Question 1172304: How many non-congruent triangles can be formed having integer sides and perimeter equals 20 units?
Answer by math_helper(2461)   (Show Source):
You can put this solution on YOUR website! 
The side lengths are (a,b,c) and they must obey a+b>c, a+c>b, b+c>a (the triangle inequality theorem)
With that, looking just at choices for a, and counting down from a=20, you don't get to a valid triangle until you reach a=9:
Values of (a,b,c) that satisfy the triangle inequality theorem:
(9,9,2),(9,8,3),(9,7,4),(9,6,5)
(8,8,4),(8,7,5),(8,6,6)
(7,7,6)
If you now try a<7, e.g. (6,b,c), you will find you get congruency with a previously listed triangle.
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