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The problem asks for how many different (non-congruent) triangles can be drawn with sides that are integers, where the perimeter is 12 cm.

### Solution Outline:
We are looking for integer side lengths \(a\), \(b\), and \(c\) such that:
1. \( a + b + c = 12 \) (since the perimeter is 12 cm)
2. The sides must satisfy the triangle inequality:
   - \( a + b > c \)
   - \( a + c > b \)
   - \( b + c > a \)

### Step-by-Step Solution:

Given the perimeter is 12, we can write the equation:

\[
a + b + c = 12
\]
From this, \(c = 12 - a - b\). Now, substitute this into the triangle inequality conditions:

1. \(a + b > c \Rightarrow a + b > 12 - a - b \Rightarrow 2a + 2b > 12 \Rightarrow a + b > 6\)
2. \(a + c > b \Rightarrow a + (12 - a - b) > b \Rightarrow 12 - b > b \Rightarrow b < 6\)
3. Similarly, \(b + c > a \Rightarrow b + (12 - a - b) > a \Rightarrow 12 - a > a \Rightarrow a < 6\)

So, we need to find pairs of integers \(a\), \(b\), and \(c = 12 - a - b\) such that \(a + b + c = 12\), and \(a + b > 6\).

We now check all possible combinations of \(a\), \(b\), and \(c\) under these conditions:

| \(a\) | \(b\) | \(c\) | Triangle Inequality Satisfied? |
|------|------|------|--------------------------------|
| 1    | 5    | 6    | Yes                          |
| 2    | 4    | 6    | Yes                          |
| 2    | 5    | 5    | Yes                          |
| 3    | 3    | 6    | Yes                          |
| 3    | 4    | 5    | Yes                          |
| 4    | 4    | 4    | Yes                          |

Thus, there are **5 different (non-congruent) triangles**.

### Answer:
The correct answer is **5**.

Would you like further details or explanations?

### Related Questions:
1. How can we determine the number of non-congruent triangles for a different perimeter?
2. What are the triangle inequality conditions?
3. Can two triangles with the same perimeter and different side lengths be congruent?
4. What are the properties of integer-sided triangles?
5. How does the triangle inequality relate to different types of triangles?

**Tip:** Always check the triangle inequality after finding the side lengths to ensure a valid triangle!

A triangle is to be drawn with sides that are integer lengths in centimeters, and a total perimeter of 12 cm. How many different (non-congruent) triangles can be drawn?

Learn how to determine how many different (non-congruent) triangles can be formed with integer side lengths and a perimeter of 12 cm. This problem explores key geometry concepts such as triangle inequalities and integer-sided triangles. Suitable for high school students (Grades 9-12).

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