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)
[tex]\implies MR\cong RP[/tex] (Because, the diagonals of parallelogram bisects each other)
[tex]\angle MRQ\cong \angle QRP[/tex] (Right angles )
[tex]QR\cong QR[/tex] (Reflexive)
Thus, By SAS postulate of congruence,
[tex]\triangle MRQ\cong \triangle PRQ[/tex]
By CPCTC,
[tex]MQ\cong QP[/tex]
Similarly,
We can prove, [tex]\triangle MRN\cong \triangle PRN[/tex]
By CPCTC,
[tex]MN\cong NP[/tex]
But, By the definition of parallelogram,
[tex]MN\cong QP[/tex] and [tex]MQ\cong NP[/tex]
⇒ [tex]MN\cong NP\cong PQ\cong MQ[/tex]
All four side of parallelogram MNQP are congruent.
⇒ Parallelogram MNPQ is a rhombus.
Hence, proved.
