Two sides of a triangle have lengths of 8 and 17. What is the range of possible vlaus of the length of the thrid side?

Answer

To find the range of possible values for the length of the third side of a triangle when the lengths of the other two sides are given as 8 and 17, we can use the triangle inequality theorem.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**: Let the lengths of the two known sides be \( a = 8 \) and \( b = 17 \). We need to find the possible values for the length of the third side, which we will denote as \( x \).

2. **Apply the triangle inequality theorem**: The triangle inequality states that for any triangle with sides \( a \), \( b \), and \( x \):
- The sum of the lengths of any two sides must be greater than the length of the third side.
- This gives us two inequalities:
- \( a + b > x \)
- \( x + a > b \)
- \( x + b > a \)

3. **Set up the inequalities**:
- From \( a + b > x \):
\[ 8 + 17 > x \implies 25 > x \implies x < 25 \]
- From \( x + a > b \):
\[ x + 8 > 17 \implies x > 17 - 8 \implies x > 9 \]
- From \( x + b > a \):
\[ x + 17 > 8 \implies x > 8 - 17 \implies x > -9 \quad (\text{This inequality is always true since } x > 9) \]

4. **Combine the inequalities**:
- From the inequalities derived, we have:
\[ 9 < x < 25 \]

5. **Conclusion**:
- Therefore, the range of possible values for the length of the third side \( x \) is:
\[ (9, 25) \]

### Final Answer:
The range of possible values for the length of the third side is \( x \) such that \( 9 < x < 25 \).