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]

$\mathrm{(A) \ } 6\qquad \mathrm{(B) \ } 4\sqrt{3}\qquad \mathrm{(C) \ } 8\qquad \mathrm{(D) \ } 9\qquad \mathrm{(E) \ } 6\sqrt{3}$

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  • 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.
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