# Difference between revisions of "2006 AMC 10B Problems/Problem 15"

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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 <math>24</math> and <math> \angle BAD = 60^\circ </math>. What is the area of rhombus <math>BFDE</math>? | ||

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+ | [[Image:2006amc10b15.gif]] | ||

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+ | <math> \mathrm{(A) \ } 6\qquad \mathrm{(B) \ } 4\sqrt{3}\qquad \mathrm{(C) \ } 8\qquad \mathrm{(D) \ } 9\qquad \mathrm{(E) \ } 6\sqrt{3} </math> | ||

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== Solution == | == Solution == | ||

+ | Using properties of a [[rhombus]]: | ||

+ | |||

+ | <math> \angle DAB = \angle DCB = 60 ^\circ </math>. | ||

+ | |||

+ | <math> \angle ADC = \angle ABC = 120 ^\circ </math>. | ||

+ | |||

+ | It is easy to see that rhombus <math>ABCD</math> is made up of equilateral triangles <math>DAB</math> and <math>DCB</math>. | ||

+ | |||

+ | Let the lengths of the sides of rhombus <math>ABCD</math> be <math>s</math>. | ||

+ | |||

+ | The longer diagonal of rhombus <math>BFDE</math> is <math>BD</math>. Since <math>BD</math> is a side of an equilateral triangle with a side length of <math>s</math>, <math> BD = s </math>. | ||

+ | |||

+ | The longer diagonal of rhombus <math>ABCD</math> is <math>AC</math>. Since <math>AC</math> is twice the length of an altitude of of an equilateral triangle with a side length of <math>s</math>, <math> AC = 2 \cdot \frac{s\sqrt{3}}{2} = s\sqrt{3} </math> | ||

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+ | The ratio of the longer diagonal of rhombus <math>BFDE</math> to rhombus <math>ABCD</math> is <math> \frac{s}{s\sqrt{3}} = \frac{\sqrt{3}}{3} </math> | ||

+ | |||

+ | Therefore, the ratio of the area of rhombus <math>BFDE</math> to rhombus <math>ABCD</math> is <math> \left( \frac{\sqrt{3}}{3} \right) ^2 = \frac{1}{3} </math> | ||

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+ | Let <math>x</math> be the area of rhombus <math>BFDE</math>. | ||

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+ | <math> \frac{x}{24} = \frac{1}{3} </math> | ||

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+ | <math> x = 8 \Rightarrow C </math> | ||

+ | |||

== See Also == | == See Also == | ||

*[[2006 AMC 10B Problems]] | *[[2006 AMC 10B Problems]] |

## Revision as of 15:24, 19 July 2006

## Problem

Rhombus is similar to rhombus . The area of rhombus is and . What is the area of rhombus ?

## Solution

Using properties of a rhombus:

.

.

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 .