Difference between revisions of "2006 AMC 10B Problems/Problem 15"
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== Solution 2 == | == Solution 2 == | ||
− | + | Triangle DAB is equilateral so triangles <math>DEA</math>, <math>AEB</math>, <math>BED</math>, <math>BFD</math>, <math>BFC</math> and <math>CFD</math> are all congruent with angles <math>30^\circ</math>, <math>30^\circ</math> and <math>120^\circ</math> from which it follows that rhombus <math>BFDE</math> has one third the area of rhombus <math>ABCD</math> i.e. <math>8 \Longrightarrow \boxed{\mathrm{(C)}}</math>. | |
== See Also == | == See Also == |
Revision as of 07:32, 17 February 2016
Contents
Problem
Rhombus is similar to rhombus . The area of rhombus is and . What is the area of rhombus ?
Solution
Using properties of a rhombus, and . It is easy to see that rhombus is made up of equilateral triangles and . Let the lengths of the sides of rhombus be .
The longer diagonal of rhombus is . Since is a side of an equilateral triangle with a side length of , . The longer diagonal of rhombus is . Since is twice the length of an altitude of of an equilateral triangle with a side length of ,
The ratio of the longer diagonal of rhombus to rhombus is . Therefore, the ratio of the area of rhombus to rhombus is
Let be the area of rhombus . Then , so .
Solution 2
Triangle DAB is equilateral so triangles , , , , and are all congruent with angles , and from which it follows that rhombus has one third the area of rhombus i.e. .
See Also
2006 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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 10 Problems and Solutions |
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