Prove that If the diagonals of a parallelogram are perpendicular, then it is a rhombus.

Explanation

Let ABCD be the given parallelogram.
Consider
`/_\AOD` and `/_\COD`
AO=CO
(Diagonals of a parallelogram bisect each other)
`/_AOD=/_COD=90^@`
DO=DO
`:. /_\AOD ≅ /_\COD` [SAS congruency rule]
⟹AD=CD [CPCT]
Since adjacent sides of the parallelogram are equal, we can conclude that all four sides are equal.
Hence, a parallelogram whose diagonals are perpendicular to each other is a rhombus