Question

If diagonals of a cyclic quadrilateral are diameters of the circle through opposite vertices of the quadrilateral, prove that the quadrilateral is a rectangle.


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Answer 1

Step-by-step explanation:

Given:A cyclic quad.ABCD,AC and BD are diameters of the circle where they meet at centre O of the circle.

To prove:ABCD is a rectangle

Proof:InΔAOD and ΔBOC,

OA=OC(both are radii of same circle)

∠AOD=∠BOC(vert.opp∠s)

OD=OB(both are radii of same circle)

∴ΔAOD≅ΔBOC⇒AD=BC(C.P.C.T)

Similarly,by taking ΔAOB and ΔCOD,AB=DC

Also,∠BAD=∠ABC=∠BCD=∠ADC=90°(angle in a semicircle)

∴ABCD is a rectangle.

Answer 2

Answer is in the attachments