# 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

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 ;     