Difference between revisions of "2006 AMC 12B Problems/Problem 13"
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− | 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 21:22, 19 June 2015
Problem
Rhombus is similar to rhombus . The area of rhombus is 24, and . What is the area of rhombus ?
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.