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

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<math> \angle ADC = \angle ABC = 120 ^\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 | + | It is easy to see that rhombus <math>ABCD</math> is made up of [[equilateral triangle]]s <math>DAB</math> and <math>DCB</math>. |

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

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

## Revision as of 10:31, 29 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 .