The perimeter of a sector OAB of a circle of centre O is 291 cm. If the radius is 63 cm, calculate, correct to one decimal place, the: (a) angle subtended by the sector; (b) length of the chord AB; (c) area of the sector. [Take π = 22/7]

Here's the solution for Question 10: Step 1: Calculate the arc length of the sector. The perimeter of a sector is given by the formula P = 2r + L, where r is the radius and L is the arc length. Given P = 291 cm and r = 63 cm. 291 = 2(63) + L 291 = 126 + L L = 291 - 126 = 165 cm Step 2: Calculate the angle subtended by the sector in radians. The arc length is also given by L = r, where is the angle in radians. 165 = 63 = (165)/(63) radians Step 3: Convert the angle to degrees for part (a). To convert radians to degrees, multiply by (180^)/(). Use = (22)/(7). = (165)/(63) × (180^)/(22)7 = (165)/(63) × (180^ × 7)/(22) = (165 × 180^ × 7)/(63 × 22) = (165 × 180^)/(9 × 22) = (15 × 180^)/(9 × 2) = (15 × 90^)/(9) = 15 × 10^ = 150^ a) The angle subtended by the sector is 150.0^. Step 4: Calculate the length of the chord AB for part (b). The length of a chord in a circle is given by 2r (()/(2)), where is the angle in degrees. Length of chord AB = 2 × 63 × ((150^)/(2)) Length of chord AB = 126 × (75^) Using a calculator, (75^) ≈ 0.9659258. Length of chord AB = 126 × 0.9659258 ≈ 121.69965 cm Correct to one decimal place: b) The length of the chord AB is 121.7 cm. Step 5: Calculate the area of the sector for part (c). The area of a sector is given by A = ()/(360^) × r^2, where is in degrees. A = (150^)/(360^) × (22)/(7) × (63)^2 A = (5)/(12) × (22)/(7) × 63 × 63 A = (5)/(12) × 22 × 9 × 63 A = (5 × 11 × 2 × 9 × 63)/(12) A = (5 × 11 × 9 × 63)/(6) A = (5 × 11 × 3 × 63)/(2) A = (10395)/(2) = 5197.5 cm^2 Correct to one decimal place: c) The area of the sector is 5197.5 cm^2. Last free one today — make it count tomorrow, or type /upgrade for unlimited.