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?
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