2006 AMC 10B Problems/Problem 15
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 .