4B. Given a square $A B C D$. The points $X$ and $Y$ lie on the sides $B C$ and $C D$, respectively. The lengths of the segments $X Y, A X$ and $A Y$ are 3, 4, and 5, respectively. Calculate the length of the side of the square.
Solution. Let the side length of the square be $a$. Triangle AXY is a right triangle (sides are 3, 4, and 5) with a right angle at vertex $X$. Since $\measuredangle C X Y = \measuredangle X A B$ (angles with perpendicular sides), it follows that $\triangle C X Y$ is similar to $\triangle B A X$. Therefore, the sides are proportional, i.e., $\frac{\overline{C X}}{\overline{B A}} = \frac{\overline{X Y}}{\overline{A X}}$ or $\overline{C X} = \frac{\overline{X Y} \cdot \overline{B A}}{\overline{A X}} = \frac{3a}{4}$. We directly obtain that

![](https://cdn.mathpix.com/cropped/2024_06_05_c86343eb2e31158b731fg-03.jpg?height=265&width=274&top_left_y=418&top_left_x=1204)
$\overline{B X} = \frac{a}{4}$.

From the Pythagorean theorem for $\triangle A B X$, we get $a^{2} + \left(\frac{a}{4}\right)^{2} = 4^{2}$, from which $a = \frac{16 \sqrt{17}}{17}$.

## 2nd year