Question

A polygon of nine sides, each side of length $2$, is inscribed in a circle. The radius of the circle is:

• A. $\sec \left(\frac{\pi }{9}\right)$
• B. $\text{cosec}\left(\frac{\pi }{9}\right)$
• C. $\cot \left(\frac{\pi }{9}\right)$
• D. $\tan \left(\frac{\pi }{9}\right)$

Answer 1

The correct option is B $\text{cosec}\left(\frac{\pi }{9}\right)$


Let $A_1{A}_2=2$ units be one of the sides of the polygon.
Central angle $=\frac{{360}^0}{9}={40}^0$
In $\Delta A_{1}MC,$
$\sin {20}^0=\frac{A_{1}M}{A_{1}C}=\frac{1}{r}$
$\therefore$ Radius of the circle is,
$r=\frac{1}{\sin {20}^0}=\text{cosec}\left(\frac{\pi }{9}\right)$