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
.