Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2001 AMC 12 Problems/Problem 14"
m (→‎Solution: clarification)
 
(2 intermediate revisions by 2 users not shown)
Line 16: Line 16:
  
 
== Solution ==
 
== Solution ==
 +
Each of the <math>\binom{9}{2} = 36</math> 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 <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\choose 2}=36</math> pairs of vertices determines two equilateral triangles, one on each side of the segment. This would give us <math>72</math> triangles. However, note that there are three equilateral triangles that have all three vertices among the vertices of the polygon. These are the triangles <math>A_1A_4A_7</math>, <math>A_2A_5A_8</math>, and <math>A_3A_6A_9</math>. We counted each of these three times (once for each side). Hence we overcounted by <math>2</math> for each of these triangles for a total of <math>6</math> overcounted, and the correct number of equilateral triangles is <math>72-6=\boxed{66}</math>.
+
== Video Solution ==
 +
https://youtu.be/gMWJisI9Ulk
  
 
== See Also ==
 
== See Also ==

Latest revision as of 01:21, 10 November 2024

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

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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2001_AMC_12_Problems/Problem_14&oldid=232977"