The construction of a `Delta LMN` in which LM = 8 cm, `angleL = 45^(@)` is possible when `(MN + LN)` is ____

Options

A. 6cm
B. 7cm
C. 9cm
D. 5cm

Answer

To solve the problem of determining the possible value of \( MN + LN \) for the triangle \( \Delta LMN \) given that \( LM = 8 \, \text{cm} \) and \( \angle L = 45^\circ \), we will use the triangle inequality theorem.

### Step-by-Step Solution:

1. **Understand the Triangle Inequality Theorem**:
The triangle inequality 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. For triangle \( \Delta LMN \), we have:
\[ LM + MN > LN \]
\[ LM + LN > MN \]
\[ MN + LN > LM \]

2. **Identify Known Values**:
From the problem, we know:
- \( LM = 8 \, \text{cm} \)
- We need to find the condition for \( MN + LN \).

3. **Apply the Triangle Inequality**:
We can apply the triangle inequality specifically for \( MN + LN > LM \):
\[ MN + LN > 8 \, \text{cm} \]

4. **Evaluate the Options**:
The options given are:
- 5 cm
- 6 cm
- 7 cm
- 9 cm

We need to find which of these options satisfies the inequality \( MN + LN > 8 \, \text{cm} \).

5. **Check Each Option**:
- For **5 cm**: \( MN + LN = 5 \, \text{cm} \) (not greater than 8)
- For **6 cm**: \( MN + LN = 6 \, \text{cm} \) (not greater than 8)
- For **7 cm**: \( MN + LN = 7 \, \text{cm} \) (not greater than 8)
- For **9 cm**: \( MN + LN = 9 \, \text{cm} \) (greater than 8)

6. **Conclusion**:
The only option that satisfies the condition \( MN + LN > 8 \, \text{cm} \) is **9 cm**.

### Final Answer:
The construction of triangle \( \Delta LMN \) is possible when \( MN + LN \) is **greater than 8 cm**, and the only suitable option provided is **9 cm**.