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

In this geometric problem, we are working with a triangle \( ABC \), where certain midpoints are given and a specific length is known. Let’s break down the information to solve for the length of \( GH \).

### Step-by-Step Analysis:

1. **Key Points Defined**:
   - \( M \) is the midpoint of \( BC \), so \( BM = MC \).
   - \( N \) is the midpoint of \( MC \), so \( MN = NC \).
   - \( E \) is the midpoint of \( AC \), so \( AE = EC \).

2. **BE as a Median**:  
   The line \( BE \) is the median from vertex \( B \) to side \( AC \). It's given that \( BE = 42 \, \text{cm} \).

3. **Relationship Between GH and BE**:
   From the figure, \( G \) and \( H \) are points on the line \( AM \), which connects the vertex \( A \) to the midpoint \( M \) of side \( BC \), and \( N \) is the midpoint of \( MC \).
   
   In such geometric configurations, \( GH \) often represents a portion of the medians, particularly when working with centroid-related geometry. In triangle centroid properties:
   - The centroid divides each median into a 2:1 ratio.
   - Here, if \( G \) is the centroid, then \( H \) would typically lie along the median \( AM \), and \( GH \) would be one-third the length of the median from vertex \( B \) (i.e., \( BE \)).

4. **Finding \( GH \)**:
   Given the centroid property, \( GH \) is one-third of \( BE \). Therefore:
   \[
   GH = \frac{1}{3} \times BE = \frac{1}{3} \times 42 \, \text{cm} = 14 \, \text{cm}.
   \]

Thus, the length of \( GH \) is **14 cm**.

Would you like further clarification or additional details on any of these steps?

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Here are some related questions to expand your understanding:

1. How do we know that \( G \) is the centroid of the triangle?
2. What other properties can we derive from knowing the midpoints of a triangle's sides?
3. Can the lengths of other segments, like \( AM \) or \( MN \), be calculated in this scenario?
4. How does the centroid divide the medians in a triangle?
5. What role does symmetry play in solving triangle midpoint problems like this?

**Tip**: In triangle geometry, the centroid (where medians intersect) divides each median into a 2:1 ratio, with the longer part being closer to the vertex. This is a useful property to remember when dealing with triangle medians.

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

Learn how to find the length of GH in a triangle where midpoints are given. This problem explores triangle geometry, specifically the Centroid Theorem and Midpoint Theorem. With BE = 42 cm, discover how GH is calculated. Suitable for students in Grades 9-10.

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