = 0

⇒ p² – 11p – 7p + 77 = 0

⇒ p(p – 11) – 7(p – 11)= 0

⇒ (p – 11)(p – 7) = 0

p – 11 = 0 or p – 7 = 0

p = 11 or p = 7