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 = \(16.094 cm^{2} - 12 cm^{2} = 4.094 cm^{2}\)