Try this beautiful problem from Geometry based on Ratio of the area of circle and semi-circles.

## Area of circles and semi-circles – AMC-8, 2010 – Problem 23

Semicircles POQ and ROS pass through the center O. What is the ratio of the combined areas of the two semicircles to the area of circle O?

- $\frac{1}{2}$
- $\frac{2}{\pi}$
- $ \frac{3}{2} $

**Key Concepts**

Geometry

Circle

co-ordinate geometry

## Check the Answer

But try the problem first…

Answer:$\frac{1}{2}$

AMC-8 (2010) Problem 23

Pre College Mathematics

## Try with Hints

First hint

Find the radius of the circle

Can you now finish the problem ……….

Second Hint

Join O and Q

can you finish the problem……..

Final Step

The co-ordinate of Q is (1,1), So OB=1 and BQ=1

By the Pythagorean Theorem, the radius of the larger circle i.e OQ=\(\sqrt{1^2+1^2}\)=\(\sqrt 2\).

Therefore the area of the larger circle be \(\pi (\sqrt 2)^2=2\pi\)

Now for the semicircles, radius OB=OC=1(as co-ordinate of P=(1,1) and S=(1,-1))

So, the area of the two semicircles is \(2\times\frac{\pi(1)^2}{2}=\pi\)

Finally, the ratio of the combined areas of the two semicircles to the area of circle O is

\(\frac{\pi}{2\pi}\)=\(\frac{1}{2}\)

## Other useful links

- https://www.cheenta.com/area-of-square-and-circle-amc-8-2011-problem-25/
- https://www.youtube.com/watch?v=W9XdZd8zXPA

Google