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Try this beautiful problem on area of circle from SMO, Singapore Mathematics Olympiad, 2010.

## Problem – Area of Circle (SMO Test)

Let ABCD be a rectangle with AB = 10 . Draw circles $C_1$ and $C_2$ with diameters AB and CD respectively. Let P,Q be the intersection points of $C_1$ and $C_2$ . If the circle with diameter PQ is tangent to AB and CD , then what is the area of the shaded region ?

• 25
• 20
• 22
• 23

### Key Concepts

Area of Circle

2D – Geometry

Area of Rectangle

But try the problem first…

Source

Challenges and thrills – Pre – college Mathematics

## Try with Hints

First Hint………………………

If you are really got stuck with this sum then we can start from here:

The diagram will be like this . So let ‘N’ be the midpoint of CD .

so $\angle {PNQ} = 90^\circ$

so PQ = 5 $\sqrt {2}$

Second Hint …………………………….

Now let us try to find the area of the shaded region

A = $2 [ \frac {1}{2}\pi (\frac {PQ)}{2})^2 + \frac {1}{2}(PN)^2 – \frac {1}{4} \pi (PN)^2]$

= $2 [\frac {1}{2}\pi (\frac {5\sqrt{2}}{2})^2+\frac {1}{2}.5^2 – \frac {1}{4}\pi . 5^2]$

= 25