Question

In the diagram below, MNPQ is a parallelogram whose diagonals are perpendicular. Prove: MNPQ is a rhombus.

Answer 1

Answer:

Given : MNPQ is a parallelogram whose diagonals are perpendicular.

To prove :  MNPQ is a rhombus.

Proof:

In parallelogram MNPQ,

R is the intersection point of the diagonals MP and NQ( shown in below diagram)

$\implies MR\cong RP$ (Because, the diagonals of parallelogram bisects each other)

$\angle MRQ\cong \angle QRP$ (Right angles )

$QR\cong QR$  (Reflexive)

Thus, By SAS postulate of congruence,

$\triangle MRQ\cong \triangle PRQ$

By CPCTC,

$MQ\cong QP$

Similarly,

We can prove, $\triangle MRN\cong \triangle PRN$

By CPCTC,

$MN\cong NP$

But, By the definition of parallelogram,

$MN\cong QP$ and $MQ\cong NP$

⇒ $MN\cong NP\cong PQ\cong MQ$

All four side of parallelogram MNQP are congruent.

⇒ Parallelogram MNPQ is a rhombus.

Hence, proved.