# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#5

Question

The diagram shows a major arc AB of a circle with centre O and radius 6 cm. Points C and D on OA  and OB respectively are such that the line AB is a tangent at E to the arc CED of a smaller  circle also with centre O. Angle COD = 1.8 radians.

i.       Show that the radius of the arc CED is 3.73 cm, correct to 3 significant figures.

ii.       Find the area of the shaded region.

Solution

i.

It is evident from the diagram that radius of arc CED is;

Consider the diagram below.

We recognize that triangle OAB is an isosceles triangle with equal sides;

In triangle OAB, we draw a perpendicular from O to AB which meets line AB at point E.

Since AB line is perpendicular to arc CED, OE is also radius of arc CED like OC and OD.

This perpendicular OE also bisects the angle AOB. Therefore;

Now consider right triangle OEB to find OE.

Expression for  trigonometric ratio in right-triangle is;

We have found above that  and we are given that , where OB is radius of major  arc AB with center O. Therefore;

ii.

It is evident from the diagram that;

Let’s find areas of these sectors one-by-one.

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

Area of sector OAB;

For this sector;

Therefore;

Now we find area of sector OCED.

For this sector;

Therefore;

Hence;