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# Circles and semi-circles| AMC 8, 2010|Problem 23

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?

### Key Concepts

Geometry

Circle

co-ordinate geometry

## Check the Answer

AMC-8 (2010) Problem 23

Pre College Mathematics

## Try with Hints

Find the radius of the circle

Can you now finish the problem ..........

Join O and Q

can you finish the problem........

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==.

Therefore the area of the larger circle be

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

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

=

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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?

### Key Concepts

Geometry

Circle

co-ordinate geometry

## Check the Answer

AMC-8 (2010) Problem 23

Pre College Mathematics

## Try with Hints

Find the radius of the circle

Can you now finish the problem ..........

Join O and Q

can you finish the problem........

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==.

Therefore the area of the larger circle be

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

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

=

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Cheenta is a knowledge partner of Aditya Birla Education Academy