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?

Guest Oct 24, 2019

#1**+1 **

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?

Omi67 Oct 24, 2019

#2**+1 **

Area of sector = (1/2)r^2 * theta

Perimeter of sector = 2r + r *theta = r [ 2 + theta]

Where theta is the measure of the sector's central angle in radians

So

49 = (1/2)r^2 theta ⇒ 49 / [(1/2) r^2] = theta (1)

28 = r [ 2 + theta] (2)

Subbing (1) into (2) we have that

28 = r [ 2 + 49 / [(1/2) r^2] ] simplify

28 = 2r + 98/r

14 = r + 49/r multiply through by r

14r = r^2 + 49

r^2 - 14r + 49 = 0

(r - 7)^2 = 0

r - 7 = 0

r = 7

So....the arc length of the sector =

28 - 2(7) =

28 - 14 =

14 cm

CPhill Oct 24, 2019