Problem 4. In $\triangle A B C, \overline{A C}=5 \text{ cm}$ and $\overline{B C}=9 \text{ cm}$. From the midpoint of side $A B$, a perpendicular is drawn which intersects side $B C$ at point $E$. Point $E$ is connected to vertex $A$. Calculate the perimeter of $\triangle A E C$.
Solution. If $D$ is the midpoint of $A B$, then from $D E \perp A B$, it follows that

![](https://cdn.mathpix.com/cropped/2024_06_05_9d1696c7fd1a2cbf5bb5g-08.jpg?height=378&width=851&top_left_y=1551&top_left_x=476)

$\triangle A B E$ is isosceles, with base $A B$, so $\overline{A E}=\overline{E B}$. Therefore, we have

$$
L_{A C E}=\overline{A C}+(\overline{C E}+\overline{E A})=5+9=14 \mathrm{~cm}
$$