# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#2

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**Question**

In the diagram, AYB is a semicircle with AB as diameter and OAXB is a sector of a circle with centre O and radius r. Angle AOB = 2 radians. Find an expression, in terms of r and , for the area of the shaded region.

**Solution**

It is evident from the diagram that;

First we calculate area of semi-circle AYB.

Expression for area of semi-circle with radius is;

Therefore, we need to find radius of semicircle AYB.

We are given that AB is the diameter of the circle AYB. Hence;

It is evident from the diagram that is an isosceles triangle with congruent sides and base . The vertex angle of this isosceles triangle is .

Now consider the diagram below to find the radius of the circle AYB.

In an isosceles triangle, the bisector of the vertex angle (the angle which is opposite to a side that might not be congruent to another side) is also perpendicular bisector of the base.

Therefore, is an angle bisector of the and is also perpendicular bisector of base . Hence;

Now consider the two triangles AOS and BOS.

It is evident that and and .

Law of Sines;

Applying law of sines to ;

Now we can find area of semi-circle AYB.

Next we calculate area of segment AXB.

It is evident from the diagram that;

Let’s first find area of circular sector AXB and then area of triangle AOB.

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

It is evident that sector AXB is sector of the circle with center O which has radius and angle of circular sector is . Therefore;

Lastly, we need area of triangle AOB.

Expression for the area of a triangle for which two sides (a and b) and the included angle (C ) is given;

For the given case, in ;

Finally;

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## abdullah2004

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