Question

a triangle has side lengths of 8 and 15. What could be the length of the third side, 22 or 24?

Answer 1

To determine if a triangle with sides of lengths 8, 15, and a third side (let's call it \( x \)) can exist, we can use the triangle inequality theorem. The triangle inequality states that for any triangle with sides of lengths \( a \), \( b \), and \( c \):

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

In this case, the sides we know are 8 and 15, and we are testing the potential lengths of the third side, \( x \).

### Testing \( x = 22 \):

1. \( 8 + 15 > 22 \) \
\( 23 > 22 \), which is true.

2. \( 8 + 22 > 15 \) \
\( 30 > 15 \), which is true.

3. \( 15 + 22 > 8 \) \
\( 37 > 8 \), which is true.

All conditions are satisfied, so a triangle with sides 8, 15, and 22 can exist.

### Testing \( x = 24 \):

1. \( 8 + 15 > 24 \) \
\( 23 > 24 \), which is false.

2. \( 8 + 24 > 15 \) \
\( 32 > 15 \), which is true.

3. \( 15 + 24 > 8 \) \
\( 39 > 8 \), which is true.

Since the first condition fails, a triangle with sides 8, 15, and 24 cannot exist.

### Conclusion:
The possible length of the third side is **22**.