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 = ________.

Answer 1

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 _.