Question

The perimeter of a certain sector of a circle is equal to half that of the circle of which it is a sector. The circular measure of one angle of the sector is

• A. $(\pi -2)$ radian
• B. $(\pi +2)$ radian
• C. $\pi$ radian
• D. $(\pi -3)$ radian

Answer 1

The correct option is A $(\pi -2)$ radian

Perimeter of sector $OPQ=\frac{1}{2}\times 2\pi r$

$\Rightarrow \frac{\theta }{{360}^0}\times 2\pi r+2r=\pi r$

$\Rightarrow \frac{\theta }{{360}^0}=\frac{(\pi -2)r}{2\pi r}\Rightarrow \theta =\frac{-{360}^0(\pi -2)}{2\pi }=\frac{180(\pi -2)}{\pi }$degree

$\Rightarrow \theta =\frac{{180}^0(\pi -2)}{\pi }\times \frac{\pi }{{180}^0}=(\pi -2)$radians