A chord subtends an angle $120^{\circ}$ at the centre of a unit circle. What is the length of the chord?

• A

  $\sqrt{2}-1$ units
• B

  $\sqrt{3}-1$ units
• Correct answer
  C

  $\sqrt{2}$ units
• D

  $\sqrt{3}$ units

Solution

Here $A B$ is the chord and $\angle A O B=120^{\circ}$ $A O=O B=1$ units (Radius)
From cosine rule, $\cos (\angle A O B)=\frac{(O A)^2+(O B)^2-(A B)^2}{2(A O) \cdot(O B)}$ $\Rightarrow \cos \left(120^{\circ}\right)=\frac{1^2+1^2-(A B)^2}{2(1) \cdot(1)}$ $\begin{aligned} & \Rightarrow \frac{1}{2}=\frac{2-(A B)^2}{2} \\ & \therefore A B=\sqrt{2} \text { units. }\end{aligned}$

Hence, the correct answer is option: 'C'.