MathDB
1997 AJHSME #24 - Ratio of Regions

Source:

June 30, 2011
ratiogeometryAMC

Problem Statement

Diameter ACEACE is divided at CC in the ratio 2:32:3. The two semicircles, ABCABC and CDECDE, divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region is
[asy]pair A,B,C,D,EE; A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0); fill(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)--arc((5,0),EE,A,CCW)--cycle,gray); draw(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)); draw(circle((5,0),5));
dot(A); dot(B); dot(C); dot(D); dot(EE); label("AA",A,W); label("BB",B,N); label("CC",C,E); label("DD",D,N); label("EE",EE,W); [/asy]
<spanclass=latexbold>(A)</span> 2:3<spanclass=latexbold>(B)</span> 1:1<spanclass=latexbold>(C)</span> 3:2<spanclass=latexbold>(D)</span> 9:4<spanclass=latexbold>(E)</span> 5:2<span class='latex-bold'>(A)</span>\ 2:3 \qquad <span class='latex-bold'>(B)</span>\ 1:1 \qquad <span class='latex-bold'>(C)</span>\ 3:2 \qquad <span class='latex-bold'>(D)</span>\ 9:4 \qquad <span class='latex-bold'>(E)</span>\ 5:2