Difference between revisions of "2001 AMC 12 Problems/Problem 14"
(New page: == Problem == Given the nine-sided regular polygon <math>A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9</math>, how many distinct equilateral triangles in the plane of the polygon have at least two ...) |
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Revision as of 21:03, 3 July 2013
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, one on each side of the segment. This would give us 
triangles. However, note that there are three equilateral triangles that have all three vertices among the vertices of the polygon. These are the triangles 
, 
, and 
. We counted each of these three times (once for each side). Hence we overcounted by 
, and the correct number of equilateral triangles is 
.
See Also
| 2001 AMC 12 (Problems • Answer Key • Resources) | |
| Preceded by Problem 13 |
Followed by Problem 15 |
| 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 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
