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The problem is asking whether the line segment \( LN \) is perpendicular to line segment \( MO \), based on the given information.

### Given:
1. \( \triangle LON \) is isosceles.
2. \( LM = NM \) (Line segments \( LM \) and \( NM \) are equal).
3. \( LO = NO \) (Line segments \( LO \) and \( NO \) are equal).
4. \( MO = MO \) (This is trivial, stating that the line segment \( MO \) is equal to itself).

### Analyzing the information:

- From the given equalities, we can infer that the triangle formed by points \( L \), \( O \), and \( N \) is symmetric.
- Since \( LO = NO \), the triangle \( LON \) is isosceles.
- The line \( MO \) acts as a median, because \( LM = NM \) suggests that \( M \) is the midpoint of \( LN \).
  
In an isosceles triangle, the median from the vertex opposite the base (here, \( M \) is the midpoint of the base \( LN \)) also acts as the altitude. This means the median is perpendicular to the base.

### Conclusion:
Thus, \( LN \perp MO \). The correct answer is:

**A. True**

Would you like a detailed explanation on any step or further clarification? Here are 5 related questions to explore the topic further:

1. What is the general property of medians in isosceles triangles?
2. Can a median be perpendicular in non-isosceles triangles as well?
3. How do you prove a line is perpendicular using slopes in coordinate geometry?
4. Why does symmetry in isosceles triangles guarantee perpendicularity?
5. Can you describe the role of midpoints in triangle geometry?

**Tip**: In isosceles triangles, the altitude, median, and angle bisector from the vertex angle are always the same line!

Based only on the given information, is it guaranteed that LN is perpendicular to MO?

Explore how to determine whether the median of an isosceles triangle is perpendicular to its base. This problem involves properties of isosceles triangles, the median formula, and the theorem connecting medians and altitudes. Suitable for students in Grades 8-10.

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