Problem Analysis:

We are given:

1. $D$ is the midpoint of $AC$, so $AD = DC$.
2. $C$ is the midpoint of $AB$, so $AC = CB$.
3. $AD = 3 \, \text{cm}$, and we need to find the length of $BC$.

Step 1: Relationships Between Segments

1. Since $D$ is the midpoint of $AC$, the length of $AC$ is: $AC = 2 \times AD = 2 \times 3 = 6 \, \text{cm}.$

2. Since $C$ is the midpoint of $AB$, the length of $AB$ is: $AB = 2 \times AC = 2 \times 6 = 12 \, \text{cm}.$

3. The segment $BC$ is half of $AB$, because $C$ is the midpoint: $BC = \frac{AB}{2} = \frac{12}{2} = 6 \, \text{cm}.$

Final Answer:

The length of $BC$ is: $\boxed{6 \, \text{cm}}.$

Let me know if you'd like further clarification or help with similar problems! Here are five related questions you might find interesting:

1. How does the concept of midpoints generalize to coordinate geometry?
2. What happens if the lengths of the segments are expressed in terms of variables?
3. How can you verify that $C$ is the midpoint using coordinate points?
4. What is the relationship between $AD$ and $AB$ in this problem?
5. Can you solve for $BC$ if $AD$ is expressed as a variable $x$?

Tip: When solving midpoint problems, always relate smaller segments to larger segments using the given conditions step by step!