# 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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