Heron of Alexandria/Problems/Pythagorean Triangle

Problem

 * In a Pythagorean triangle, the sum of the area and the perimeter is $280$.


 * Find the sides and the area.

Solution
The Pythagorean triangle has sides $20$, $21$ and $29$.

Its area is $210$.

Proof
There is no simple way to solve this except by trial and error.

First we note that:
 * the area of the $20-21-29$ Pythagorean triangle is $\dfrac {20 \times 21} 2 = 210$

and we have that:
 * $210 + 20 + 21 + 29 = 280$

Then from Sequence of Solutions of Pythagorean Equation:

from which it can soon be seen that the $20-21-29$ triangle is the one required.