To find the number of different triangles with integer side lengths that have a perimeter of 12 cm, we can follow these steps:
### Step 1: Understand the Triangle Inequality
The triangle inequality states that for any triangle with sides \(A\), \(B\), and \(C\):
1. \(A + B > C\)
2. \(A + C > B\)
3. \(B + C > A\)
### Step 2: Set Up the Equation
Since the perimeter of the triangle is 12 cm, we can express this as:
\[ A + B + C = 12 \]
### Step 3: Express One Side in Terms of the Others
We can express \(C\) in terms of \(A\) and \(B\):
\[ C = 12 - A - B \]
### Step 4: Apply the Triangle Inequality
Now we substitute \(C\) into the triangle inequalities:
1. From \(A + B > C\):
\[ A + B > 12 - A - B \]
\[ 2A + 2B > 12 \]
\[ A + B > 6 \]
2. From \(A + C > B\):
\[ A + (12 - A - B) > B \]
\[ 12 - B > B \]
\[ 12 > 2B \]
\[ B < 6 \]
3. From \(B + C > A\):
\[ B + (12 - A - B) > A \]
\[ 12 - A > A \]
\[ 12 > 2A \]
\[ A < 6 \]
### Step 5: Determine Possible Values
From the inequalities, we have:
- \(A + B > 6\)
- \(A < 6\)
- \(B < 6\)
### Step 6: List Possible Integer Combinations
Now we can list the integer combinations of \(A\), \(B\), and \(C\) that satisfy these inequalities:
1. **If \(A = 1\)**:
- \(B + C = 11\) (not possible since \(B\) and \(C\) must be less than 6)
2. **If \(A = 2\)**:
- \(B + C = 10\) (not possible)
3. **If \(A = 3\)**:
- \(B + C = 9\) (not possible)
4. **If \(A = 4\)**:
- \(B + C = 8\) (not possible)
5. **If \(A = 5\)**:
- \(B + C = 7\) (possible combinations: \(B = 5, C = 2\) or \(B = 4, C = 3\))
6. **If \(A = 6\)**:
- \(B + C = 6\) (possible combinations: \(B = 3, C = 3\))
### Step 7: Valid Combinations
From the above steps, the valid combinations that satisfy the triangle inequality and the perimeter condition are:
1. \(A = 5, B = 5, C = 2\)
2. \(A = 5, B = 4, C = 3\)
3. \(A = 4, B = 4, C = 4\)
### Conclusion
Thus, the different triangles that can be formed with integer side lengths summing to 12 cm are:
- \( (5, 5, 2) \)
- \( (5, 4, 3) \)
- \( (4, 4, 4) \)
Therefore, the total number of different triangles possible is **3**.
