Problem

2005 AMC 12A Problem 15

Let \overline{AB} be a diameter of a circle and C be a point on \overline{AB} with 2 \cdot AC = BC. Let D and E be points on the circle such that \overline{DC} \perp \overline{AB} and \overline{DE} is a second diameter. What is the ratio of the area of \triangle DCE to the area of \triangle ABD?

[asy] unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); pair D=dir(aCos(C.x)), E=(-D.x,-D.y); draw(A--B--D--cycle); draw(D--E--C); draw(unitcircle,white); drawline(D,C); dot(O); clip(unitcircle); draw(unitcircle); label("$E$",E,SSE); label("$B$",B,E); label("$A$",A,W); label("$D$",D,NNW); label("$C$",C,SW); draw(rightanglemark(D,C,B,2));[/asy]

\textbf{(A) } \frac{1}{6} \qquad \textbf{(B) } \frac{1}{4} \qquad \textbf{(C) } \frac{1}{3} \qquad \textbf{(D) } \frac{1}{2} \qquad \textbf{(E) } \frac{2}{3}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.


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Category: AMC 12A
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