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

Options

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

Answer

To solve the problem, we need to determine the condition under which the triangle \( \Delta LMN \) can be constructed given that \( LM = 8 \) cm and \( \angle L = 45^\circ \). The question specifically asks for the sum \( MN + LN \).

### Step-by-Step Solution:

1. **Understanding Triangle Inequality**:
The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This is known as the triangle inequality theorem.

2. **Identifying the Sides**:
In triangle \( \Delta LMN \):
- Let \( LM = 8 \) cm (given).
- Let \( MN = a \) cm.
- Let \( LN = b \) cm.

We need to find the condition for \( a + b \).

3. **Applying the Triangle Inequality**:
According to the triangle inequality:
- \( MN + LN > LM \)
- This can be written as:
\[ a + b > 8 \]

4. **Conclusion**:
Therefore, for the triangle \( \Delta LMN \) to be constructed, the sum \( MN + LN \) must be greater than \( 8 \) cm.

Thus, we can conclude that \( MN + LN \) must be greater than \( 8 \) cm.

### Final Answer:
The construction of triangle \( \Delta LMN \) is possible when \( MN + LN > 8 \) cm.