Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  MayJun  (P19709/13)  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;
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