Question

If the diagonals of a cyclic quadrilateral are diameter of the circle through the opposite vertices of the quadrilateral. Prove that the quadrilateral is a rectangle

Answer 1

Let ABCD be a cyclic quadrilateral having diagonals BD and AC, intersecting each other at point O.

(Consider BD as a chord)

∠BCD + ∠BAD = 180° (Cyclic quadrilateral)

∠BCD = 180° − 90° = 90°

(Considering AC as a chord)

∠ADC + ∠ABC = 180° (Cyclic quadrilateral)

90° + ∠ABC = 180°

∠ABC = 90°

Each interior angle of a cyclic quadrilateral is of 90°. Hence, it is a rectangle.

Answer 2

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.