# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#6

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Question The diagram shows a circle with radius r cm and centre O. The line PT is the tangent to the circle at  P and angle radians. The line OT meets the circle at Q.

i.       Express the perimeter of the shaded region PQT in terms of r and .

ii.       In the case where and , find the area of the shaded region correct to 2 significant  figures.

Solution

i.

It is evident from the diagram that; Let’s first of all find arc PQ.

Expression for length of a circular arc with radius and angle rad is; For arc PQ;   Now we find PT.

Tangent at any point of the circle is perpendicular to the radius through the point of contact.

Consider . It is evident that is a right angle.

Expression for trigonometric ratio in right-triangle is; For right angled triangle OPT;    Next, we find QT.

It is evident from the diagram that;   To find OT, consider right angled triangle OPT.

Pythagorean Theorem For right angled triangle OPT;   We know that and from above  we have . Therefore; Finally;  ii.

It is evident from the diagram that; Let’s first of all find area of triangle OPT.

Expression for the area of the triangle is; For arc triangle OPT;  We know that and from above  we have . Therefore;  Next we find area of sector OPQ.

Expression for area of a circular sector with radius and angle rad is; For sector OPQ;  Hence, we can find area of shaded region;   We are given that and ;    