Difference between revisions of "2001 AMC 12 Problems/Problem 14"
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== Solution == | == Solution == | ||
− | + | Each of the <math>\binom{9}{2} = 36</math> pairs of vertices determines two equilateral triangles, for a total of 72 triangles. However, the three triangles <math>A_1A_4A_7</math>, <math>A_2A_5A_8</math>, and <math>A_3A_6A_9</math> are each counted 3 times, resulting in an overcount of 6. Thus, there are <math>\boxed{66}</math> distinct equilateral triangles. | |
− | Each of the <math>{9 | ||
== See Also == | == See Also == |
Latest revision as of 13:51, 29 March 2020
Problem
Given the nine-sided regular polygon , how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set ?
Solution
Each of the pairs of vertices determines two equilateral triangles, for a total of 72 triangles. However, the three triangles , , and are each counted 3 times, resulting in an overcount of 6. Thus, there are distinct equilateral triangles.
See Also
2001 AMC 12 (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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All AMC 12 Problems and Solutions |
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