# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#7

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**Question**

In the diagram, AOB is a quarter circle with centre O and radius r. The point C lies on the arc AB and the point D lies on OB. The line CD is parallel to AO and angle radians.

** i. **Express the perimeter of the shaded region in terms of r, and .

** ii. **For the case where r = 5 cm and , find the area of the shaded region.

**Solution**

i.

It is evident from the diagram that;

Let’s first find length of arc BC.

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

It is evident from the diagram that;

We can see from the diagram that;

Therefore;

Let us now find BD.

Consider the right-triangle COD in the diagram where and .

Expression for trigonometric ratio in right-triangle is;

Therefore;

We have trigonometric ratio;

Now let us find CD.

It is evident from the diagram that;

For OD we again consider the right-triangle COD in the diagram where and

.

Expression for trigonometric ratio in right-triangle is;

Therefore;

We have trigonometric ratio;

Therefore, we can write;

Finally, we can find the perimeter of the shaded region;

ii.

It is evident from the diagram that;

Let us first find area of sector OCB.

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

It is evident from the diagram that;

It is evident from the diagram and ;

Now we find area of triangle COD.

Expression for the area of the triangle is;

It is evident from the diagram that;

From (i) we have found that;

Therefore;

Now we can find area of shaded region;

We are given that and ;

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