AB is the tangent on the circle at point A. The line BC meets the circle at points C and E. Line AD bisects the angle EAC. If angle EAC = 60° and angle BAC : angle ACB = 2: 5. Find angle ABC.
Explanation:
Let ∠CAB and ∠ACB be 2x and 5x respectively.
From the alternate segment thorem (In any circle, the angle between a chord and a tangent through one of the end points of the chord is equal to the angle in the alternate segment), we can deduce that ∠AEC = ∠CAB = 2x. In ∆AEC, ∠AEC = 2x, ∠EAC = 30 × 2 = 60° and ∠ACE = 180 − 5x. ∴ 2x + 60 + (180 − 5x) = 180. ∴ x = 20. ∠ABC = 180 − (∠BAC + ∠ACB) = 180 − (2x + 5x) = 180 − 7x = 180 − (7 × 20) = 40°. Hence option (a).
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