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