Question

Show that the diagonals of a square are equal and bisect each other at right angles.

Answer 1

Let ABCD be a quadrilateral, whose diagonals AC and BD bisect each other at right angle i.e., OA = OC, OB = OD, and ∠AOB = ∠BOC = ∠COD = ∠AOD = 90º. To prove ABCD a rhombus, we have to prove ABCD is a parallelogram and all the sides of ABCD are equal. 

In ∆AOD and ∆COD, 

OA = OC (Diagonals bisect each other) 

∠AOD = ∠COD (Given) 

OD = OD (Common) 

∠∆AOD ∠∆COD (By SAS congruence rule) 

∴ AD = CD (1) 

Similarly, it can be proved that 

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.