Q74 / 180 · KCET 2008 · Mathematics

The perimeter of a certain sector of a circle is equal to the length of the arc of the semicircle. Then, the angle at the centre of the sector in radians is

  • Correct answer
    A
    $\pi-2$
  • B
    $\pi+2$
  • C
    $\frac{\pi}{3}$
  • D
    $\frac{2 \pi}{3}$

Solution

Let the radius of circle be $r$. $\therefore$ Length of an arc $=\frac{\theta}{360^{\circ}} \times 2 \pi \mathrm{r}$ Since, perimeter of a sector of a circle $$ \begin{array}{lc} & =\text { length of the arc of the semicircle } \\ \therefore & \frac{\theta}{360^{\circ}} \times 2 \pi r+2 r=\pi r \\ \Rightarrow & \theta+2=\pi \\ \Rightarrow & \theta=\pi-2 \end{array} $$

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

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