Question

Prove that if the diagonals of a parallelogram are perpendicular, then it is a rhombus. [3 MARKS]

Answer 1

Concept: 1 Mark
Proof: 2 Marks



Let ABCD be the given parallelogram.

Consider $\Delta AOD$ and $\Delta COD$ .

$AO=CO$ (Diagonals of a parallelogram bisect each other)

$\angle AOD=\angle COD={90}^{\circ }$

$DO=DO$

$\therefore \Delta AOD\cong \Delta COD$ [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.