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
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