A triangle is not possible with sides of length (in cm)

Options

A. `6, 4, 10`
B. ` 5, 3, 7`
C. `7, 8, 9`
D. `3 . 6 , 5 . 4, 8 `

Answer

To determine which set of side lengths cannot form a triangle, we will use the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following conditions must be satisfied:

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

Let's analyze each option step by step.

### Step 1: Analyze Option 1 (6, 4, 10)
- Check if \( 6 + 4 > 10 \):
- \( 10 \) is not greater than \( 10 \) (fails).
- Check if \( 6 + 10 > 4 \):
- \( 16 > 4 \) (holds).
- Check if \( 4 + 10 > 6 \):
- \( 14 > 6 \) (holds).

Since the first condition fails, a triangle cannot be formed with sides of lengths 6, 4, and 10 cm.

### Step 2: Analyze Option 2 (5, 7, 3)
- Check if \( 5 + 7 > 3 \):
- \( 12 > 3 \) (holds).
- Check if \( 5 + 3 > 7 \):
- \( 8 > 7 \) (holds).
- Check if \( 7 + 3 > 5 \):
- \( 10 > 5 \) (holds).

All conditions are satisfied, so a triangle can be formed with sides of lengths 5, 7, and 3 cm.

### Step 3: Analyze Option 3 (7, 8, 9)
- Check if \( 7 + 8 > 9 \):
- \( 15 > 9 \) (holds).
- Check if \( 7 + 9 > 8 \):
- \( 16 > 8 \) (holds).
- Check if \( 8 + 9 > 7 \):
- \( 17 > 7 \) (holds).

All conditions are satisfied, so a triangle can be formed with sides of lengths 7, 8, and 9 cm.

### Step 4: Analyze Option 4 (3.6, 5.4, 8)
- Check if \( 3.6 + 5.4 > 8 \):
- \( 9 > 8 \) (holds).
- Check if \( 3.6 + 8 > 5.4 \):
- \( 11.6 > 5.4 \) (holds).
- Check if \( 5.4 + 8 > 3.6 \):
- \( 13.4 > 3.6 \) (holds).

All conditions are satisfied, so a triangle can be formed with sides of lengths 3.6, 5.4, and 8 cm.

### Conclusion
The only option that does not satisfy the triangle inequality theorem is Option 1 (6, 4, 10). Therefore, a triangle cannot be formed with these side lengths.