Difference between revisions of "2006 AMC 10A Problems/Problem 19"
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== Problem == | == Problem == | ||
− | How many non-similar triangle have | + | 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 == | ||
− | + | 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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− | + | == See also == | |
+ | {{AMC10 box|year=2006|ab=A|num-b=18|num-a=20}} | ||
[[Category:Introductory Geometry Problems]] | [[Category:Introductory Geometry Problems]] |
Revision as of 17:10, 26 February 2007
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 possibly 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 |