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To solve this problem, we need to find the length of \( GH \) in the given triangle. Here’s a step-by-step approach:

### Given:
1. \( M \), \( N \), and \( E \) are midpoints of \( BC \), \( MC \), and \( AC \), respectively.
2. The length of \( BE \) is given as 42 cm.

### Explanation:
Since \( M \), \( N \), and \( E \) are midpoints:
- \( BE \) connects a vertex to the midpoint of the opposite side, making \( BE \) a median of triangle \( ABC \).

#### Step 1: Properties of Medians and Midpoints
- The line segments connecting the midpoints of two sides of a triangle (like \( MN \)) are parallel to the third side and equal to half its length.
- Hence, \( MN \parallel AB \) and \( MN = \frac{1}{2} AB \).

#### Step 2: Median Intersections
The centroid \( G \) of triangle \( ABC \) divides each median in the ratio 2:1, where the longer segment is from the vertex to the centroid.

#### Step 3: Relation between \( GH \) and \( BE \)
Point \( H \) lies on median \( BE \), and \( G \) divides \( BE \) in the ratio 2:1. Therefore, \( GH \) is one-third the length of \( BE \).

### Calculation:
Given:
\[
BE = 42 \text{ cm}
\]
Thus, the length of \( GH \) is:
\[
GH = \frac{1}{3} \times 42 = 14 \text{ cm}
\]

The length of \( GH \) is 14 cm.

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

Here are five related questions to extend your understanding:
1. How do the properties of midpoints help in solving problems involving triangles?
2. What are other significant properties of the centroid in a triangle?
3. Can the method used here be applied to any triangle, or does it require specific types like equilateral?
4. How would the problem change if point \( H \) was located at a different position on the median?
5. How does the centroid's location affect the triangle's area distribution?

**Tip:** When dealing with triangle medians, remember that the centroid divides each median in a 2:1 ratio, providing a helpful geometric property for solving related problems.

In the figure below, M, N, and E are the mid-points of BC, MC, and AC respectively. If the length of BE is 42 cm, find the length of GH in cm.

Solve a geometry problem where M, N, and E are the midpoints of triangle sides, and BE is a median with a given length. Learn how to find the length of GH using the properties of triangle medians and centroids. This problem is suitable for high school students in Grades 9-12.

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