# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#7

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**Question**

The diagram shows a metal plate ABCDEF which has been made by removing the two shaded regions from a circle of radius 10 cm and centre O. The parallel edges AB and ED are both of length 12 cm.

** i. **Show that angle DOE is 1.287 radians, correct to 4 significant figures.

** ii. **Find the perimeter of the metal plate.

** iii. **Find the area of the metal plate.

**Solution**

** i.
**

Consider the diagram below.

Line OG bisects both the angle DOE and the chord DE. If we consider the triangle OGD, we can write;

Expression for trigonometric ratio in right-triangle is;

ii.

It is evident from the diagram that;

We are given that;

Now we find out length of arcs BCD & EFA.

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

For the given case;

It is evident from the diagram below that;

Hence;

Similarly, by symmetry, we can find;

Therefore;

iii.

It is evident from the given diagram that;

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

For the given case;

It is evident from the diagram that;

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

Therefore;

Expression for the area of the triangle is;

Consider the .

Pythagorean Theorem

Therefore;

Hence,

By symmetry, area of other shaded region is also same i.e.16.35.

Finally;

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