Two sides of a triangle are of length 4 cm and 10 cm. If the length of the third side is a cm, then

Options

A. `a gt 5`
B. `6 le a le 12`
C. `a lt 6`
D. `6 lt a lt 14`

Answer

To determine the possible lengths of the third side \( a \) of a triangle with the other two sides measuring 4 cm and 10 cm, 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.

### Step-by-step Solution:

1. **Identify the sides of the triangle**:
- Let the lengths of the two known sides be \( b = 4 \) cm and \( c = 10 \) cm.
- Let the length of the third side be \( a \) cm.

2. **Apply the triangle inequality theorem**:
- The triangle inequality theorem gives us three conditions:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

3. **Substituting the known values into the inequalities**:
- From the first inequality \( a + 4 > 10 \):
\[ a > 10 - 4 \implies a > 6 \]
- From the second inequality \( a + 10 > 4 \):
\[ a > 4 - 10 \implies a > -6 \quad (\text{This condition is always true since } a \text{ is positive}) \]
- From the third inequality \( 4 + 10 > a \):
\[ 14 > a \implies a < 14 \]

4. **Combine the results**:
- From the inequalities derived, we have:
\[ 6 < a < 14 \]

### Conclusion:
The length of the third side \( a \) must be greater than 6 cm and less than 14 cm. Thus, the possible range for \( a \) is:
\[ a \in (6, 14) \]