The problem presents a diagram of triangle $LMN$ with an altitude $LP$ drawn from vertex $L$ to side $NM$. We are given that $NM = 9$ and $LM = 6$. Also, angle $NLM$ is a right angle and angle $LPN$ is a right angle. The problem asks us to: i) Name two triangles which are similar to triangle $LMN$. ii) Find the length of $PM$.

1. Problem Description

The problem presents a diagram of triangle $LMN$ with an altitude $LP$ drawn from vertex $L$ to side $NM$. We are given that $NM = 9$ and $LM = 6$. Also, angle $NLM$ is a right angle and angle $LPN$ is a right angle.

The problem asks us to:

i) Name two triangles which are similar to triangle $LMN$.

ii) Find the length of $PM$.

2. Solution Steps

i) Similarity of triangles:

Since $LP$ is an altitude, $\angle LPN = 90^\circ$. Also, $\angle NLM = 90^\circ$.

Consider $\triangle LMN$ and $\triangle LPM$:

$\angle LPM = 90^\circ$ and $\angle LMN = \angle LMN$

Thus, by Angle-Angle (AA) similarity, $\triangle LMN \sim \triangle LPM$

Consider $\triangle LMN$ and $\triangle LNP$:

$\angle LNP = \angle LNP$ and $\angle LPN = 90^\circ = \angle NLM$.

Thus, by Angle-Angle (AA) similarity, $\triangle LMN \sim \triangle LNP$

Therefore, two triangles similar to $\triangle LMN$ are $\triangle LNP$ and $\triangle LPM$.

ii) Finding $PM$:

We know $\triangle LMN \sim \triangle LPM$, so we can set up the following ratios:

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

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

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

3. Final Answer

i) The two triangles similar to $\triangle LMN$ are $\triangle LNP$ and $\triangle LPM$.

ii) The length of $PM$ is $4$.