Difference between revisions of "2006 AMC 10A Problems/Problem 19"
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<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 == | ||
| − | Let us begin by first realizing that the sum of the | + | 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> |
| − | Highest: 1-60-119 | + | Highest: 1-60-119<br> |
| − | Lowest: 59-60-61 | + | Lowest: 59-60-61<br> |
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) | 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 == | == See Also == | ||
*[[2006 AMC 10A Problems]] | *[[2006 AMC 10A Problems]] | ||
Revision as of 14:35, 18 July 2006
Problem
How many non-similar triangle have angles whose degree measures are distinct positive integers in arithmetic progression?
Solution
Let us begin by first realizing that the sum of the angles must add up to 180 degrees. Then let us consider the highest and lowest sets of angles that satisfy the conditions of the problem.
Highest: 1-60-119
Lowest: 59-60-61
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)
