3. In a right triangle $A B C$ with a right angle at $C$, it is given that $|A C|=4$ and $|B C|=8$. On the side $B C$, we choose a point $D$ such that $|C D|=5$. How far is point $D$ from the side $A B$?
I/3. By the Pythagorean theorem, $|A B|=\sqrt{4^{2}+8^{2}}=$ $4 \sqrt{5}$. From the similarity of triangles $A B C$ and $D B E$, it follows that $\frac{|A B|}{|A C|}=$ $\frac{|D B|}{|D E|}$. Therefore,

$$
|D E|=\frac{|D B| \cdot|A C|}{|A B|}=\frac{3 \cdot 4}{4 \sqrt{5}}=\frac{3 \sqrt{5}}{5}
$$

![](https://cdn.mathpix.com/cropped/2024_06_07_fb5898f0472fc84ca82bg-06.jpg?height=280&width=551&top_left_y=1916&top_left_x=1301)

Calculation of $|A B|=4 \sqrt{5}$

2 points

![](https://cdn.mathpix.com/cropped/2024_06_07_fb5898f0472fc84ca82bg-06.jpg?height=51&width=1633&top_left_y=2302&top_left_x=220)

![](https://cdn.mathpix.com/cropped/2024_06_07_fb5898f0472fc84ca82bg-06.jpg?height=77&width=1631&top_left_y=2337&top_left_x=218)

Calculation of $|D E|=\frac{3 \sqrt{5}}{5}$ (or in the form $\frac{3}{\sqrt{5}}$ ) ....................................................................