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In the figure given below (not drawn to scale) A, B and C are three points on a circle with center O. The chord BA is extended to a point T such that CT becomes a tangent to the circle at point C. If ∠ATC = 30° and ∠ACT = 50°, then ∠ BOA is

(a) 100° 

(b) 150° 

(c) 80° 

(d) not possible to determine

1 Answer

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Answer : (a) 100º 

In ∆ ATC, ∠CAT = 180° – (∠ACT + ∠ATC) 

= 180° – (50° + 30°) = 100°

Also, ∠CBA = ∠ACT = 50° (Alternate Segment Theorem)

∴  ext ∠CAT = int. opp. ∠s  (∠CBA + ∠BCA) 

⇒ 100° = 50° + ∠BCA 

⇒ ∠BCA = 50° 

⇒ ∠BOA = 2 × ∠BCA = 100°.

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