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To solve for the length of \( LM \) in the given triangle, we'll use trigonometry. Here's the step-by-step process:

### Given:
- \( \angle LNM = 90^\circ \) (Right angle at \( N \))
- \( \angle KLM = 52^\circ 25' \) (Angle \( KLM \))
- \( \angle NLM = 25^\circ 30' \) (Angle \( NLM \))
- \( LN = 48 \) units

### Steps to Solve:
1. **Calculate \( \angle KML \):**
   - Since \( \angle KML \) is the exterior angle to \( \triangle LNM \):
   \[
   \angle KML = \angle KLM - \angle NLM = 52^\circ 25' - 25^\circ 30' = 26^\circ 55'
   \]

2. **Use the sine rule in \( \triangle LNM \):**
   Since \( \angle LNM = 90^\circ \), we can use basic trigonometry:
   \[
   \sin(\angle NLM) = \frac{LM}{LN}
   \]
   \[
   \sin(25^\circ 30') = \frac{LM}{48}
   \]

   To find \( LM \):
   \[
   LM = 48 \times \sin(25^\circ 30')
   \]

3. **Calculate \( LM \):**
   Use the sine value for \( 25^\circ 30' \):
   \[
   \sin(25^\circ 30') \approx 0.4305
   \]
   \[
   LM = 48 \times 0.4305 \approx 20.66 \text{ units}
   \]

So, the length of \( LM \) is approximately **20.66 units**.

Would you like further details or have any questions about this solution?

Here are 5 related questions:
1. How do you calculate the area of \( \triangle LNM \)?
2. What is the length of \( MN \) using the given angles and sides?
3. How do you determine \( \angle LKN \) in the full triangle?
4. How would you apply the cosine rule in this triangle?
5. What would the problem look like if the given triangle was scalene?

**Tip:** In trigonometry, the sine rule is particularly useful when you know two angles and one side of a triangle, which allows you to find unknown sides.

Learn how to solve for the length of LM in a triangle using trigonometric principles. This problem involves applying the sine rule and exterior angle theorem to find unknown sides in a right triangle. Ideal for students in Grades 10-12.

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