4. Let triangle $ABC$ be an equilateral triangle with side length $8 \, \text{cm}$ and point $D$ on side $\overline{BC}$ such that $|DC|=2 \, \text{cm}$. From point $D$, perpendiculars $\overline{DE}$ and $\overline{DF}$ are drawn to sides $\overline{AB}$ and $\overline{AC}$, respectively. Determine the lengths of segments $\overline{BE}$ and $\overline{CF}$.
4. Sketch:

1 POINT

![](https://cdn.mathpix.com/cropped/2024_05_30_e1c3024da477c84dc560g-5.jpg?height=953&width=1082&top_left_y=283&top_left_x=287)

In an equilateral triangle $ABC$, the measure of angle $\angle ABC$ is $60^{\circ}$, and triangle $BDE$ is a right triangle, so $|\angle EDB|=90^{\circ}-60^{\circ}=30^{\circ}$.

Let point $B'$ be the point symmetric to point $B$ with respect to point $E$.

Then triangle $BDB'$ is equilateral.

$\left|BB'\right|=|BD|=6 \text{ cm}, \text{ and } |BE|=\frac{1}{2}\left|BB'\right|=3 \text{ cm}$.

Similarly, triangle $DCC'$ is equilateral, where $C'$ is the point symmetric to point $C$ with respect to point $F$, so $\left|CC'\right|=|DC|=2 \text{ cm}$, and $|CF|=\frac{1}{2}\left|CC'\right|=1 \text{ cm}$. $\qquad$