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.