215. Given the area of a right-angled triangle and its hypotenuse, find the legs.

Problems by Leonardo Fibonacci from "Liber Abaci".

Problem "De duobus hominibus" (About two men).
215. If $x$ and $y$ are the legs, $a$ is the hypotenuse, and 1 is the area, we have: $x y=2 \Delta ; x^{2}+y^{2}=a^{2}$.

Herbert's Solution:

$$
\begin{gathered}
2 x y=4 \Delta \\
x^{2}+y^{2}=a^{2}
\end{gathered}
$$

Adding and subtracting give:

$$
\begin{aligned}
& (x+y)^{2}=a^{2}+4 \Delta ; x+y=\sqrt{a^{2}+4 \Delta} \\
& (x-y)^{2}=a^{2}-4 \Delta ; x-y=\sqrt{a^{2}-4 \Delta}
\end{aligned}
$$

128

From (1) and (2) we find:

$$
\begin{aligned}
& x=\frac{\sqrt{a^{2}+4 \Delta}+\sqrt{a^{2}-4 \Delta}}{2} \\
& y=\frac{\sqrt{a^{2}+4 \Delta}-\sqrt{a^{2}-4 \Delta}}{2}
\end{aligned}
$$

Leonardo of Pisa (born in 1180), author of two treatises: "Liber abaci" (1202) and "Practica geometriae" (1220). The first was written under strong influence of Arabic mathematical literature, the second is a revision of "Liber embadorum" by Savasorda, though expanded. Leonardo's works enjoyed great popularity.