PRMO 2012 question 20

20. PS is a line segment of length 4 and O is the midpoint of PS. A semicircular arc is drawn with PS as diameter. Let X be the midpoint of this arc. Q and R are points on the arc PXS such that QR is parallel to PS and the semicircular arc drawn with QR as diameter is tangent to PS. How can I get the area of the region QXROQ bounded by the two semicircular arcs?

Answer: 2*PI - 2

Solution:
Image taken from here.



Now, the big circle has area 4*Pi.

Since RQ is the diameter of the smaller circle, it subtends 90 degree on its circumference. So Angle QOR is right angle.

So the line segments QO and OR will capture 1/4th the area of the completed blue circle which is Pi.--------[1]

Now we just need to find the area between green  lines and the red circular arcs below them.

If you see the area captured by line segments YQ and YO in the red circle, it will be 1/4th of the completed red circle. Again the same reason that angle QYQ is right angle.

Radius of the red circle is sqrt(2). So the complete area is 2*Pi.
1/4th the area is Pi/2. The right triangle QYO has area 1. So area between green dotted line and the red circle below it would be Pi/2 - 1. There are 2 such parts. So their area is Pi - 2.

Add it to [1] to get the final answer 2*Pi - 2 .




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