1.

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

Answer»

Data : ABCD is a parallelogram and diagonals AC and BD bisects at right angles at O’. 

To Prove: ABCD is a rhombus. 

Proof: Here, AC and BD bisect each other at right angles. 

∴ AO = OC 

BO = OD 

and ∠AOB = ∠BOC = ∠COD = ∠AOD = 90° 

If sides are euqal to each other, then ABCD is said to be a rhombus. 

Now, ∆AOD and ∆COD, AO = OC (Data) 

∠AOD = ∠COD = 90° (Data) OD is common. 

∴ ∆AOD ≅ ∆COD (SAS Postulate) 

∴ AD = CD …………… (i) 

Similarly, ∆AOD = ∆AOB 

AD = AB ………… (ii) 

∆AOB ≅ ∆COB 

∴ AB = BC ……….. (iii) 

∆COB ≅ ∆COD 

∴ BC = CD ……………. (iv) 

From (i), (ii), (iii) and (iv), 

AB = BC = CD = AD 

All 4 sides of parallelogram ABCD are equal, then it is rhombus.


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