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### Question 15
We need to find the length of \( LM \) in the triangle diagram.

- The sides \( LM \) and \( MN \) are given as algebraic expressions:
  \[
  LM = 7y - 6
  \]
  \[
  MN = 2y + 9
  \]

Typically, in such problems, the sides are set equal to each other or to a known length if there's an indication of congruent segments or a relationship in the triangle. However, since there’s no additional relationship provided here, we might be missing some specific instruction. Please confirm if there's additional information or if it's assumed that \( LM = MN \), which would allow us to solve for \( y \).

If we assume \( LM = MN \), we set up the equation:
\[
7y - 6 = 2y + 9
\]
Solving for \( y \) would give the length of \( LM \).

### Question 16
In this problem, triangle \( \triangle ABC \) is given with \( L \), \( M \), and \( N \) as midpoints. 

- Since \( L \), \( M \), and \( N \) are midpoints, \( \triangle LMN \) is a medial triangle of \( \triangle ABC \).
- A key property of the medial triangle is that each side of the medial triangle is half the length of the corresponding side of the larger triangle.

Given:
\[
NC = 3
\]
Since \( N \) is the midpoint, \( NC \) is half of \( AC \). Therefore:
\[
AC = 2 \times 3 = 6
\]
Since \( LM \) corresponds to \( AC \) in the medial triangle, we find:
\[
LM = \frac{AC}{2} = \frac{6}{2} = 3
\]

So, the length of \( LM \) is \( 3 \).

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### Related Questions to Explore
1. What are the properties of a medial triangle in any given triangle?
2. How can we solve for unknowns when two algebraic expressions represent equal line segments?
3. What is the process for solving a system of equations if two triangle side lengths are given as algebraic expressions?
4. How do we determine relationships between sides in isosceles or similar triangles?
5. How does identifying midpoints in a triangle help simplify calculations for side lengths?

### Tip:
Always look for given geometric properties, like midpoints or congruent sides, as they often imply useful relationships or equations.

What is the length of the side LM in the triangle with sides represented by algebraic expressions 7y - 6 and 2y + 9, and in triangle ABC, where L, M, and N are midpoints and NC = 3, what is the length of LM?

Step-by-step solution to find the length of side LM in a triangle with sides represented by algebraic expressions. This problem includes concepts in algebra and geometry, with a focus on medial triangles and midpoints, suitable for students in Grades 9-10.

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