Try this beautiful problem from Geometry based on Ratio of the area of circle and semi-circles.
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?
Geometry
Circle
co-ordinate geometry
Answer:$\frac{1}{2}$
AMC-8 (2010) Problem 23
Pre College Mathematics
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=\(\sqrt{1^2+1^2}\)=\(\sqrt 2\).
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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