how many different triangles  with side lengths that are whole numbers have a perimeter of 15 cm 

To find the number of different triangles with whole-number side lengths and a perimeter of 15 cm, we need to use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's set the side lengths as \(a\), \(b\), and \(c\) where \(a \leq b \leq c\), and their sum (the perimeter) is 15:
\[
a + b + c = 15
\]

Given this setup, we have two conditions:
1. \(a + b + c = 15\),
2. \(a + b > c\), \(a + c > b\), and \(b + c > a\).

Since \(a\), \(b\), and \(c\) are whole numbers and \(a \leq b \leq c\), we can start by exploring values of \(c\) from 5 to 7 (as increasing \(c\) further will not satisfy the triangle inequality):

### Case by Case Analysis
1. **If \(c = 7\):**
   - \(a + b = 15 - 7 = 8\)
   - Possible pairs \((a, b)\) where \(a \leq b\) and \(a + b = 8\):
     - \(a = 3\), \(b = 5\)
     - \(a = 4\), \(b = 4\)
   - These pairs satisfy the triangle inequality.

2. **If \(c = 6\):**
   - \(a + b = 15 - 6 = 9\)
   - Possible pairs \((a, b)\) where \(a \leq b\) and \(a + b = 9\):
     - \(a = 3\), \(b = 6\)
     - \(a = 4\), \(b = 5\)
   - These pairs satisfy the triangle inequality.

3. **If \(c = 5\):**
   - \(a + b = 15 - 5 = 10\)
   - Possible pairs \((a, b)\) where \(a \leq b\) and \(a + b = 10\):
     - \(a = 5\), \(b = 5\)
   - This pair satisfies the triangle inequality.

### Summary of Valid Triangles
The valid triangles with side lengths in whole numbers that have a perimeter of 15 cm are:
- \((3, 5, 7)\)
- \((4, 4, 7)\)
- \((3, 6, 6)\)
- \((4, 5, 6)\)
- \((5, 5, 5)\)

Thus, there are **5 different triangles** with whole-number side lengths and a perimeter of 15 cm.

Would you like a detailed explanation of each case? 

Here are five additional questions related to this topic:

1. How would the answer change if the perimeter were 20 cm instead of 15 cm?
2. How can we verify that all cases satisfy the triangle inequality without calculating each one?
3. What happens if we allow for non-integer side lengths but still require the perimeter to be 15 cm?
4. How can we generalize this approach to find triangles with any given perimeter?
5. Can we create a formula to find the number of triangles given any specific integer perimeter?

**Tip:** For problems with a fixed perimeter, always check constraints from the triangle inequality to avoid impossible side length combinations.

how many different triangles with side lengths that are whole numbers have a perimeter of 15 cm

Learn how to find the number of different triangles with whole number side lengths and a perimeter of 15 cm, using the triangle inequality theorem. This detailed explanation covers five valid triangles and explores key geometric principles. Suitable for students in Grades 7-9.

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