The image shows a right triangle $\triangle LMN$ with a right angle at $L$. Point $P$ is on $NM$ such that $LP \perp NM$. Given $NM = 9$ and $LM = 6$, we need to find two triangles that are similar to $\triangle LMN$ and find the length of $PM$.

1. Problem Description

The image shows a right triangle $\triangle LMN$ with a right angle at $L$. Point $P$ is on $NM$ such that $LP \perp NM$. Given $NM = 9$ and $LM = 6$, we need to find two triangles that are similar to $\triangle LMN$ and find the length of $PM$.

2. Solution Steps

First, we need to identify the similar triangles. Since $\triangle LMN$ is a right triangle, $\angle LNM + \angle NML = 90^\circ$.

In $\triangle LPN$, $\angle LNP + \angle NLP = 90^\circ$. Since $\angle LNM = \angle LNP$, we have $\angle NML = \angle NLP$. Also, both $\triangle LPN$ and $\triangle LPM$ are right triangles.

Thus, $\triangle LMN \sim \triangle LPN$ (Angle-Angle similarity).

Similarly, $\triangle LMN \sim \triangle LPM$ (Angle-Angle similarity).

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

Now, we need to find the length of $PM$. Let $PM = x$. Since $NM = 9$, we have $NP = 9 - x$.

Since $\triangle LMN \sim \triangle LPM$, we have the proportion $\frac{LM}{PM} = \frac{NM}{LM}$.

Substituting the given values, we get $\frac{6}{x} = \frac{9}{6}$.

Cross-multiplying, we have $9x = 36$.

Dividing both sides by 9, we have $x = 4$.

3. Final Answer

The two triangles similar to $\triangle LMN$ are $\triangle LPN$ and $\triangle LPM$. The length of $PM$ is 4.