Question

only genius can answer show that if the diagonals of a quadilateral bisect each other at right angles, then it is a rhombus.

Answer 1

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.