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

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

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

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+ | *[[2006 AMC 10B Problems/Problem 14|Previous Problem]] | ||

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+ | *[[2006 AMC 10B Problems/Problem 16|Next Problem]] |

## Revision as of 15:01, 2 August 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 .