Difference between revisions of "2006 AMC 10B Problems/Problem 15"
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== Solutions == | == Solutions == | ||
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− | Using | + | Using the property that opposite angles are equal in a [[rhombus]], <math> \angle DAB = \angle DCB = 60 ^\circ </math> and <math> \angle ADC = \angle ABC = 120 ^\circ </math>. It is easy to see that rhombus <math>ABCD</math> is made up of [[equilateral triangle]]s <math>DAB</math> and <math>DCB</math>. Let the lengths of the sides of rhombus <math>ABCD</math> be <math>s</math>. |
The longer [[diagonal]] of rhombus <math>BFDE</math> is <math>BD</math>. Since <math>BD</math> is a side of an equilateral triangle with a side length of <math>s</math>, <math> BD = s </math>. The longer diagonal of rhombus <math>ABCD</math> is <math>AC</math>. Since <math>AC</math> is twice the length of an altitude of of an equilateral triangle with a side length of <math>s</math>, <math> AC = 2 \cdot \frac{s\sqrt{3}}{2} = s\sqrt{3} </math>. | The longer [[diagonal]] of rhombus <math>BFDE</math> is <math>BD</math>. Since <math>BD</math> is a side of an equilateral triangle with a side length of <math>s</math>, <math> BD = s </math>. The longer diagonal of rhombus <math>ABCD</math> is <math>AC</math>. Since <math>AC</math> is twice the length of an altitude of of an equilateral triangle with a side length of <math>s</math>, <math> AC = 2 \cdot \frac{s\sqrt{3}}{2} = s\sqrt{3} </math>. |
Revision as of 21:35, 2 June 2021
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 | |
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 |
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