2001 AMC 12 Problems/Problem 14

Problem

Given the nine-sided regular polygon $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9$ , how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $\{A_1,A_2,\dots,A_9\}$ ?

$\text{(A) }30 \qquad \text{(B) }36 \qquad \text{(C) }63 \qquad \text{(D) }66 \qquad \text{(E) }72$

Solution

Each of the $\binom{9}{2} = 36$ pairs of vertices determines two equilateral triangles — one facing towards the center, and one outwards — for a total of 72 triangles. However, the three triangles $A_1A_4A_7$ , $A_2A_5A_8$ , and $A_3A_6A_9$ are each counted 3 times, resulting in an overcount of 6. Thus, there are $\boxed{66}$ distinct equilateral triangles.

Video Solution

https://youtu.be/gMWJisI9Ulk

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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2001 AMC 12 Problems/Problem 14

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Problem

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Video Solution

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