The perimeter of the shaded region is 18√15 cm (6π + 3).
Divide the shaded region into smaller parts:
Divide OPQ into 5 equal sectors by drawing radii from O to points on PQ.
Divide OAB into 5 equal sectors by drawing radii from O to points on AB.
Area of each small sector:
Let r be the radius of the circle.
Area of sector OPQ = 1/5 * π * r^2
Area of sector OAB = 1/5 * π * r^2
Area of shaded region:
Area of shaded region = (4/5 * π * r^2) - (1/5 * π * r^2)
Area of shaded region = 3/5 * π * r^2
Perimeter of shaded region:
Perimeter = length of PO + length of OB + length of arc PQ - length of arc AB
Calculate each length:
PO = 3/5 * r
OB = 3/5 * r
Arc PQ = 4/5 * 2πr (4/5 of the circumference)
Arc AB = 1/5 * 2πr (1/5 of the circumference)
Perimeter of shaded region:
Perimeter = 3/5 * r + 3/5 * r + 4/5 * 2πr - 1/5 * 2πr
Perimeter = 6/5 * r + 6πr/5
Perimeter = 6πr/5 + 6r/5
Substitute the given information:
Area of shaded region = 3/5 * π * r^2 = 81 cm²
Solve for r: π * r^2 = (81 cm² * 5) / 3 r^2 = 135 cm² r = √135 cm = 3√15 cm
Substitute r back into the perimeter formula:
Perimeter = 6π * 3√15 cm / 5 + 6 * 3√15 cm / 5
Perimeter = 6π√15 cm + 18√15 cm
Perimeter = 18√15 cm (6π + 3)
Therefore, the perimeter of the shaded region is 18√15 cm (6π + 3).
Complete question:
OPQ is a sector of a circle, centre O OAB is a sector of a circle, centre O A is the point on OP such that OA:AP=3:2 B is the point on OQ such that OB:BQ=3:1 Angle POQ=45° The area of the shaded region is 81/2 π cm^2 Work out the perimeter of the shaded region Give your answer in terms ofπ.