If two sides of a tringle are of length 5 cm and 1.5 cm, then the length of third side of the triangle cannot be

Options

A. 3.6 cm
B. 4.1 cm
C. 3.8 cm
D. 3.4 cm

Explanation

To determine the length of the third side of a triangle when the lengths of the other two sides are given, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side, and the difference of the lengths of any two sides must be less than the length of the third side.

### Step-by-Step Solution:

1. **Identify the lengths of the two sides**:
- Let the lengths of the two sides be \( a = 5 \, \text{cm} \) and \( b = 1.5 \, \text{cm} \).

2. **Apply the triangle inequality theorem**:
- According to the triangle inequality theorem:
- The sum of the two sides must be greater than the third side.
- The difference of the two sides must be less than the third side.

3. **Calculate the sum of the two sides**:
- \( a + b = 5 \, \text{cm} + 1.5 \, \text{cm} = 6.5 \, \text{cm} \).
- This means the third side \( c \) must be less than \( 6.5 \, \text{cm} \):
\[ c < 6.5 \, \text{cm} \]

4. **Calculate the difference of the two sides**:
- \( a - b = 5 \, \text{cm} - 1.5 \, \text{cm} = 3.5 \, \text{cm} \).
- This means the third side \( c \) must be greater than \( 3.5 \, \text{cm} \):
\[ c > 3.5 \, \text{cm} \]

5. **Combine the inequalities**:
- From the above calculations, we conclude:
\[ 3.5 \, \text{cm} < c < 6.5 \, \text{cm} \]

6. **Determine the length that cannot be the third side**:
- Any length \( c \) that does not satisfy the above inequalities cannot be the length of the third side.
- For example, if \( c = 3.4 \, \text{cm} \), it does not satisfy \( c > 3.5 \, \text{cm} \), thus it cannot be the length of the third side.

### Conclusion:
The length of the third side of the triangle cannot be \( 3.4 \, \text{cm} \).