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) ang...

Answer Details

From the diagram, \(O\) is the centre, radius \(OX = OY = 5\ \text{cm}\), chord \(|XY| = 6\ \text{cm}\), and the shaded portion is the minor segment cut off by the chord \(XY\) (the region between the chord and the minor arc on the right).

(a) Angle subtended at the centre, \(\angle XOY\):

Drop the perpendicular from \(O\) to the midpoint \(M\) of \(XY\). Then \(XM = 3\ \text{cm}\) and:

\[\sin\left(\frac{\angle XOY}{2}\right) = \frac{XM}{OX} = \frac{3}{5} = 0.6\]\[\frac{\angle XOY}{2} = \sin^{-1}(0.6) = 36.8699^\circ\]\[\angle XOY = 2 \times 36.8699^\circ = 73.7398^\circ\]

\(\angle XOY \approx 73.74^\circ\)

(b) Area of the shaded (minor) segment \(=\) area of sector \(XOY\) \(-\) area of triangle \(XOY\).

Area of sector:

\[\frac{\theta}{360^\circ} \times \pi r^2 = \frac{73.7398}{360} \times \frac{22}{7} \times 5^2\]\[= 0.204833 \times 78.5714 = 16.09\ \text{cm}^2\]

Area of triangle \(XOY\):

\[\frac{1}{2} r^2 \sin\theta = \frac{1}{2} \times 25 \times \sin 73.7398^\circ = \frac{1}{2} \times 25 \times 0.96 = 12.00\ \text{cm}^2\]

Area of shaded segment:

\[16.09 - 12.00 = 4.09\ \text{cm}^2\]

Shaded area \(\approx 4.09\ \text{cm}^2\) (to 2 decimal places).