We are given a triangle $LMN$ with a right angle at $L$. $LP$ is perpendicular to $NM$. We are given that $NM = 9$ and $LM = 6$. We are asked to find two triangles that are similar to $\triangle LMN$, and to find the length of $PM$.

1. Problem Description

We are given a triangle $LMN$ with a right angle at $L$. $LP$ is perpendicular to $NM$. We are given that $NM = 9$ and $LM = 6$. We are asked to find two triangles that are similar to $\triangle LMN$, and to find the length of $PM$.

2. Solution Steps

First, we need to identify the similar triangles. Since $\triangle LMN$ has a right angle at $L$, and $LP$ is perpendicular to $NM$, we have two other right triangles, $\triangle NLP$ and $\triangle LPM$.

Similarity can be shown by AA (Angle-Angle) similarity.

$\triangle LMN$: $\angle L = 90^\circ$

$\triangle NLP$: $\angle P = 90^\circ$, $\angle N = \angle N$. Therefore, $\triangle LMN \sim \triangle NLP$.

$\triangle LPM$: $\angle P = 90^\circ$. Since $\angle N + \angle M = 90^\circ$ in $\triangle LMN$, and $\angle NLP + \angle PLM = 90^\circ$, then $\angle PLM = \angle M$. Therefore $\triangle LMN \sim \triangle LPM$.

Thus, the two triangles similar to $\triangle LMN$ are $\triangle NLP$ and $\triangle LPM$.

Next, we need to find $PM$.

Since $\triangle LMN \sim \triangle LPM$, we have the following proportion:

$\frac{LM}{NM} = \frac{PM}{LM}$

Substituting the given values, we have

$\frac{6}{9} = \frac{PM}{6}$

$PM = \frac{6 \times 6}{9} = \frac{36}{9} = 4$

3. Final Answer

The two triangles similar to $\triangle LMN$ are $\triangle NLP$ and $\triangle LPM$.

$PM = 4$.