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

1. Problem Description

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

2. Solution Steps

a) Similarity of Triangles:

Since $\angle LNM$ is common to both $\triangle LMN$ and $\triangle LPN$, and $\angle MLN = \angle LPN = 90^\circ$, then $\triangle LMN \sim \triangle LPN$ by the AA (angle-angle) similarity criterion.

Similarly, since $\angle NML$ is common to both $\triangle LMN$ and $\triangle LPM$, and $\angle MLN = \angle LPM = 90^\circ$, then $\triangle LMN \sim \triangle LPM$ by the AA (angle-angle) similarity criterion.

Thus, $\triangle LPN \sim \triangle LPM$.

b) Finding $PM$:

Since $\triangle LMN \sim \triangle LPM$, we can write the ratio of corresponding sides as:

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

Substituting the given values, we get:

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

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

3. Final Answer

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

The length of $PM$ is 4.