# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#7

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**Question**

The diagram shows two circles with centres A and B having radii 8 cm and 10 cm respectively. The two circles intersect at C and D where CAD is a straight line and AB is perpendicular to CD.

**(i) **Find angle ABC in radians.

**(ii) **Find the area of the shaded region.

**Solution**

**(i)
**

It is evident from the diagram that triangle ABC is a right-triangle.

Expression for trigonometric ratio in right-triangle is;

For the given triangle ABC;

Using calculator;

**(ii)
**

It is evident from the diagram that;

Let’s find all these area one-by-one.

First we find area of circle with center A.

Expression for area of a circle with radius is;

Therefore, for circle with center A with ;

Next we find area of sector BCD.

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

For circular sector BCD, it is evident that (because triangles ABC and ABD are congruent) and ; therefore;

Lastly, we need area of triangle BCD.

Expression for the area of the triangle is;

It is evident from the diagram that;

We can find CD as follows.

Since both AD and AC represent the radius of circle with center A, , therefore;

We can find by considering right-triangle ABC.

Pythagorean Theorem

For right triangle ABC;

Now we can find area of triangle BCD.

Finally;

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