# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/13) | Q#6

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Question The diagram shows triangle ABC where AB=5cm, AC=4cm and BC=3cm. Three circles with centres  at A, B and C have radii 3 cm, 2 cm and 1 cm respectively. The circles touch each other at  points E, F and G, lying on AB, AC and BC respectively. Find the area of the shaded region EFG.

Solution

It is evident from the diagram that; Let’s first find area of ;

Expression for the area of a triangle for which two sides (a and b) and the included angle (C) is given; We know lengths of all three sides of .   Law of cosines is given as;   It is evident we can use law of cosines to find any angle of the .

Let’s find angle .         Using calculator; Now we can find area of .    Now we need areas of sectors.

Expression for area of a circular sector with radius and angle rad is; To find areas of sectors;   We are given that;   We have already found that; Therefore, we need to find and .

Law of cosines is given as;   We are given that;   Therefore;

 For ; For ;                      Therefore; to find areas of sectors;      Finally, we can find area of shaded region is;   