To find how many isosceles triangles with integral sides are possible with a perimeter of 12 cm, we can follow these steps:
### Step-by-Step Solution:
1. **Define the sides of the triangle**:
For an isosceles triangle, we can denote the lengths of the two equal sides as \( A \) and the base as \( B \). Therefore, the perimeter can be expressed as:
\[
2A + B = 12
\]
2. **Express \( B \) in terms of \( A \)**:
Rearranging the equation gives:
\[
B = 12 - 2A
\]
3. **Determine the conditions for triangle inequality**:
For a triangle to be valid, it must satisfy the triangle inequality, which states that the sum of the lengths of any two sides must be greater than the length of the third side. For our isosceles triangle, we have the following conditions:
- \( A + A > B \) (which simplifies to \( 2A > B \))
- \( A + B > A \) (which simplifies to \( B > 0 \))
- \( A + B > A \) (this is always true since \( B > 0 \))
4. **Substituting \( B \)**:
From \( B = 12 - 2A \), we substitute into the inequality \( 2A > B \):
\[
2A > 12 - 2A
\]
Simplifying this gives:
\[
4A > 12 \quad \Rightarrow \quad A > 3
\]
5. **Condition for \( B > 0 \)**:
We also need \( B > 0 \):
\[
12 - 2A > 0 \quad \Rightarrow \quad 12 > 2A \quad \Rightarrow \quad A < 6
\]
6. **Determine possible integer values for \( A \)**:
From the inequalities \( 3 < A < 6 \), the possible integer values for \( A \) are:
- \( A = 4 \)
- \( A = 5 \)
7. **Calculate corresponding \( B \) values**:
- If \( A = 4 \):
\[
B = 12 - 2(4) = 12 - 8 = 4
\]
This gives the triangle sides \( 4, 4, 4 \) (equilateral).
- If \( A = 5 \):
\[
B = 12 - 2(5) = 12 - 10 = 2
\]
This gives the triangle sides \( 5, 5, 2 \) (isosceles).
8. **Count valid triangles**:
- The triangle \( 4, 4, 4 \) is equilateral, which does not count as isosceles for this problem.
- The triangle \( 5, 5, 2 \) is valid and isosceles.
Thus, the only isosceles triangle with integral sides and a perimeter of 12 cm is \( 5, 5, 2 \).
### Final Answer:
There is **1 isosceles triangle** with integral sides possible with a perimeter of 12 cm.
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