MathDB
Ratio of Triangles in a Circle

Source:

June 13, 2009
ratiogeometry

Problem Statement

In the accompanying figure CE \overline{CE} and DE \overline{DE} are equal chords of a circle with center O O. Arc AB AB is a quarter-circle. Then the ratio of the area of triangle CED CED to the area of triangle AOB AOB is: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm);
pair O = origin; pair C = (-1,0); pair D = (1,0); pair E = (0,1); pair A = dir(-135); pair B = dir(-60);
draw(Circle(O,1)); draw(C--E--D--cycle); draw(A--O--B--cycle);
label("AA",A,SW); label("CC",C,W); label("EE",E,N); label("DD",D,NE); label("BB",B,SE); label("OO",O,N);[/asy] <spanclass=latexbold>(A)</span> 2:1<spanclass=latexbold>(B)</span> 3:1<spanclass=latexbold>(C)</span> 4:1<spanclass=latexbold>(D)</span> 3:1<spanclass=latexbold>(E)</span> 2:1 <span class='latex-bold'>(A)</span>\ \sqrt {2} : 1\qquad <span class='latex-bold'>(B)</span>\ \sqrt {3} : 1\qquad <span class='latex-bold'>(C)</span>\ 4 : 1\qquad <span class='latex-bold'>(D)</span>\ 3 : 1\qquad <span class='latex-bold'>(E)</span>\ 2 : 1