Difference between revisions of "2006 AMC 12B Problems/Problem 13"
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== Problem == | == Problem == | ||
| − | Rhombus <math> ABCD</math> is similar to rhombus <math> BFDE</math>. The area of rhombus <math> ABCD</math> is 24, and <math> \angle BAD | + | Rhombus <math> ABCD</math> is similar to rhombus <math> BFDE</math>. The area of rhombus <math> ABCD</math> is 24, and <math> \angle BAD = 60^\circ</math>. What is the area of rhombus <math> BFDE</math>? |
<asy> defaultpen(linewidth(0.7)+fontsize(11)); | <asy> defaultpen(linewidth(0.7)+fontsize(11)); | ||
Revision as of 20:22, 19 June 2015
Problem
Rhombus 
is similar to rhombus 
. The area of rhombus is 24, and 
. What is the area of rhombus ?
![[asy] defaultpen(linewidth(0.7)+fontsize(11)); pair A=origin, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/sqrt(3)); pair point=(3/2, sqrt(3)/2); draw(B--C--D--A--B--F--D--E--B); 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]](http://latex.artofproblemsolving.com/4/3/e/43e4a11b576cfc60bb546ede67ebb9668484c340.png)
Solution
The ratio of any length on ABCD to a corresponding length on BFDE squared is equal to the ratio of their areas. Since 
, 
and 
are equilateral. DB, which is equal to AB, is the diagonal of rhombus ABCD. Therefore, 
. DB and AC are the longer diagonal of rhombuses BEDF and ABCD, respectively. So the ratio of their areas is 
or 
. One-third the area of ABCD is equal to 8. So the answer is C
See also
| 2006 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 12 |
Followed by Problem 14 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
