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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.
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Solution
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°
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