Three sides of a triangle are 5 cm, 9 cm and x cm. The mini mum integral value of xis :

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the minimum integral value of \( x \) such that the sides of a triangle are 5 cm, 9 cm, and \( x \) 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**: The sides are given as 5 cm, 9 cm, and \( x \) cm.

2. **Apply the triangle inequality theorem**: We need to satisfy the following inequalities:
- \( 5 + 9 > x \)
- \( 5 + x > 9 \)
- \( 9 + x > 5 \)

3. **Solve each inequality**:
- From \( 5 + 9 > x \):
\[ 14 > x \quad \text{or} \quad x < 14 \]
- From \( 5 + x > 9 \):
\[ x > 9 - 5 \quad \Rightarrow \quad x > 4 \]
- From \( 9 + x > 5 \):
\[ x > 5 - 9 \quad \Rightarrow \quad x > -4 \quad \text{(This is always true since \( x \) is positive)} \]

4. **Combine the results**: The inequalities we need to satisfy are:
- \( x < 14 \)
- \( x > 4 \)

5. **Determine the minimum integral value of \( x \)**: The minimum integral value of \( x \) that satisfies \( x > 4 \) is:
\[ x = 5 \]

### Conclusion:
The minimum integral value of \( x \) is **5 cm**.