Prove that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.

Given:- ABCD is a rectangle i.e AB = CD and AD = BC

To Prove:- ABCD is a square only if its diagonal are perpendicular

Proof:- Let A be at the origin

Now,

By parallelogram law of vector addition,

⇒

Since in rectangle opposite sides are equal BC = AD

⇒

⇒

and

⇒

Negative sign as vector is opposite

⇒

⇒

Diagonals are perpendicular to each other only

⇒

⇒

⇒

⇒

⇒ AB² = AD²

⇒ AB = AD

Hence all sides are equal if diagonals are perpendicular to each

other

ABCD is a square

Hence proved