MathDB

14

Part of 2003 AMC 12-AHSME

Problems(1)

Square Around Equilateral Triangles

Source:

2/19/2008
Points K K, L L, M M, and N N lie in the plane of the square ABCD ABCD so that AKB AKB, BLC BLC, CMD CMD, and DNA DNA are equilateral triangles. If ABCD ABCD has an area of 16 16, find the area of KLMN KLMN.
[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("AA",A,SE); label("BB",B,SW); label("CC",C,NW); label("DD",D,NE); label("KK",K,NNW); label("LL",L,E); label("MM",M,S); label("NN",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
geometrytrigonometry