Question

AB is a diameter and AC is a chord of a circle such that angle BAC = 30°. If the tangent at Christmas intersects AB produced at D , then prove that BC = BD

Answer 1

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Answer 2

Answer:

AB is a diameter and AC is a chord of a circle. BC = BD is proved

Solution:

The diagram for this sum is attached below

We have to prove that BC = BD

∠ACB = 90° (Since AB is the diameter of the circle)

In ∆ACB

∠A + ∠C + ∠B = 180°

30°+ 90° + ∠B = 180°

120° + ∠B = 180°

∠B = 180°- 120°

∠B = 60°

∠CBA = 60°…………….(Equation 1)

Now in ∆OCB,

OC = OB

So ∠OCB = ∠OBC

∴∠OCB = 60°

Now ∠OCD = 90° (Perpendicular from center of tangent)

∠OCB + ∠BCD = 90°

60°+ ∠BCD = 90°

∠BCD = 90°- 60°

∠BCD = 30°………………..(Equation 2)

∠CBO = ∠BCD + ∠CDB

60°=30° + ∠CDB

∠CDB= 30°……………..(Equation 3)

From equation 2 and equation 3

BC =BD

Thus BC = BD is proved