Given that $\triangle ABC \sim \triangle PQR$, their corresponding sides are proportional. We are also given the ratio of the sides of $\triangle ABC$:

$BC : CA : AB = 3:4:5$

Step 1: Define the scaling factor

Since the triangles are similar, let the sides of $\triangle ABC$ be:

$BC = 3x, \quad CA = 4x, \quad AB = 5x$

Given that the corresponding side $PR$ of $\triangle PQR$ is 12, and it corresponds to $AB = 5x$, we set up the proportion:

$\frac{PR}{AB} = \frac{12}{5x}$

Since $AB = 5x$, equating:

$5x = 12$

Solving for $x$:

$x = \frac{12}{5}$

Step 2: Find the perimeter of $\triangle PQR$

The perimeter of $\triangle ABC$ is:

$BC + CA + AB = 3x + 4x + 5x = 12x$

Since $\triangle PQR$ is similar to $\triangle ABC$ with a scaling factor of $\frac{12}{5x} = \frac{12}{12} = \frac{12}{5}$, the perimeter of $\triangle PQR$ is:

$\frac{12}{5} \times 12 = 36$

Final Answer:

$\boxed{36}$

---

Would you like a breakdown of any step? Here are some related questions:

1. What is the concept of similar triangles and how do they relate to proportionality?
2. How do you determine the scaling factor in similar triangles?
3. Can you solve similar problems with different side ratios?
4. How do you check if two triangles are similar using different criteria?
5. What happens if a different side of $\triangle ABC$ was given instead of $PR$?

Tip: When dealing with similar triangles, always match corresponding sides correctly to set up the right proportions.