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All the sides of a triangle are integers.  If the perimeter of the triangle is $3,$ then how many different possible triangles are there?  (Assume that the triangle is non-degenerate.  Two triangles are considered the same if they are congruent.)

 Dec 10, 2023
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By the Triangle Inequality, the lengths of any two sides of the triangle must sum to a value greater than the length of the third side. Therefore, no side of the triangle can have a length of 3, as this would violate the Triangle Inequality. The possible combinations for the lengths of the sides are:

 

(1,1,1). There is only one triangle with side lengths (1,1,1).

 

(2,1,1). Each leg of a right isosceles triangle with hypotenuse of length 2 has length 1. Since the triangle is non-degenerate, this is the only triangle with side lengths (2,1,1).

 

Therefore, there are 2​ possible triangles.

 Dec 10, 2023

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