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Question

Prove that if the diagonals of a parallelogram are equal, then it is a rectangle. [4 MARKS]

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Solution

Concept : 1 Mark
Application : 1 Mark
Proof : 2 Marks



Let ABCD be the parallelogram with diagonal AC = diagonal BD. Let O be their point of intersection.

We know that diagonals bisect each other in a parallelogram. It is also given that these diagonals are equal.

AO=BO=CO=DO.

Let us assume that ADO=x and CDO=y

DAO=x and DCO=y
[Since ΔCOD,ΔDOA are isosceles.]


Also, DAO=BCO=x and DCO=BAO=y
[Pair of alternate angles]

CBO=x and ABO=y
[Since ΔCOB,ΔBOA are isosceles.]

A+B+C+D=360 [By angle sum property]

i.e., 4x+4y=360

x+y=90.

Thus, the given parallelogram is a rectangle.


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