In the diagram, O is the centre of the circleand XY is a chord. If the radius is 5 cm and /XY/ = 6 cm, calculate, correct to 2 decimal places, the :
(a) angle which XY subtends at the centre O ;
(b) area of the shaded portion.
Explanation
(a) Let the angle subtended at the centre by XY = (theta)
Length of chord = (2r sin frac{theta}{2})
(implies 6 cm = 2 times 5 times frac{theta}{2})
(sin frac{theta}{2} = frac{6}{10} = 0.6)
(frac{theta}{2} = sin^{-1} (0.6) = 36.87°)
(theta = 2 times 36.87° = 73.74°)
(b) Area of sector = (frac{theta}{360°} times pi r^{2})
= (frac{73.74}{360} times frac{22}{7} times 5^{2})
= (16.094 cm^{2})
Area of triangle = (sqrt{s(s – a)(s – b)(s – c)})
where (s = frac{a + b + c}{2})
( a = 5 cm ; b = 5 cm ; c = 6 cm therefore s = frac{5 + 5 + 6}{2} = frac{16}{2} = 8)
(therefore Area = sqrt{8(8 – 5)(8 – 5)(8 – 6)})
= (sqrt{8(3)(3)(2)})
= (sqrt{144})
= (12 cm^{2})
Area of shaded portion = (16.094 cm^{2} – 12 cm^{2} = 4.094 cm^{2})