## Task A-2.3.

Let $A B C$ be a right triangle and $\overline{C N}$ its altitude. If $|A C|=|B N|=1$, what is the length of the hypotenuse $\overline{A B}$?
## First solution.

Let $|A N|=x$ and $|C N|=v$.

![](https://cdn.mathpix.com/cropped/2024_05_30_abcf4bcd4243d7db1a46g-08.jpg?height=443&width=791&top_left_y=2074&top_left_x=564)

According to Euclid's theorem, we have $|A N| \cdot |N B|=|N C|^{2}$, which means $x \cdot 1=v^{2}$.

1 point

According to the Pythagorean theorem, we have $|C N|^{2}=|A C|^{2}-|A N|^{2}$, which means $v^{2}=1-x^{2}$. 1 point

It follows that $x=1-x^{2}$, or $x^{2}+x-1=0$.

The solution to this equation is $x=\frac{-1+\sqrt{5}}{2}$ (the other solution is negative),

so the length of the hypotenuse $|A B|=|A N|+|N B|=x+1=\frac{1+\sqrt{5}}{2}$.

1 point