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 aperimeter of 28 cm and an area of 49 sq cm. What is the length of the arc of this sector?

Guest Nov 17, 2019

#1**+1 **

The area is given by

49 = (1/2)r^2 (theta) where theta is in radians

Multiply both sides by 2 and divide by r^2 and we have that

98 / r^2 = theta (1)

The perimeter of the sector = 2r + r (theta)

So

28 = 2r + r (theta) (2)

Sub (1) into (2) and we have that

28 = 2r + r (98 / r^2) simplify

28 = 2r + 98/r divide through by 2

14 = r + 49/r multiply through by r

14r = r^2 + 49 rearrange as

r^2 - 14r + 49 = 0 factor

(r - 7)^2 = 0 take the square root

r - 7 = 0

r = 7

And theta = 98 / 7^2 = 98 /49 = 2

So....the arc length of the sector is

r * (theta) = 7 (2) = 14 cm

CPhill Nov 17, 2019