How many non-congruent triangles with only integer side legnths have a perimeter of 15 units?
These triangles:
4,4,7
4,5,6
5,5,5
7,7,1
6,6,3
2,6,7
3,5,7
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how many non-congruent triangles with only integers side lengths have a perimeter of 15 units
To find the number of non-congruent triangles with integer side lengths and a perimeter of 15 units, we can use the concept of triangle inequality. According to this rule, in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
We can start by considering the smallest possible side lengths for the triangle. Since all side lengths must be integers, the smallest possible side lengths are 1, 1, and 1. However, the sum of these side lengths is only 3, which is less than the required perimeter of 15 units. Therefore, it is not possible to have a triangle with side lengths of 1, 1, and 1.
Next, we can consider the possibility of having side lengths of 1, 1, and 13. In this case, the sum of the two smaller sides is 2, which is smaller than the third side length. Thus, it does not satisfy the triangle inequality, and this combination is not possible.
We can continue this process of trying different combinations of side lengths until we find all the possible triangles. Here is a list of all the non-congruent triangles that satisfy the given conditions:
1. Side lengths: 2, 6, 7
2. Side lengths: 3, 5, 7
So, there are two non-congruent triangles with integer side lengths and a perimeter of 15 units.
In a triangle, the lengths of any two sides must add up to a value larger than the third length's side. This is known as the Triangle Inequality. Keeping this in mind, we list out cases based on the length of the shortest side.
Case 1: shortest side has length $1$. Then the other two sides must have lengths $7$ and $7$. This leads to the set $\{1,7,7\}$.
Case 2: shortest side has length $2$. Then the other two sides must have lengths $6$ and $7$. This leads to the set $\{2,6,7\}$.
Case 3: shortest side has length $3$. Then the other two sides can have lengths $6$ and $6$ or $5$ and $7$. This leads to the sets $\{3,6,6\}$ and $\{3,5,7\}$.
Case 4: shortest side has length $4$. Then the other two sides can have lengths $5$ and $6$ or $4$ and $7$. This leads to the sets $\{4,5,6\}$ and $\{4,4,7\}$.
Case 5: shortest side has length $5$. Then the other two sides must have lengths $5$ and $5$. This leads to the set $\{5,5,5\}$.
Hence there are $\boxed{7}$ sets of non-congruent triangles with a perimeter of $15$ units.