We are given a triangle $LMN$ with a right angle at vertex $L$. $LP$ is an altitude to the hypotenuse $MN$. We are given that $NM = 9$ and $LN = 6$. We need to name two triangles similar to $\triangle LMN$ and then find the length of $PM$.

1. Problem Description

We are given a triangle $LMN$ with a right angle at vertex $L$. $LP$ is an altitude to the hypotenuse $MN$. We are given that $NM = 9$ and $LN = 6$. We need to name two triangles similar to $\triangle LMN$ and then find the length of $PM$.

2. Solution Steps

a) Identification of similar triangles:

Since $\triangle LMN$ is a right triangle with the altitude $LP$, we can say that $\triangle LMN \sim \triangle NLP \sim \triangle LMP$. This is because all three triangles share angles.

$\triangle LMN$ has angles at vertices $L, M, N$.

$\triangle NLP$ has angles at vertices $N, L=90^{\circ}, P$. Since $\angle NLP + \angle MLP = 90^{\circ}$, and $\angle LMN + \angle LNM = 90^{\circ}$. $\angle NLP=\angle LMN$. Hence $\triangle NLP \sim \triangle LMN$.

Similarly, $\triangle LMP$ has angles at vertices $L, M, P=90^{\circ}$.

So, we know $\triangle LMN \sim \triangle NLP \sim \triangle LMP$.

b) Finding $PM$:

We are given $NM=9$ and $LN=6$. We want to find $PM$.

Let $PM = x$. Then $NP = NM - PM = 9-x$.

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

$\frac{LM}{LN} = \frac{LP}{LM} = \frac{MP}{LP}$

Since $\triangle LMN \sim \triangle NLP$, we have

$\frac{LM}{LN} = \frac{LP}{NP} = \frac{MN}{LP}$.

Using the fact that $\triangle NLP \sim \triangle LMP$, we have $\frac{NP}{LP} = \frac{LP}{MP} = \frac{NL}{LM}$.

From $\triangle LMN$, $LM^2+LN^2 = MN^2$, and $MN=9, LN=6$, so we can find $LM$.

$LM^2 + 6^2 = 9^2$

$LM^2 + 36 = 81$

$LM^2 = 81 - 36 = 45$

$LM = \sqrt{45} = 3\sqrt{5}$

We use the similarity $\triangle NLP \sim \triangle LMP$. So, $\frac{NP}{LP} = \frac{LP}{MP}$.

So $LP^2 = NP \cdot MP = (9-x)x$.

Also $\triangle LMN \sim \triangle LMP$. Hence $\frac{MN}{LM} = \frac{LN}{LP} = \frac{LM}{MP}$. So $\frac{9}{3\sqrt{5}} = \frac{6}{LP} = \frac{3\sqrt{5}}{x}$.

From $\frac{6}{LP} = \frac{3\sqrt{5}}{x}$, we have $LP = \frac{6x}{3\sqrt{5}} = \frac{2x}{\sqrt{5}}$.

Now $LP^2 = \frac{4x^2}{5}$.

Then $(9-x)x = \frac{4x^2}{5}$.

Since $x \ne 0$, we can divide by $x$:

$9-x = \frac{4x}{5}$

$45 - 5x = 4x$

$45 = 9x$

$x = 5$

Therefore, $PM = 5$.

3. Final Answer

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

$PM = 5$.