# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#4

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Question In the diagram, D lies on the side AB of triangle ABC and CD is an arc of a circle with centre A and radius 2 cm. The line BC is of length 23 cm and is perpendicular to AC. Find the area of the shaded region BDC, giving your answer in terms of and .

Solution

It is evident from the diagram that; First we find the area of the triangle ABC.

Expression for the area of the triangle is; For the given case;    Now we find area of circular sector ADC.

Expression for area of a circular sector with radius and angle rad is; We are given that radius of this circular section is 2. We need to find the area of circular sector ACD.

We know that is a right angled triangle with .

Expression for trigonometric ratio in right-triangle is; For we take and therefore;       Hence, area of circular sector ACD;   Finally, we can find area of shaded region;  