The image shows a triangle $LMN$ with a right angle at $L$. $LP$ is an altitude from $L$ to the side $NM$. We are given that $NP = 9$ and $NM=56$ cm. We need to identify two triangles similar to $\triangle LMN$ and find the length of $PM$.

1. Problem Description

The image shows a triangle $LMN$ with a right angle at $L$. $LP$ is an altitude from $L$ to the side $NM$. We are given that $NP = 9$ and $NM=56$ cm. We need to identify two triangles similar to $\triangle LMN$ and find the length of $PM$.

2. Solution Steps

First, we need to identify the similar triangles.

$\triangle LMN$ is a right triangle. Since $LP$ is the altitude to the hypotenuse $NM$, $\triangle NLP \sim \triangle LPM \sim \triangle LMN$.

Therefore, $\triangle NLP$ and $\triangle LPM$ are similar to $\triangle LMN$.

Now, we need to find the length of $PM$. Since $NM = 56$ and $NP = 9$, we have

$PM = NM - NP = 56 - 9 = 47$.

3. Final Answer

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

The length of $PM$ is $47$ cm.