Asked by K
How many non-congruent triangles with only integer side legnths have a perimeter of 15 units?
Answers
Answered by
mike
how many non-congruent triangles with only integers side lengths have a perimeter of 15 units
Answered by
Anonymous
These triangles:
4,4,7
4,5,6
5,5,5
7,7,1
6,6,3
2,6,7
3,5,7
4,4,7
4,5,6
5,5,5
7,7,1
6,6,3
2,6,7
3,5,7
Answered by
Alcumus
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.
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.
Answered by
Alcumus
This is because you cheated, we gave you the answers