Difference between revisions of "2006 AMC 10B Problems/Problem 15"
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− | Using | + | Using kiouonp of 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> |
Latest revision as of 14:01, 15 December 2020
Problem
Rhombus is similar to rhombus . The area of rhombus is and . What is the area of rhombus ?
Solutions
Solution 1
Using kiouonp 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 |
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