Question

Perimeter of a segment is $5(\pi +2)\text{m}.$ If the angle subtended by a segment at the centre of the circle is ${180}^{\circ },$ the the radius of the circle is m.

Answer 1

Let, the given segment is $ACB.$
Given, the perimeter of the segment $=5(\pi +2)\text{m}$
and the angle subtentded by segment at the centre of the circle$\left(\theta \right)={180}^{\circ }$

Let, the radius of the circle be $r,$ then
perimeter of the segment
$=\stackrel{⌢}{ACB}+BA$
$=\frac{\theta }{{360}^{\circ }}\times \pi \times d+2r(\because BA\text{is the diameter of the circle)}=\frac{{180}^{\circ }}{{360}^{\circ }}\times \pi \times 2\times r+2r(\because d=2r)$
$=\frac{1}{2}\times \pi \times 2r+2r$
$=\pi r+2r$
Taking $r$ common
$=r(\pi +2)$
But the perimeter is given as $5(\pi +2)$
$\therefore r(\pi +2)=5(\pi +2)$
Cancel out $(\pi +2)$ from both sides
$\Rightarrow r=5\text{m}$

Hence, the radius of the circle is $5\text{m}.$