(a) In the diagram, XY is a chord of a circle of radius 5cm. The chord subtends an angle 96° at the centre. Calculate, correct to three significant figures,...

Answer Details

(a) Area of the minor segment

The chord \(XY\) subtends \(\theta = 96^\circ\) at the centre and the radius is \(r = 5\text{ cm}\). The minor segment is the minor sector minus triangle \(OXY\).

Area of minor sector (with \(\pi = \tfrac{22}{7}\)):

\[\frac{\theta}{360}\pi r^2 = \frac{96}{360}\times\frac{22}{7}\times 5^2 = 20.95\text{ cm}^2\]

Area of triangle OXY:

\[\frac{1}{2}r^2\sin\theta = \frac{1}{2}\times 5^2\times\sin 96^\circ = 12.43\text{ cm}^2\]

Area of minor segment:

\[20.95 - 12.43 = 8.52\text{ cm}^2\]

Correct to three significant figures, the area of the minor segment is \(\mathbf{8.52\text{ cm}^2}\).

(b) Total area of the shaded portions

The square has side 14 cm, so the inscribed circle has radius \(r = 7\text{ cm}\).

Shaded sector of the circle (angle \(80^\circ\)):

\[\frac{80}{360}\times\frac{22}{7}\times 7^2 = 34.22\text{ cm}^2\]

Shaded portion of the square (the four corners between the square and the circle):

\[\text{Area of square} = 14^2 = 196\text{ cm}^2\]

\[\text{Area of circle} = \frac{22}{7}\times 7^2 = 154\text{ cm}^2\]

\[196 - 154 = 42\text{ cm}^2\]

Total shaded area:

\[42 + 34.22 = 76.22 \approx 76.2\text{ cm}^2\]