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 \( x \) is the largest side), we will follow these steps:

### Step 1: Understand the properties of a triangle
For any triangle with sides \( a \), \( b \), and \( c \) (where \( c \) is the largest side), the following inequalities must hold:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

In our case, the sides are 8 cm, 15 cm, and \( x \) cm, with \( 15 \) being the largest side.

### Step 2: Apply the triangle inequality
Since \( 15 \) is the largest side, we need to satisfy the triangle inequality:
1. \( 8 + x > 15 \)
2. \( 8 + 15 > x \)
3. \( 15 + x > 8 \)

From the first inequality:
\[ 8 + x > 15 \implies x > 7 \]

From the second inequality:
\[ 8 + 15 > x \implies 23 > x \implies x < 23 \]

From the third inequality:
\[ 15 + x > 8 \implies x > -7 \text{ (always true since } x > 7 \text{)} \]

Thus, the relevant inequalities we need to consider are:
\[ 7 < x < 23 \]

### Step 3: Determine the integer values for \( x \)
Since \( x \) must be an integer, we can list the possible integer values for \( x \):
\[ x = 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 \]

This gives us a total of \( 22 - 8 + 1 = 15 \) integer values.

### Step 4: Check for obtuse triangle condition
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:
\[ 15^2 > 8^2 + x^2 \]
Calculating the squares:
\[ 225 > 64 + x^2 \]
This simplifies to:
\[ x^2 < 161 \]

### Step 5: Find the integer values of \( x \) satisfying the obtuse condition
Now we need to find the integer values of \( x \) such that:
\[ x < \sqrt{161} \approx 12.688 \]
Thus, the possible integer values of \( x \) are:
\[ x = 8, 9, 10, 11, 12 \]

### Step 6: Count the valid integer values
The valid integer values of \( x \) that satisfy both the triangle inequality and the obtuse condition are:
- \( 8, 9, 10, 11, 12 \)

This gives us a total of \( 5 \) possible triangles.

### Final Answer
The number of possible obtuse triangles with sides 8 cm, 15 cm, and \( x \) cm is **5**.