The question asks for the length of $PQ$ given that triangles $\triangle ABC$ and $\triangle PQR$ are similar. Here is how to solve it step by step:

Given Data:

1. $BC = 2$, $AB = 3$ in $\triangle ABC$.
2. $QR = 3$ in $\triangle PQR$.
3. $\triangle ABC \sim \triangle PQR$ (the triangles are similar).

Property of Similar Triangles:

If two triangles are similar, the corresponding sides are in proportion: $\frac{AB}{PQ} = \frac{BC}{QR}.$

Substitute the values:

$\frac{3}{PQ} = \frac{2}{3}.$

Solve for $PQ$:

$PQ = \frac{3 \times 3}{2} = 4.5 \, \text{units}.$

Final Answer:

The length of $PQ$ is 4.5 cm.

Let me know if you need further clarification or have questions!

Expanding Questions:

1. What are the basic criteria for two triangles to be similar?
2. How do proportionality and similarity help in practical applications?
3. Can you find the ratio of the areas of these triangles based on similarity?
4. What other information could have been used to verify similarity in this case?
5. How do the angles correspond in similar triangles?

Tip: Always verify the corresponding sides carefully when working with similar triangles!