KCET · Maths · Circle
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
- A \(\pi-2\)
- B \(\pi+2\)
- C \(\frac{\pi}{3}\)
- D \(\frac{2 \pi}{3}\)
Answer & Solution
Correct Answer
(A) \(\pi-2\)
Step-by-step Solution
Detailed explanation
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
\[
\begin{array}{lc}
& =\text { length of the arc of the semicircle } \\
\therefore & \frac{\theta}{360^{\circ}} \times 2 \pi r+2 r=\pi r \\
\Rightarrow & \theta+2=\pi \\
\Rightarrow & \theta=\pi-2
\end{array}
\]
\(\therefore\) Length of an arc \(=\frac{\theta}{360^{\circ}} \times 2 \pi \mathrm{r}\)
Since, perimeter of a sector of a circle
\[
\begin{array}{lc}
& =\text { length of the arc of the semicircle } \\
\therefore & \frac{\theta}{360^{\circ}} \times 2 \pi r+2 r=\pi r \\
\Rightarrow & \theta+2=\pi \\
\Rightarrow & \theta=\pi-2
\end{array}
\]
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