Question

If a circle of unit radius is divided into two parts by an arc of another circle subtending an angle \(60^\circ\) on the circumference of the first circle, then the radius of the arc is:

• A. \(\sqrt3\)
• B. \(\frac12\)
• C. 1
• D. None of these

Hint:

The common chord has length \(2\sin60^\circ\) in the unit circle; apply the sine rule in the second circle.

Answer 1

The Correct Option is C

Let the common points be \(A,B\). The angle \(60^\circ\) subtended on the circumference of the unit circle gives chord \(AB=2\cdot1\cdot\sin60^\circ=\sqrt3\).
The dividing arc also subtends \(60^\circ\) at its own circumference, so for its radius \(R\): \(AB=2R\sin60^\circ\Rightarrow\sqrt3=2R\cdot\frac{\sqrt3}{2}\Rightarrow R=1\).
Answer: (3) 1