4. Given a square $A B C D$ with side $\overline{A B}=5$. On the side $C D$ and the diagonal $A C$, points $E$ and $F$ are chosen respectively, such that $\frac{\overline{A F}}{\overline{F C}}=\frac{\overline{C E}}{\overline{D E}}=\frac{2}{3}$. Calculate $\overline{E F}$.
Solution. From $\frac{\overline{C E}}{\overline{D E}}=\frac{2}{3}$ it follows that $\overline{D E}=\frac{3}{5} \overline{C D}$, similarly $\overline{F C}=\frac{3}{5} \overline{A C}$.

We will divide the square into 25 unit squares each with side 2, then from the above we have that $E$ and $F$ are as shown in the diagram. Now it is easy to see that

$$
\overline{E F}^{2}=3^{2}+1^{2}=10 \text { i.e. } \overline{E F}=\sqrt{10}
$$

![](https://cdn.mathpix.com/cropped/2024_06_05_50edc896fa4080786845g-5.jpg?height=574&width=576&top_left_y=208&top_left_x=904)