Which of the following cannot be the sides of a triangle?

Options

A. 3 cm, 4 cm, 5 cm
B. 2 cm, 4 cm, 6 cm
C. 2.5 cm, 3.5 cm, 4.5 cm
D. 2.3 cm, 6.4 cm, 5.2 cm

Answer

To determine which of the given options cannot be the sides of 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. \( B + C > A \)

If any of these conditions are not satisfied, the three lengths cannot form a triangle.

Let's check each option step by step.

### Step 1: Check Option 1 (3, 4, 5)
- **Condition 1**: \( 3 + 4 > 5 \) → \( 7 > 5 \) (True)
- **Condition 2**: \( 3 + 5 > 4 \) → \( 8 > 4 \) (True)
- **Condition 3**: \( 4 + 5 > 3 \) → \( 9 > 3 \) (True)

Since all conditions are satisfied, **Option 1 can be the sides of a triangle**.

### Step 2: Check Option 2 (2, 4, 6)
- **Condition 1**: \( 2 + 4 > 6 \) → \( 6 > 6 \) (False)
- **Condition 2**: \( 2 + 6 > 4 \) → \( 8 > 4 \) (True)
- **Condition 3**: \( 4 + 6 > 2 \) → \( 10 > 2 \) (True)

Since Condition 1 is not satisfied, **Option 2 cannot be the sides of a triangle**.

### Step 3: Check Option 3 (5, 6, 7)
- **Condition 1**: \( 5 + 6 > 7 \) → \( 11 > 7 \) (True)
- **Condition 2**: \( 5 + 7 > 6 \) → \( 12 > 6 \) (True)
- **Condition 3**: \( 6 + 7 > 5 \) → \( 13 > 5 \) (True)

Since all conditions are satisfied, **Option 3 can be the sides of a triangle**.

### Step 4: Check Option 4 (8, 10, 12)
- **Condition 1**: \( 8 + 10 > 12 \) → \( 18 > 12 \) (True)
- **Condition 2**: \( 8 + 12 > 10 \) → \( 20 > 10 \) (True)
- **Condition 3**: \( 10 + 12 > 8 \) → \( 22 > 8 \) (True)

Since all conditions are satisfied, **Option 4 can be the sides of a triangle**.

### Conclusion
The only option that cannot be the sides of a triangle is **Option 2 (2, 4, 6)**.

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