Question

all the sides of a triangle are integers, and the perimeter is $12.$ how many different possible triangles are there? (assume that the triangle is non-degenerate. two triangles are considered the same if they are congruent.)

Answer 1

To find the number of different possible triangles with integer side lengths and a perimeter of 12, we need to consider the restrictions on triangle side lengths. There are three different possible triangles that satisfy the given conditions.

Step-by-step explanation:

To find the number of different possible triangles with integer side lengths and a perimeter of 12, we need to consider the restrictions on triangle side lengths. In a non-degenerate triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Based on this constraint, we can try all possible combinations of integer side lengths that add up to 12. The possible triangles are:

1. Side lengths: 1, 5, 6
2. Side lengths: 2, 4, 6
3. Side lengths: 3, 4, 5

Therefore, there are three different possible triangles that satisfy the given conditions.