Question

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 $\{A_1, A_2, \ldots, A_9\}?$

$30$

$36$

$63$

$66$

$72$

Answer: D

Concepts:counting shapes in figures regular polygon double counting

Difficulty rating: 1710

Small Hint:

Each pair of vertices is a side of exactly two equilateral triangles

Big Hint:

This counts $2\binom{9}{2}$ triangles, but the three all-vertex triangles get counted three times each

Solution:

Each of the $\binom{9}{2} = 36$ pairs of vertices is a side of exactly two equilateral triangles, giving $72$ triangles counted with multiplicity.

The triangles $A_1 A_4 A_7,$ $A_2 A_5 A_8,$ and $A_3 A_6 A_9$ have all three vertices in the set, so each is counted three times instead of once, an overcount of $2$ apiece.

The number of distinct triangles is $72 - 3 \cdot 2 = 66.$

Thus, the correct answer is D.

2001 AMC 12 Problem 14

Answer: D

Difficulty rating: 1710

Small Hint:

Big Hint:

Solution:

Problem 14 in Other Years