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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