# 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;