Problem 507

307. Similarly: given the hypotenuse $c$ and the sum of the legs

$$S=a+b$$

10 Popov. Problem Collection.

Since

$$c^{2}=a^{2}+b^{2}$$

and

$$S^{2}=a^{2}+b^{2}+2 a b$$

Then

$$S^{2}-c^{2}=2 a b$$

from which

$$a b=\frac{S^{2}-c^{2}}{2}$$

The legs $a$ and $b$ can be found from the sum and the product.

Adam Riese (1492-1559), a famous German mathematician, teacher in Annaberg, and author of an algebraic treatise (Die Coss) (1524).