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Problem 228
228
The diagram shows a circle with radius r cm and centre O. Points A and B lie on the circle and ABCD is a rectangle. Angle AOB = 2θ radians and AD = r cm.
(i) Express the perimeter of the shaded region in terms of r and θ.
(ii) In the case where r = 5 and θ = \(\frac{1}{6} \pi\), find the area of the shaded region.
Solution
(i) To find the perimeter of the shaded region, we need to consider the lengths of AB, the arc AB, and the sides AD and DC of the rectangle.
The length of AB can be found using the formula: \(AB = 2r\sin\theta\).
The length of the arc AB is given by: \(2r\theta\).
The perimeter P of the shaded region is: \(P = AD + DC + AB + \text{arc } AB = 2r + 2r\theta + 2r\sin\theta\).
(ii) For r = 5 and θ = \(\frac{1}{6} \pi\), we calculate the area of the shaded region.
The area of sector AOB is \(\frac{1}{2} r^2 (2\theta) = \frac{25\pi}{6}\) or 13.1.
The area of triangle AOB is \(\frac{1}{2} \times 2r\sin\theta \times r\cos\theta = \frac{25\sqrt{3}}{4}\) or 10.8.
The area of rectangle ABCD is \(r \times 2r\sin\theta = 25\).
The area of the shaded region is: \(25 - (\frac{25\pi}{6} - \frac{25\sqrt{3}}{4})\) or 22.7.
