In this triangle $\triangle LMN$, point $C$ is the centroid. By definition, the centroid divides each median into two segments such that the segment closer to the vertex is twice the length of the segment closer to the midpoint of the opposite side.

Observations:

• The given lengths are $LG = 3x + 1$ and $GC = x + 2$.
• Since $C$ is the centroid, the relationship between $LG$ and $GC$ is: $LG = 2 \cdot GC.$

Solve for $x$:

Substitute $LG = 3x + 1$ and $GC = x + 2$ into the relationship: $3x + 1 = 2(x + 2).$

Simplify: $3x + 1 = 2x + 4.$

$x = 3.$

Find the length of $LH$:

The full median $LH$ is: $LH = LG + GC.$

Substitute $x = 3$ into $LG = 3x + 1$ and $GC = x + 2$: $LG = 3(3) + 1 = 10, \quad GC = 3 + 2 = 5.$

$LH = LG + GC = 10 + 5 = 15.$

Final Answer:

The length of $LH$ is: $\boxed{15}.$

Do you want me to explain centroid properties further or verify other parts of this question?

Here are related questions for deeper understanding:

1. How do centroids divide medians in triangles, and why?
2. What is the formula for the centroid when given coordinates of the triangle vertices?
3. How can you verify centroid division ratios geometrically?
4. If $LMN$ were an equilateral triangle, how would the centroid affect symmetry?
5. What happens to centroid properties if the triangle becomes isosceles?

Tip: Always check that the given segments satisfy the centroid division ratio before finalizing calculations.