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?
+12528 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?
+128460 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
