To solve this problem, we must verify that the triangles are similar through dilation, which preserves proportionality between corresponding sides. Let's calculate $x$ by using the ratios of the corresponding sides.

Step-by-Step Solution:

1. Identify corresponding sides between triangles A and B:

   • $15 \, \text{cm}$ in $\triangle A$ corresponds to $x \, \text{cm}$ in $\triangle B$.
   • $12 \, \text{cm}$ in $\triangle A$ corresponds to $16 \, \text{cm}$ in $\triangle B$.
   • $6 \, \text{cm}$ in $\triangle A$ corresponds to $8 \, \text{cm}$ in $\triangle B$.

2. Use the proportionality condition: Since the triangles are similar: $\frac{\text{Side of } \triangle A}{\text{Corresponding side of } \triangle B} = \text{constant ratio}.$ Using the corresponding sides $12 \, \text{cm}$ and $16 \, \text{cm}$, we compute the ratio: $\text{Ratio} = \frac{12}{16} = \frac{3}{4}.$

3. Set up the proportion for $x$: Using the corresponding sides $15 \, \text{cm}$ and $x \, \text{cm}$: $\frac{15}{x} = \frac{3}{4}.$

4. Solve for $x$: Cross-multiply: $3x = 15 \cdot 4.$ Simplify: $3x = 60 \quad \Rightarrow \quad x = \frac{60}{3} = 20.$

Final Answer:

The value of $x$ is 20 cm.

Would you like a more detailed explanation or have further questions?
Here are 5 related questions for practice:

1. What does it mean for two triangles to be similar?
2. How do you verify similarity between two triangles?
3. Can the ratio of corresponding sides help find unknown lengths in all types of triangles?
4. How does dilation affect the angles of a triangle?
5. If the scale factor between two similar figures is $k$, what happens to their areas?

Tip: Always confirm the proportionality of all sides when verifying triangle similarity.