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AMC-8 Geometry Math Olympiad USA Math Olympiad

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.You may use sequential hints to solve the problem.

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?

circles and semi-circles
  • $\frac{1}{2}$
  • $\frac{2}{\pi}$
  • $ \frac{3}{2} $

Key Concepts


Geometry

Circle

co-ordinate geometry

Check the Answer


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

AMC-8 (2010) Problem 23

Pre College Mathematics

Try with Hints


Find the radius of the circle

Can you now finish the problem ……….

ratio of the areas

Join O and Q

can you finish the problem……..

ratio of the areas of circles and semi-circles

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

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

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