Problem 4. Calculate the area of the square inscribed in a right-angled triangle with hypotenuse $13 \mathrm{~cm}$ and leg $5 \mathrm{~cm}$.
Solution. Let $c=13, a=5$, then from the Pythagorean theorem we have $5^{2}+b^{2}=13^{2}$, which means $b^{2}=144$. Thus, we get $b=12$. Let $E \in A B, D \in A C$ and $F \in B C$ be such that $C D E F$ is a square. From the similarity of triangles $A E D$ and $E B F$, we obtain the proportion

![](https://cdn.mathpix.com/cropped/2024_06_05_06e08f885cfce663e921g-10.jpg?height=291&width=404&top_left_y=388&top_left_x=1055)
$(12-x): x=x:(5-x)$. In this way, we have $(12-x) \cdot(5-x)=x^{2}$, from which we get $x=\frac{60}{17}$. Using the formula for the area of a square, we get $P=x^{2}=\frac{3600}{289}$.