To find $PN$, let's analyze the given information in the figure.

From the diagram:

1. $L$, $P$, $Q$, and $M$ are points along two parallel lines, with arrows indicating parallel lines.
2. We have triangles $LPN$ and $QMN$, and it appears that these triangles are similar by AA similarity (since the lines are parallel and they share angle $N$).

Since the triangles $LPN$ and $QMN$ are similar, we can set up a proportion based on their corresponding sides. The sides $LP$ and $QM$ are given as 3 and 2, respectively. Let $PN$ represent the segment we need to find, and let $NM$ be the corresponding segment in triangle $QMN$.

Solution

Using the similarity of the triangles, we can set up the proportion: $\frac{PN}{NM} = \frac{LP}{QM}$

Substitute the values: $\frac{PN}{NM} = \frac{3}{2}$

If the length of $NM$ were known, we could solve for $PN$ directly by rearranging this equation to: $PN = \frac{3}{2} \times NM$

Please confirm if there is a given length for $NM$ or any additional information, as it would allow me to calculate $PN$ directly.

Would you like further clarification or additional assistance on this question?

Here are 5 related questions that might help deepen your understanding:

1. What properties make two triangles similar?
2. How can we use the similarity of triangles to solve for unknown side lengths?
3. How do parallel lines affect angles in geometry?
4. What are other ways to prove triangles are similar besides AA (Angle-Angle)?
5. How is a proportion set up between corresponding sides in similar triangles?

Tip: When working with similar triangles, always check if corresponding angles are equal and if a proportion between corresponding sides can be established.