Difference between revisions of "2006 AMC 10B Problems/Problem 15"
(→Solution) |
(→Solution 2) |
||
Line 27: | Line 27: | ||
== Solution 2 == | == Solution 2 == | ||
− | + | Triangle DAB is equilateral so triangles <math>DEA</math>, <math>AEB</math>, <math>BED</math>, <math>BFD</math>, <math>BFC</math> and <math>CFD</math> are all congruent with angles <math>30^\circ</math>, <math>30^\circ</math> and <math>120^\circ</math> from which it follows that rhombus <math>BFDE</math> has one third the area of rhombus <math>ABCD</math> i.e. <math>8 \Longrightarrow \boxed{\mathrm{(C)}}</math>. | |
== See Also == | == See Also == |
Revision as of 07:32, 17 February 2016
Contents
Problem
Rhombus is similar to rhombus
. The area of rhombus
is
and
. What is the area of rhombus
?
Solution
Using properties of 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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.