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Difference between revisions of "2006 AMC 10A Problems/Problem 19"

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== Problem ==
 
== Problem ==
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How many non-similar triangle have angles whose degree measures are distinct positive integers in arithmetic progression?  
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How many non-[[similar]] [[triangle]]s have [[angle]]s whose [[degree]] measures are distinct positive integers in [[arithmetic progression]]?  
  
 
<math>\mathrm{(A) \ } 0\qquad\mathrm{(B) \ } 1\qquad\mathrm{(C) \ } 59\qquad\mathrm{(D) \ } 89\qquad\mathrm{(E) \ } 178\qquad</math>
 
<math>\mathrm{(A) \ } 0\qquad\mathrm{(B) \ } 1\qquad\mathrm{(C) \ } 59\qquad\mathrm{(D) \ } 89\qquad\mathrm{(E) \ } 178\qquad</math>
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== Solution ==
 
== Solution ==
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Let us begin by first realizing that the sum of the [[angle]]s must add up to 180 degrees. Then let us consider the highest and lowest sets of angles that satisfy the conditions of the problem.<br>
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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 possibly 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>.
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Highest: 1-60-119<br>
 
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Lowest: 59-60-61<br>
 
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The increment in the highest set is 59, while the increment in the lowest set is 1. Therefore, any increment between 1 and 59 would create a set of angles that work. Therefore, there are 59 possibilities. (c)
 
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== See Also ==
 
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*[[2006 AMC 10A Problems]]
 
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*[[2006 AMC 10A Problems/Problem 18|Previous Problem]]
 
  
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*[[2006 AMC 10A Problems/Problem 20|Next Problem]]
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== See also ==
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{{AMC10 box|year=2006|ab=A|num-b=18|num-a=20}}
  
 
[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]

Revision as of 16:10, 26 February 2007

Problem

How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?

$\mathrm{(A) \ } 0\qquad\mathrm{(B) \ } 1\qquad\mathrm{(C) \ } 59\qquad\mathrm{(D) \ } 89\qquad\mathrm{(E) \ } 178\qquad$

Solution

The sum of the angles of a triangle is $180$ 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 $\frac{180}{3} = 60$ degrees. The minimum possibly value for the smallest angle is $1$ and the highest possible is $59$ (since the numbers are distinct), so there are $59$ possibilities $\Longrightarrow \mathrm{C}$.

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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