The triangle diagram represents a situation where the line $MN$ is drawn parallel to the base $AB$ of the triangle $ABC$, and it passes through the centroid $G$. Here’s the step-by-step solution to find the length of $MN$:

Step 1: Understand the centroid properties

The centroid $G$ of a triangle divides each median into two segments in the ratio $2:1$. Since $MN$ is parallel to $AB$, and it passes through $G$, the triangles $ABC$ and $AMN$ are similar by the Basic Proportionality Theorem (also known as Thales' theorem).

Step 2: Relationship between the sides of similar triangles

The ratio of the sides of the similar triangles $AMN$ and $ABC$ is equal to the ratio of their corresponding heights. Since $MN$ passes through the centroid $G$, the ratio of similarity is $2:3$ (as the centroid divides each median in this ratio).

Thus: $\frac{\text{Length of } MN}{\text{Length of } AB} = \frac{2}{3}.$

Step 3: Substitute the value of $AB$

The length of $AB$ is given as $18 \, \text{cm}$. Using the similarity ratio: $\text{Length of } MN = \frac{2}{3} \times \text{Length of } AB = \frac{2}{3} \times 18 = 12 \, \text{cm}.$

Final Answer:

The length of $MN$ is 12 cm.

Would you like further clarification or details? Here are some related questions to deepen understanding:

1. How does the centroid divide the medians of a triangle?
2. What is the Basic Proportionality Theorem, and how is it applied here?
3. Can you explain why $MN$ is proportional to $AB$ using triangle similarity?
4. How does the centroid affect the proportions of areas between $AMN$ and $ABC$?
5. How would the solution change if $MN$ passed through a different point?

Tip:

In any triangle, when a line is drawn parallel to one side and intersects the other two sides, it always divides the triangle into similar triangles.