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 angle triangles with sides 8 cm, 15 cm, and \( x \) cm (where the largest side is 15 cm), we can follow these steps:

### Step 1: Determine the range for \( x \)
For any triangle with sides \( a \), \( b \), and \( c \) (where \( c \) is the longest side), the triangle inequality states that:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

In our case, we have:
- \( a = 8 \) cm
- \( b = x \) cm
- \( c = 15 \) cm

Applying the triangle inequality:
1. \( 8 + x > 15 \) → \( x > 7 \)
2. \( 8 + 15 > x \) → \( x < 23 \)
3. \( x + 15 > 8 \) → This condition is always satisfied since \( x \) is positive.

From the first two inequalities, we find:
\[ 7 < x < 23 \]

Since \( x \) must be an integer, the possible integer values for \( x \) are from 8 to 22.

### Step 2: Determine conditions for the triangle to be obtuse
For the triangle to be obtuse with \( c \) as the longest side, the following condition must hold:
\[ a^2 + b^2 < c^2 \]

Substituting the values:
\[ 8^2 + x^2 < 15^2 \]
\[ 64 + x^2 < 225 \]
\[ x^2 < 225 - 64 \]
\[ x^2 < 161 \]
\[ x < \sqrt{161} \approx 12.688 \]

Since \( x \) is an integer, we can take the largest integer less than 12.688, which is 12. Therefore:
\[ x \leq 12 \]

### Step 3: Combine the conditions
From Step 1, we found \( 7 < x < 23 \). From Step 2, we found \( x \leq 12 \). Thus, we combine these conditions:
\[ 7 < x \leq 12 \]

The possible integer values for \( x \) are:
- 8
- 9
- 10
- 11
- 12

### Step 4: Count the possible values
The integers that satisfy the conditions are 8, 9, 10, 11, and 12. Therefore, there are a total of 5 possible values for \( x \).

### Final Answer
Thus, the number of possible obtuse angle triangles is **5**.

---