Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/11) | Q#8

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Question

The diagram shows a sector OAC of a circle with centre O. Tangents AB and CB to the circle meet  at B. The arc AC is of length 6 cm and angle  radians.


i.       
Find the length of OA correct to 4 significant figures.

   ii.       Find the perimeter of the shaded region.

 iii.       Find the area of the shaded region.

Solution


i.
 

We are required to find length of OA which is radius of the circle of which we are given;

Expression for length of a circular arc with radius and angle rad is;

Therefore, for the given case;


ii.
 

We are required to find the perimeter of the shaded region.

It is evident from the diagram that;

We are given that;

We need to find AB & BC.

Consider the diagram below.

Let us consider the right-triangle BOA.

Expression for  trigonometric ratio in right-triangle is;

Therefore;

If two tangents to a circle meet at a point external to the circle, a line segments from the point of  intersection of tangents to the center of the circle bisects the angle of arc made between tangent  points on the circle.

Therefore, in the given case;

Hence;

Hence;

Next, we need to find BC.

If two tangents to a circle meet at a point external to the circle, the line segments (tangents) from  respective tangent point to the point of intersection of tangents are equal in length.

Therefore, in the given case;

Hence;

Hence;

 

 iii.
 

We are required to find the area of the shaded region.

It is evident from the diagram that;

First, we find area of triangle AOB.

Expression for the area of the triangle is;

Consider the diagram below.

As found in (i) and (ii);

If two tangents to a circle meet at a point external to the circle, a line segments from the point of  intersection of tangents to the center of the circle bisects the angle of arc made between tangent  points on the circle and triangle AOB and triangle BOC are congruent right triangles.

Therefore, in the given case;

Next, we find area of sector AOC.

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

Finally, we can find area of shaded region.

 

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