Question

Consider a circle of radius $R$. What is the length of a chord which subtends an angle $\theta$ at the centre?

• A. $2R\sin \frac{\theta }{2}$
• B. $2R\sin \theta$
• C. $2R\tan \frac{\theta }{2}$
• D. $2R\tan \theta$

Answer 1

The correct option is A $2R\sin \frac{\theta }{2}$

In $\Delta AOD$,
$\sin \frac{\theta }{2}=\frac{AD}{AO}\Rightarrow \sin \frac{\theta }{2}=\frac{AD}{R}$
$\Rightarrow AD=R\sin \frac{\theta }{2}$
$\therefore$ Length of chord $AB=2AD=2R\sin \frac{\theta }{2}$