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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
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Verified Answer
The correct answer is:
$\pi-2$
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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