# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#9

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Question The diagram shows a semicircle ABC with centre O and radius 8 cm. Angle radians.

i.
In the case where , calculate the area of the sector BOC.

ii.       Find the value of for which the perimeter of sector AOB is one half of the perimeter of sector BOC.

iii.       In the case where , show that the exact length of the perimeter of triangle ABC is cm.

Solution

i.

Expression for area of a circular sector with radius and angle rad is; In the given case, angle made by the sector is .  It is evident from the diagram that;   Therefore;  It is given that; & Hence;    ii.

We are given that; Expression for perimeter of a sector with radius and angle rad is;         According to the given condition;           iii.

It is evident from the diagram that; First we find .

We can see from the diagram that; Since; We can write the above equation as;  Now we find .

It is evident from given data and the diagram that in ;   If we have lengths of two sides and the included angle, we can use law of cosines to find the 3rd length of the triangle. Law of cosines is;   For the given case;        Finally, we need .

From above, we know that;  If we have the angle opposite to side , we can find the length of using cosine rule. To find the angle opposite to side , we note from the diagram that;     If we have lengths of two sides and the included angle, we can use law of cosines to find the 3rd length of the triangle. Law of cosines is;   For the given case;        Hence;   