Task A-3.3. (4 points)

Point $D$ is the foot of the altitude from vertex $A$, and point $E$ is the foot of the altitude from vertex $B$ of triangle $A B C$. If $|A E|=5,|C E|=3$ and $|C D|=2$, determine the length of $|B D|$.
## First solution.

Notice that triangles $A C D$ and $B C E$ are similar (two angles).

Therefore, $|A C|:|C D|=|B C|:|C E|$.

Let $|B D|=x$.

Now (*) can be written in the form $\frac{5+3}{2}=\frac{x+2}{3}$,

from which it follows that $x+2=3 \cdot \frac{8}{2}=12$ so the desired length $|B D|=x=10$.