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Problem: 1012

Points \(E\) and \(F\) are located on square \(ABCD\) so that \(\Delta BEF\) is equilateral. What is the ratio of the area of \(\Delta DEF\) to that of \(\Delta ABE\)? [asy] pair A=origin, B=(1,0), C=(1,1), D=(0,1), X=B+2*dir(165), E=intersectionpoint(B--X, A--D), Y=B+2*dir(105), F=intersectionpoint(B--Y, D--C); draw(B--C--D--A--B--F--E--B); pair point=(0.5,0.5); label("\(A\)", A, dir(point--A)); label("\(B\)", B, dir(point--B)); label("\(C\)", C, dir(point--C)); label("\(D\)", D, dir(point--D)); label("\(E\)", E, dir(point--E)); label("\(F\)", F, dir(point--F));[/asy] $$\textbf{(A)}\; \frac43\qquad \textbf{(B)}\; \frac32\qquad \textbf{(C)}\; \sqrt3\qquad \textbf{(D)}\; 2\qquad \textbf{(E)}\; 1+\sqrt3\qquad$$

Tags: geometry,area

Source: AMC 10 2004 A Q20

Elo Rating: 1514.03

Problem: 2292

How many triangles have area \( 10\) and vertices at \( (-5,0)\), \( (5,0)\), and \( (5\cos \theta, 5\sin \theta)\) for some angle \( \theta\)? $$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 8$$

Tags: trigonometry,area-of-triangle,area-formula,coordinates,triangles,geometry

Source: AMC 12 1998 A Q19

Elo Rating: 1500.00

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