# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/13) | Q#5

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Question

The diagram shows a triangle OAB in which angle OAB = 90o and OA = 5 cm. The  arc AC is part of a circle with centre O. The arc has length 6 cm and it meets OB at  C. Find the area of the shaded region.

Solution

It is evident from the diagram that;

Therefore, we find areas of triangle OAB and circular sector AOC one-by-one.

First we find area of triangle OAB.

Expression for the area of the triangle is;

Therefore;

We can see from the diagram that;

Hence;

We need to find height of triangle OAB.

It is given that triangle OAB is a right-angled triangle in which angle OAB = 90o.

Expression for  trigonometric ratio in right-triangle is;

Therefore;

We need to find .

It is evident that in the triangle OAB is the same as in circular sector  AOC.

Expression for length of a circular arc with radius and angle  rad is;

For the given case;

Hence;

Therefore;

Next we need to find area of circular sector AOC.

Expression for area of a circular sector with radius and angle  rad is;

For the given case;

As demonstrated above ; hence,

Finally;