If two sides of an obtuse angled triangle are 8 cm and 15 cm and third sides is x, then

Options

A. `7ltxlt23`
B. `7ltxltsqrt(161)`
C. `17ltxlt23`
D. (b) or, (c )

Answer

To solve the problem, we need to establish the relationship between the sides of an obtuse-angled triangle. We are given two sides of the triangle, which are 8 cm and 15 cm, and we need to find the possible values for the third side, denoted as \( x \).

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
Let the sides of the triangle be:
- \( A = x \) (the third side)
- \( B = 8 \) cm
- \( C = 15 \) cm

2. **Use the property of obtuse triangles**:
In an obtuse triangle, the square of the longest side is greater than the sum of the squares of the other two sides. This can be expressed as:
\[ C^2 > A^2 + B^2 \]
Here, \( C \) is the longest side, which is 15 cm.

3. **Set up the inequality**:
\[ 15^2 > x^2 + 8^2 \]
This simplifies to:
\[ 225 > x^2 + 64 \]

4. **Rearrange the inequality**:
Subtract 64 from both sides:
\[ 225 - 64 > x^2 \]
This gives us:
\[ 161 > x^2 \]
or
\[ x^2 < 161 \]

5. **Find the upper limit for \( x \)**:
Taking the square root of both sides, we find:
\[ x < \sqrt{161} \]

6. **Determine the lower limit for \( x \)**:
For the triangle inequality to hold, the sum of the lengths of any two sides must be greater than the length of the third side. Therefore, we need to check:
\[ x + 8 > 15 \quad \Rightarrow \quad x > 15 - 8 \quad \Rightarrow \quad x > 7 \]

7. **Combine the inequalities**:
We now have two inequalities:
\[ 7 < x < \sqrt{161} \]

### Final Result:
Thus, the range for \( x \) is:
\[ 7 < x < \sqrt{161} \]