Given      A chord AB of a circle C (O,r) and perpendicular OD to the chord AB.

TO PROVE   AD = DB

CONSTRUCTION   Join OA and OB.

PROOF    In triangles AOD and BOD, we have

                          [ Radii of the same circle ]

                                   [ Common ]

and,              [ Each equal to ]

So, by RHS - criterion of congruence, we have

Hence Proved

Given      A chord AB of a circle C (O,r) and a line OD which bisects the chord AB.

TO PROVE   OD is perpendicular to AB

CONSTRUCTION   Join OA and OB.

PROOF    In triangles AOD and BOD, we have

                          [ Radii of the same circle ]

                                   [ Common ]

and,                             [ Given ]

So, by SSS - criterion of congruence, we have

 But they are linear pairs so

 = 

     Hence Proved

Illustration: If the length of a chord of a circle is 24 cm and is at a distance of 9 cm from the centre of the circle, Find the radius of the circle .

The length of a chord of a circle  AB = 24 cm

Perpendicular distance from the centre OD =  9 cm

As the perpendicular bisects the chord AD = 12 cm

OA = 15 cm