Two sides of a triangle are of lengths 5 cm and 1.5 cm. The length of the third side of the triangle cannot be

Options

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

Answer

To determine the possible lengths of the third side of a triangle when two sides are given, we can use the triangle inequality theorem. The 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.

Let's denote the sides of the triangle as follows:
- Side a = 5 cm
- Side b = 1.5 cm
- Side c = length of the third side (unknown)

### Step-by-step Solution:

1. **Apply the Triangle Inequality Theorem**:
According to the triangle inequality theorem, we have the following conditions:
- a + b > c
- a + c > b
- b + c > a

2. **Substituting the Known Values**:
Let's substitute the known values into the inequalities:
- From the first inequality:
\[ 5 + 1.5 > c \implies 6.5 > c \implies c < 6.5 \]
- From the second inequality:
\[ 5 + c > 1.5 \implies c > 1.5 - 5 \implies c > -3.5 \quad (\text{This is always true since } c \text{ must be positive}) \]
- From the third inequality:
\[ 1.5 + c > 5 \implies c > 5 - 1.5 \implies c > 3.5 \]

3. **Combining the Results**:
From the inequalities we derived:
- \( c < 6.5 \)
- \( c > 3.5 \)

Therefore, the possible range for the length of the third side \( c \) is:
\[ 3.5 < c < 6.5 \]

4. **Identifying the Lengths that Cannot Be the Third Side**:
Any length that is less than or equal to 3.5 cm or greater than or equal to 6.5 cm cannot be the length of the third side.

### Conclusion:
Thus, the length of the third side of the triangle cannot be 3.5 cm or less, and it cannot be 6.5 cm or more.