The sides of a traingle are 6 . 5 cmm 10 cm and x cm, where x is a positive number . What ist he smallest possible value ofx amongthe following

Options

A. `3.5`
B. 4
C. `4.5`
D. `2.8`

Answer

To find the smallest possible value of \( x \) such that the sides of a triangle can be formed with lengths \( 6.5 \, \text{cm} \), \( 10 \, \text{cm} \), and \( x \, \text{cm} \), we will use the triangle inequality theorem. According to this theorem, 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 sides be \( A = 6.5 \, \text{cm} \), \( B = 10 \, \text{cm} \), and \( C = x \, \text{cm} \).

2. **Apply the triangle inequality:**
- We need to check the following inequalities:
1. \( A + B > C \)
2. \( A + C > B \)
3. \( B + C > A \)

3. **Evaluate the inequalities:**
- **First inequality:**
\[ 6.5 + 10 > x \implies 16.5 > x \implies x < 16.5 \]
- **Second inequality:**
\[ 6.5 + x > 10 \implies x > 10 - 6.5 \implies x > 3.5 \]
- **Third inequality:**
\[ 10 + x > 6.5 \implies x > 6.5 - 10 \implies x > -3.5 \]
(This inequality is always true since \( x \) is positive.)

4. **Combine the inequalities:**
- From the second inequality, we have \( x > 3.5 \).
- From the first inequality, we have \( x < 16.5 \).
- Therefore, the possible range for \( x \) is:
\[ 3.5 < x < 16.5 \]

5. **Determine the smallest possible value of \( x \):**
- The smallest integer value greater than \( 3.5 \) is \( 4 \).
- Thus, the smallest possible value of \( x \) is \( 4 \, \text{cm} \).

### Conclusion:
The smallest possible value of \( x \) among the given options is \( 4 \, \text{cm} \).