Question

In the following figure, AD is the diameter of the circle with centre O. chords AB, BC and CD are equal. If ∠DEF = 110°, Calculate: ∠FAB.

Answer 1

Join AE , OB and OC

∵ Chord AB = Chord BC = Chord CD    ...[given]

∴ ∠AOB = ∠BOC = ∠COD    ...(Equal chords subtends equal angles at the centre)

But ∠AOB + ∠BOC + ∠COD = 180°    ...[AOD is a straight line ]

∠AOB = ∠BOC = ∠COD = 60°

In ∠OAB, OA = OB

∴ ∠OAB = ∠OBA    ...[radii of the same circle]

But ∠OAB + ∠OBA = 180° −  AOB

= 180° − 60°

= 120°

∴ ∠OAB = ∠OBA = 60°

In cyclic quadrilateral ADEF,

∠DEF + ∠DAF = 180°

⇒ ∠DAF = 180° − ∠DEF

= 180° − 110°

= 70°

Now, ∠FAB = ∠DAF + ∠OAB

= 70° + 60°

= 130°