2006 AMC 10B Problems/Problem 15
Problem
Rhombus is similar to rhombus
. The area of rhombus
is
and
. What is the area of rhombus
?
Solutions
Solution 1
Using the property that opposite angles are equal in a rhombus, and
. 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
. Then
, so
.
Solution 2
Triangle DAB is equilateral so triangles ,
,
,
,
and
are all congruent with angles
,
and
from which it follows that rhombus
has one third the area of rhombus
i.e.
.
See Also
2006 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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