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

Explanation:

Given:

AB is the diameter of the circle

AC is a chord and ∠BAC = 30°.

The tangent at C intersects the extended diameter AB at D.

Step 1: Apply the alternate segment theorem

The angle between a tangent (CD) and a chord through the point of contact (BC) is equal to the angle in the alternate segment (∠BAC).

∠BCD = ∠BAC = 30°

Step 2: Find the angles in ΔABC

Since AB is the diameter of the circle, the angle it subtends at any point on the semicircle is a right angle.

∠ACB = 90°

Using the angle sum property in the right-angled ΔABC:

∠ABC = 180° – (∠ACB + ∠BAC)

∠ABC = 180° – (90° + 30°) = 60°

Step 3: Use the exterior angle property for ΔBCD

For ΔBCD, the side DB is extended to A, making ∠ABC the exterior angle at vertex B. The exterior angle of a triangle is equal to the sum of the two opposite interior angles. §

∠ABC = ∠BDC + ∠BCD

Substitute the known values into the equation:

60° = ∠BDC + 30°

∠BDC = 30°

Step 4: Conclude with the isosceles triangle property

In ΔBCD, we have established that two angles are equal:

∠BCD = 30°

∠BDC = 30°

Because the base angles are equal, ΔBCD is an isosceles triangle. Therefore, the sides opposite to these equal angles must also be equal:

BC = BD