Question

The perimeter of a sector of a circle is the sum of the two sides formed by the radii and the length of the included arc. A sector of a particular circle has a perimeter of 28 cm and an area of 49 sq cm. What is the length of the arc of this sector?

Answer 1

Answer:

Length of the arc of this sector, l = 14 cm

Explanation:

It is given that, the perimeter of a sector of a circle is the sum of the two sides formed by the radii and the length of the included arc.

Perimeter of sector, P = 28 cm

Area of sector, $A=49\ cm^2$

According to figure,

2r + l = 28 ............(1)

Area of sector, $A=\dfrac{\theta}{360}\times \pi r^2$

Where, $\theta$ is in radian and $\theta=\dfrac{l}{r}$

Since, $1^{\circ}=\dfrac{\pi}{180}\ radian$

$A=\dfrac{l}{2\pi r}\times \pi r^2$

$r=\dfrac{98}{l}$

Put the value of r in equation (1) so,

$2\times (\dfrac{98}{l})+l=28$

$l^2-28l+196=0$

On solving above equation for l we get, l = 14 cm. So, the length of the arc of this sector is 14 cm. Hence, this is the required solution.