Question

# AB is a diameter of a circle and AC is its chord such that ∠BAC = 30°. If the tangent at C intersects AB extended at D, then BC = ________.

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Solution

## AB is a diameter of circle with centre O and AC is the chord of the circle such that ∠BAC = 30°. The tangent at C intersects AB extended at D. In ∆OAC, OA = OC (Radius of circle) ∴ ∠OCA = ∠OAC (In a triangle, equal sides have equal angles opposite to them) ⇒ ∠OCA = 30° Now, ∠OCD = 90° (Radius is perpendicular to the tangent at the point of contact) ∴ ∠ACD = ∠OCA + ∠OCD = 30° + 90° = 120° In ∆ACD, ∠A + ∠ACD + ∠D = 180º (Angle sum property) ⇒ 30° + 120° + ∠D = 180º ⇒ ∠D = 180º − 150° = 30° .....(1) Now, ∠ACB = 90° (Angle in a semi-circle is 90°) ∠ACD = ∠ACB + ∠BCD ⇒ 120° = 90° + ∠BCD ⇒ ∠BCD = 120º − 90° = 30° .....(2) In ∆BCD, ∠D = ∠BCD [From (1) and (2)] ⇒ BC = BD (In a triangle, equal angles have equal sides opposite to them) AB is a diameter of a circle and AC is its chord such that ∠BAC = 30°. If the tangent at C intersects AB extended at D, then BC = _ BD _.

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