Prove that if the diagonals of a quadrilateral are equal and are perpendicular bisectors to each other, then it is a square.[4 MARKS]
Concept : 1 Mark
Application : 1 Mark
Proof : 2 Marks
Let ABCD be a quadrilateral whose diagonal AC and BD are equal and are perpendicular bisector of each other.
Thus, ∠AOB=∠BOC=∠COD=∠DOA=90°
and AO=BO=CO=DO.
Consider triangles ΔAOB and ΔBOC, COD and DOA
AO=CO [Given]
BO=BO [Common]
∠AOB=∠BOC [Given]
⇒ΔAOB≅ΔCOB [SAS congruency rule]
AB=BC [CPCTC]
Similarly
ΔCOB≅ΔCOD⇒CB=CD
ΔCOD≅ΔAOD⇒CD=AD
ΔAOD≅ΔAOB⇒AD=AB
∴AB=BC=CD=DA -------(i)
Moreover, since all four are isosceles right triangles.
∠DAO=∠BAO=45∘
Or, ∠DAB=90∘ -----(ii)
Combining (i) and (ii), we can say that the quadrilateral ABCD is a square.