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2013 UNCO Math Contest II Problems/Problem 3

Revision as of 15:09, 16 October 2014 by Timneh (talk | contribs) (Created page with "== Problem == <asy> filldraw(circle((0,-sqrt(2)/2),sqrt(2)/2),grey); filldraw(circle((0,0),1),white); draw((-sqrt(2)/2,-sqrt(2)/2)--(sqrt(2)/2,-sqrt(2)/2)--(0,0)--cycle,black); ...")
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Problem

[asy] filldraw(circle((0,-sqrt(2)/2),sqrt(2)/2),grey); filldraw(circle((0,0),1),white); draw((-sqrt(2)/2,-sqrt(2)/2)--(sqrt(2)/2,-sqrt(2)/2)--(0,0)--cycle,black); MP("C",(0,0),N);MP("A",(-sqrt(2)/2,-sqrt(2)/2),W);MP("B",(sqrt(2)/2,-sqrt(2)/2),E); [/asy]

Point $C$ is the center of a large circle that passes through both $A$ and $B$, and $C$ lies on the small circle whose diameter is $AB$. The area of the small circle is $9\pi$. Find the area of the shaded $\textit{lune}$, the region inside the small circle and outside the large circle.

Solution

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