MathDB
Ratio of two equilateral triangles' area

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November 21, 2005
ratiogeometry

Problem Statement

Equilateral triangle DEFDEF is inscribed in equilateral triangle ABCABC such that DEBC\overline{DE} \perp \overline{BC}. The ratio of the area of DEF\triangle DEF to the area of ABC\triangle ABC is [asy] size(180); pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10); pair B = (0,0), C = (1,0), A = dir(60), D = C*2/3, E = (2*A+C)/3, F = (2*B+A)/3; D(D("A",A,N)--D("B",B,SW)--D("C",C,SE)--cycle); D(D("D",D)--D("E",E,NE)--D("F",F,NW)--cycle); D(rightanglemark(C,D,E,1.5));[/asy] <spanclass=latexbold>(A)</span> 16<spanclass=latexbold>(B)</span> 14<spanclass=latexbold>(C)</span> 13<spanclass=latexbold>(D)</span> 25<spanclass=latexbold>(E)</span> 12<span class='latex-bold'>(A)</span>\ \dfrac{1}{6}\qquad <span class='latex-bold'>(B)</span>\ \dfrac{1}{4} \qquad <span class='latex-bold'>(C)</span>\ \dfrac{1}{3} \qquad <span class='latex-bold'>(D)</span>\ \dfrac{2}{5} \qquad <span class='latex-bold'>(E)</span>\ \dfrac{1}{2}