# 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

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;

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;