Question

if diagonals of a cyclic quadrilateral are diameters of the circle and perpendicular to each other,prove that it is a square.​

Answer 1

Step-by-step explanation:

Given that,

A cyclic quadrilateral .

ABCD,AC and BD$ are diameters of the circle where they meet at center O of the circle.

To prove:ABCD is a rectangle.

Proof: In triangle ΔAOD 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

0

(angle in a semicircle)

∴ ABCD is a rectangle.