Question

Show that if the diagnol of a quadrilateral bisect each other at right angles,then it is a rombus​

Answer 1



Sol:   We have a quadrilateral ABCD such that the diagonals AC and BD bisect each other at right angles at O.

           ∴ In ΔAOB and ΔAOD, we have

                AO = AO

[Common]

                OB = OD

[Given that O in the mid-point of BD]

                ∠AOB = ∠AOD

[Each = 90°]

                ΔAOB ≌ ΔAOD

[SAS criteria]

           Their corresponding parts are equal.

 AB = AD...........(1)

Similarly,

AB = BC...........(2)

 

BC = CD..........(3)

 

CD = AD...........(4)

           ∴ From (1), (2), (3) and (4), we have AB = BC CD = DA

           Thus, the quadrilateral ABCD is a rhombus