Consider on obtuse angle triangles with side 8 cm, 15 cm and `xx` cm (largest side being 15 cm). If `xx ` is an integer, then find the number of possible triangels.

Answer

To solve the problem of finding the number of possible obtuse triangles with sides 8 cm, 15 cm, and \( x \) cm (where 15 cm is the largest side), we will follow these steps:

### Step 1: Apply the Triangle Inequality
The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. Since 15 cm is the largest side, we can write the inequalities:
1. \( 8 + x > 15 \)
2. \( 8 + 15 > x \)
3. \( 15 + x > 8 \)

### Step 2: Solve the Inequalities
1. From \( 8 + x > 15 \):
\[ x > 15 - 8 \implies x > 7 \]

2. From \( 8 + 15 > x \):
\[ 23 > x \implies x < 23 \]

3. From \( 15 + x > 8 \):
\[ x > 8 - 15 \implies x > -7 \quad \text{(This inequality is always satisfied since } x > 7\text{)} \]

Thus, combining the valid inequalities, we have:
\[ 7 < x < 23 \]

### Step 3: Determine Integer Values for \( x \)
Since \( x \) must be an integer, the possible integer values for \( x \) are:
\[ x = 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 \]
This gives us a total of 15 possible integer values for \( x \).

### Step 4: Ensure the Triangle is Obtuse
For the triangle to be obtuse, the square of the largest side must be greater than the sum of the squares of the other two sides. Using the cosine rule:
\[ c^2 > a^2 + b^2 \quad \text{(where \( c \) is the largest side)} \]
In our case:
\[ 15^2 > 8^2 + x^2 \]
This simplifies to:
\[ 225 > 64 + x^2 \implies 225 - 64 > x^2 \implies 161 > x^2 \]
Thus:
\[ x^2 < 161 \implies x < \sqrt{161} \approx 12.688 \]

### Step 5: Identify Valid Integer Values for \( x \)
Since \( x \) must be an integer and also less than approximately 12.688, the possible integer values for \( x \) are:
\[ x = 8, 9, 10, 11, 12 \]
This gives us a total of 5 possible integer values for \( x \) that result in an obtuse triangle.

### Final Answer
Thus, the number of possible obtuse triangles with sides 8 cm, 15 cm, and \( x \) cm is:
\[ \boxed{5} \]