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.)

tomtom Dec 10, 2023

#1**0 **

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.

BuiIderBoi Dec 10, 2023