question_answer

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

A) \[2R\sin \left( \frac{\theta }{2} \right)\]

B) \[2R\sin \theta \]

C) \[2R\tan \left( \frac{\theta }{2} \right)\]

D) \[2R\tan \theta \]

Correct Answer: A

Solution :

[a] Let there be a circle of radius R and AB a chord. \[OD\bot AB\] and \[AD=DB.\] And \[AD=2AD\] \[\angle AOB=\theta \] \[\Rightarrow \angle AOD=\frac{\theta }{2}\] In \[\Delta AOD,\] \[\sin \frac{\theta }{2}=\frac{AD}{OA}\] \[\sin \frac{\theta }{2}=\frac{AD}{R}\] \[AD=R\sin \frac{\theta }{2}\] \[\therefore \] Length of chord \[AB=2AD=2R\sin \frac{\theta }{2}.\]