To solve for the length of $BC$, we use the fact that $\triangle ABC$ is similar to $\triangle BDC$. This similarity implies that the corresponding sides of the two triangles are proportional.

Step-by-Step Solution:

1. Identify Known Values:

   • $AB = 15 \, \text{cm}$
   • $AD = 18 \, \text{cm}$
   • $CD = 6 \, \text{cm}$
   • Since $AC = AD + CD$, we can calculate: $AC = 18 + 6 = 24 \, \text{cm.}$

2. Set up the Proportions: From similarity, we know: $\frac{AB}{BD} = \frac{AC}{BC}.$

   Let $BD = x$. Then, by the segment addition property: $BC = BD + CD = x + 6.$

   Substitute into the proportion: $\frac{15}{x} = \frac{24}{x + 6}.$

3. Solve the Proportion: Cross-multiply: $15(x + 6) = 24x.$

   Expand and simplify: $15x + 90 = 24x.$

   Rearrange to isolate $x$: $24x - 15x = 90 \implies 9x = 90 \implies x = 10.$

4. Find $BC$: Since $BD = x = 10$ and $CD = 6$, the length of $BC$ is: $BC = BD + CD = 10 + 6 = 16 \, \text{cm.}$

Final Answer:

The length of $BC$ is: $\boxed{16 \, \text{cm}}$

Do you want further clarification or a breakdown of a specific step? Here are 5 follow-up questions related to this problem:

1. What are the properties of similar triangles and how do they help solve such problems?
2. Can you explain why the segment addition property was used in this problem?
3. What would change if $AB$ or $AD$ had different lengths?
4. How can we verify that our solution is correct using the proportionality rule?
5. How are similar triangles used in real-life applications?

Tip: Always write out known values clearly when solving geometry problems involving proportions or similarity to avoid mistakes!