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Question

Prove that if the diagonals of a quadrilateral are equal and are perpendicular bisectors to each other, then it is a square.[4 MARKS]

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Solution

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]

ΔAOBΔCOB [SAS congruency rule]

AB=BC [CPCTC]

Similarly

ΔCOBΔCODCB=CD

ΔCODΔAODCD=AD

ΔAODΔAOBAD=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.


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