Q. 34.3( 560 Votes )

Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

Answer :

To Prove: If diagonals of a quadrilateral bisect at 90º, it is a rhombus. 

Figure: 

Definition of Rhombus:
A parallelogram whose all sides are equal.
Given: Let ABCD be a quadrilateral whose diagonals bisect at 90º

In ΔAOD and ΔCOD,

OA = OC (Diagonals bisect each other)

AOD = COD (Given)

OD = OD (Common)

ΔAOD ΔCOD (By SAS congruence rule)

AD = CD            ..................(1)

Similarly,

AD = AB and CD = BC      ..................(2)

From equations (1) and (2),

AB = BC = CD = AD

Since opposite sides of quadrilateral ABCD are equal, it can be said that ABCD is a parallelogram. Since all sides of a parallelogram ABCD are equal, it can be said that

ABCD is a rhombus

Hence, Proved.

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