Square Around Equilateral Triangles
Source:
February 19, 2008
geometrytrigonometry
Problem Statement
Points , , , and lie in the plane of the square so that , , , and are equilateral triangles. If has an area of , find the area of .[asy]unitsize(2cm);
defaultpen(fontsize(8)+linewidth(0.8));
pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5);
pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0);
draw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A);
label("",A,SE);
label("",B,SW);
label("",C,NW);
label("",D,NE);
label("",K,NNW);
label("",L,E);
label("",M,S);
label("",N,W);[/asy] (A)\ 32 \qquad (B)\ 16 \plus{} 16\sqrt {3} \qquad (C)\ 48 \qquad (D)\ 32 \plus{} 16\sqrt {3} \qquad (E)\ 64