Task B-2.7. Let $A B C$ be a right triangle with a right angle at vertex $C$. The height $C N$ is drawn to the side $A B$ and from point $N$ we draw the height $N P$ to the side $B C$ in triangle $B C N$. If the lengths of the legs are $|A C|=3$ and $|B C|=4$, what is the length of the height $N P$?
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_247a158c2fc9e6452974g-08.jpg?height=748&width=588&top_left_y=1773&top_left_x=758)

Triangles $\triangle A B C$ and $\triangle C B N$ are similar,

so we have the following ratio:

$$
\frac{|N B|}{|B C|}=\frac{|B C|}{|A B|}
$$

However, triangles $\triangle A B C$ and $\triangle N B P$ are also similar,

so we have

$$
\frac{|N P|}{|A C|}=\frac{|N B|}{|A B|}
$$

Due to the above two ratios and the Pythagorean theorem,

we have

$$
|N P|=\frac{|A C|}{|A B|} \cdot|N B|=\frac{|A C|}{|A B|} \cdot \frac{|B C|}{|A B|} \cdot|B C|=\frac{|A C| \cdot|B C|^{2}}{|A B|^{2}}=\frac{3 \cdot 4^{2}}{3^{2}+4^{2}}=\frac{48}{25}
$$