The perimeter of a triangle is 12 cm. If all the three sides have lengths (in cm), in integers, then how many such different triangles are possible ?

Question

Accepted Answer

To determine how many different triangles are possible with a perimeter of 12 cm and integer side lengths, we need to consider the properties of triangles and the given conditions.

### Explanation for Option 2:

1. **Triangle Inequality Theorem**: For any triangle with sides $a$, $b$, and $c$, the sum of the lengths of any two sides must be greater than the length of the third side. This means:
   - $a + b > c$
   - $a + c > b$
   - $b + c > a$

2. **Perimeter Condition**: The perimeter of the triangle is given as 12 cm, so:
   $$
   a + b + c = 12
   $$

3. **Integer Side Lengths**: The sides $a$, $b$, and $c$ must be integers.

4. **Finding Possible Triangles**:
   - Let's assume $a \leq b \leq c$ to avoid repetition and ensure uniqueness.
   - Using the perimeter condition, $a + b + c = 12$, we can express $c$ as:
     $$
     c = 12 - a - b
     $$
   - Applying the triangle inequality $a + b > c$, we substitute for $c$:
     $$
     a + b > 12 - a - b
     $$
     $$
     2(a + b) > 12
     $$
     $$
     a + b > 6
     $$

5. **Checking Integer Solutions**:
   - Since $a \leq b \leq c$, we start with the smallest possible values for $a$ and $b$ that satisfy $a + b > 6$.
   - Possible combinations are:
     - $a = 3$, $b = 4$, $c = 5$ (since $3 + 4 + 5 = 12$)
     - $a = 4$, $b = 4$, $c = 4$ (since $4 + 4 + 4 = 12$)
     - $a = 3$, $b = 5$, $c = 4$ (since $3 + 5 + 4 = 12$)

6. **Verification**:
   - For each combination, verify the triangle inequality:
     - $3 + 4 > 5$, $3 + 5 > 4$, $4 + 5 > 3$
     - $4 + 4 > 4$, $4 + 4 > 4$, $4 + 4 > 4$
     - $3 + 5 > 4$, $3 + 4 > 5$, $5 + 4 > 3$

7. **Conclusion**:
   - There are exactly 3 different sets of integer side lengths that satisfy both the perimeter and the triangle inequality conditions.

### Explanation for Other Options:

- **Option 1 (2 triangles)**: This is incorrect because we have identified 3 valid triangles.
- **Option 3 (4 triangles)**: This is incorrect because only 3 combinations satisfy all conditions.
- **Option 4 (5 triangles)**: This is incorrect because there are no more than 3 valid combinations.

Thus, the correct answer is Option 2, as there are exactly 3 different triangles possible with the given conditions.