Problem #2182

 2182 Rhombus $ABCD$ is similar to rhombus $BFDE$. The area of rhombus $ABCD$ is $24$ and $\angle BAD = 60^\circ$. What is the area of rhombus $BFDE$? $[asy] unitsize(3cm); defaultpen(.8); pair A=(0,0), B=(1,0), D=dir(60), C=B+D; draw(A--B--C--D--cycle); pair Ep = intersectionpoint( B -- (B+10*dir(150)), D -- (D+10*dir(270)) ); pair F = intersectionpoint( B -- (B+10*dir(90)), D -- (D+10*dir(330)) ); draw(B--Ep--D--F--cycle); label("A",A,SW); label("B",B,SE); label("C",C,NE); label("D",D,NW); label("E",Ep,SW); label("F",F,NE); [/asy]$ $\mathrm{(A) \ } 6\qquad \mathrm{(B) \ } 4\sqrt{3}\qquad \mathrm{(C) \ } 8\qquad \mathrm{(D) \ } 9\qquad \mathrm{(E) \ } 6\sqrt{3}$ This problem is copyrighted by the American Mathematics Competitions.
Note: you aren't logged in. If you log in, we'll keep a record of which problems you've solved.

• Reduce fractions to lowest terms and enter in the form 7/9.
• Numbers involving pi should be written as 7pi or 7pi/3 as appropriate.
• Square roots should be written as sqrt(3), 5sqrt(5), sqrt(3)/2, or 7sqrt(2)/3 as appropriate.
• Exponents should be entered in the form 10^10.
• If the problem is multiple choice, enter the appropriate (capital) letter.
• Enter points with parentheses, like so: (4,5)
• Complex numbers should be entered in rectangular form unless otherwise specified, like so: 3+4i. If there is no real component, enter only the imaginary component (i.e. 2i, NOT 0+2i).