2001 AMC 12 Problems/Problem 14
Contents
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 
equilateral triangles — one facing towards the center, and one outwards — for a total of 
triangles. However, the 
triangles 
, 
, and 
are each counted times (once for each of the 
possible pairs of vertices, all of which are vertices of the 
-gon), whereas they should of course only be counted once, resulting in an overcount of 
. Thus, there are 
distinct equilateral triangles.
Video Solution
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 | |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
