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The problem involves solving for side \( n \) in a triangle \( \Delta LMN \), where:

- \( L = 5 \, \text{cm} \)
- \( M = 7 \, \text{cm} \)
- \( \angle N = 60^\circ \)

This is a case for the **Law of Cosines**, which relates the sides of a triangle to the cosine of one of its angles. The formula for the Law of Cosines is:

\[
n^2 = L^2 + M^2 - 2 \cdot L \cdot M \cdot \cos(\angle N)
\]

Substitute the given values:

- \( L = 5 \, \text{cm} \)
- \( M = 7 \, \text{cm} \)
- \( \cos(60^\circ) = 0.5 \)

Let's compute \( n \).Using the Law of Cosines, the length of side \( n \) is approximately \( 6.2 \, \text{cm} \). Therefore, the correct answer is:

**a) 6.2**

Would you like any further details or clarifications?

Here are five related questions to expand your understanding:
1. How does the Law of Cosines work when the angle is obtuse or acute?
2. What are the conditions under which the Law of Sines would be used instead?
3. Can the Law of Cosines be used to find an angle if all three sides are known?
4. How does the cosine value affect the result when the angle is more than 90 degrees?
5. What other formulas are applicable for non-right triangles?

**Tip:** Always make sure the angle is measured in degrees or radians as required when using trigonometric functions.

In ΔLMN, L = 5 cm, M = 7 cm, ∠N = 60°, then find n (side opposite to angle N) using the Law of Cosines.

Solve for the unknown side of triangle ΔLMN using the Law of Cosines, with L = 5 cm, M = 7 cm, and ∠N = 60°. Learn how to apply trigonometry and geometry concepts effectively.

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