Difference between revisions of "2006 AMC 10A Problems/Problem 19"
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== Solution == | == Solution == | ||
| − | The sum of the angles of a triangle is <math>180</math> degrees. For an arithmetic progression with an odd number of terms, the middle term is equal to the average of the sum of all of the terms, making it <math>\frac{180}{3} = 60</math> degrees. The minimum | + | The sum of the angles of a triangle is <math>180</math> degrees. For an arithmetic progression with an odd number of terms, the middle term is equal to the average of the sum of all of the terms, making it <math>\frac{180}{3} = 60</math> degrees. The minimum possible value for the smallest angle is <math>1</math> and the highest possible is <math>59</math> (since the numbers are distinct), so there are <math>59</math> possibilities <math>\Longrightarrow \mathrm{C}</math>. |
== See also == | == See also == | ||
Revision as of 12:20, 29 December 2013
Problem
How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?
Solution
The sum of the angles of a triangle is 
degrees. For an arithmetic progression with an odd number of terms, the middle term is equal to the average of the sum of all of the terms, making it 
degrees. The minimum possible value for the smallest angle is 
and the highest possible is 
(since the numbers are distinct), so there are possibilities 
.
See also
| 2006 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
| 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 10 Problems and Solutions | ||
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