Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#4

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Question

In the diagram, ABCD is a parallelogram with AB=BD=DC=10cm and angle ABD= 0.8 radians. APD and BQC are arcs of circles with centres B and D respectively.

    i.       Find the area of the parallelogram ABCD.

   ii.       Find the area of the complete figure ABQCDP.

  iii.       Find the perimeter of the complete figure ABQCDP.

Solution


i.
 

Expression for the area of the parallelogram is;

It is evident from the diagram that base of parallelogram is AB=CD=10cm. However, to find the height of the parallelogram we can modify the given diagram as follows.

Consider the .

Expression for   trigonometric ratio in right-triangle is;

Therefore;

For the given case area of parallelogram is;

 


ii.
 

It is evident from the diagram that;

Now we find area of sector  .

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

For the given case;

Since both sectors are carved out with circles of same diameter;

Hence;


iii.
 

It is evident from the diagram that;

We are given that;

Lets find arcs  and .

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

For the given arcs;

Since opposite angles are equal.

Therefore;

Hence;

Finally;

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