2006 AMC 10B Problems/Problem 15
Contents
Problem
Rhombus is similar to rhombus . The area of rhombus is and . What is the area of rhombus ?
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. .
Note: A quick way to visualize this method is to draw the line and notice the two equilateral triangles and .
See Also
2006 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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