The lengths of three line segments (in cm) are given in each of the four cases. Which one of the the following cases is not suitable to be the three sides of a triangle?

Options

A. `2, 3, 4`
B. `2, 3, 5`
C. `2, 4, 5`
D. `3, 4, 5`

Answer

To determine which set of line segments cannot form a triangle, we will apply the triangle inequality theorem. According to this theorem, for any three sides \(a\), \(b\), and \(c\) of a triangle, the following conditions must hold:

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

We will check each of the four cases provided.

### Case 1: Sides = 2, 3, 4

1. Check \(2 + 3 > 4\):
- \(5 > 4\) (True)

2. Check \(2 + 4 > 3\):
- \(6 > 3\) (True)

3. Check \(3 + 4 > 2\):
- \(7 > 2\) (True)

All conditions are satisfied. Therefore, these sides can form a triangle.

### Case 2: Sides = 2, 3, 5

1. Check \(2 + 3 > 5\):
- \(5 > 5\) (False)

2. Check \(2 + 5 > 3\):
- \(7 > 3\) (True)

3. Check \(3 + 5 > 2\):
- \(8 > 2\) (True)

Since the first condition is not satisfied, these sides cannot form a triangle.

### Case 3: Sides = 2, 4, 5

1. Check \(2 + 4 > 5\):
- \(6 > 5\) (True)

2. Check \(2 + 5 > 4\):
- \(7 > 4\) (True)

3. Check \(4 + 5 > 2\):
- \(9 > 2\) (True)

All conditions are satisfied. Therefore, these sides can form a triangle.

### Case 4: Sides = 3, 4, 5

1. Check \(3 + 4 > 5\):
- \(7 > 5\) (True)

2. Check \(3 + 5 > 4\):
- \(8 > 4\) (True)

3. Check \(4 + 5 > 3\):
- \(9 > 3\) (True)

All conditions are satisfied. Therefore, these sides can form a triangle.

### Conclusion
The only case that does not satisfy the triangle inequality theorem is Case 2: Sides = 2, 3, 5. Thus, these sides cannot form a triangle.

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