To solve the problem of how many scalene triangles exist with integer sides and a perimeter of 15 cm, we can follow these steps:
### Step 1: Understand the properties of a scalene triangle
A scalene triangle has all sides of different lengths. Therefore, if we denote the sides of the triangle as \( a \), \( b \), and \( c \), we must have \( a \neq b \neq c \).
### Step 2: Set up the equation for the perimeter
The perimeter of the triangle is given by the equation:
\[ a + b + c = 15 \]
where \( a \), \( b \), and \( c \) are the lengths of the sides.
### Step 3: Apply the triangle inequality
For any triangle, the triangle inequality must hold:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Since \( a + b + c = 15 \), we can rearrange the inequalities:
- From \( a + b > c \), we can substitute \( c = 15 - a - b \):
\[ a + b > 15 - a - b \]
\[ 2(a + b) > 15 \]
\[ a + b > 7.5 \]
Since \( a + b \) must be an integer, we have:
\[ a + b \geq 8 \]
- Similarly, from \( a + c > b \):
\[ a + (15 - a - b) > b \]
\[ 15 - b > b \]
\[ 15 > 2b \]
\[ b < 7.5 \]
Thus, \( b \leq 7 \).
- From \( b + c > a \):
\[ b + (15 - a - b) > a \]
\[ 15 - a > a \]
\[ 15 > 2a \]
\[ a < 7.5 \]
Thus, \( a \leq 7 \).
### Step 4: List possible integer combinations
Now we need to find all combinations of \( a \), \( b \), and \( c \) such that:
- \( a + b + c = 15 \)
- \( a \), \( b \), and \( c \) are all different integers
- \( a + b \geq 8 \)
- \( a, b, c \leq 7 \)
We can start by fixing \( a \) and finding valid pairs \( (b, c) \).
### Step 5: Check combinations
Let's check the possible values for \( a \):
1. **If \( a = 1 \)**:
- \( b + c = 14 \) (not possible since \( b, c \) must be different and \( \leq 7 \))
2. **If \( a = 2 \)**:
- \( b + c = 13 \) (not possible)
3. **If \( a = 3 \)**:
- \( b + c = 12 \) (not possible)
4. **If \( a = 4 \)**:
- \( b + c = 11 \) (not possible)
5. **If \( a = 5 \)**:
- \( b + c = 10 \) (valid pairs: \( (5, 5) \) not allowed)
6. **If \( a = 6 \)**:
- \( b + c = 9 \) (valid pairs: \( (6, 3) \), \( (4, 5) \))
7. **If \( a = 7 \)**:
- \( b + c = 8 \) (valid pairs: \( (1, 7) \), \( (2, 6) \), \( (3, 5) \))
### Step 6: Count valid combinations
The valid combinations that satisfy all conditions are:
- \( (6, 4, 5) \)
- \( (7, 6, 2) \)
- \( (7, 5, 3) \)
- \( (7, 4, 4) \) (not valid since sides must be different)
### Conclusion
After checking all combinations, we find that there are **4 valid scalene triangles** with integer sides and a perimeter of 15 cm.
