Chapter A.2, Problem 56AYU

To determine

$a$, $b$, and $c$ are the lengths of the sides of a right triangle. (This formula can be used to find the sides of a right triangle that are integers, such as $3,4,5$; $5,12,13$; and so on. Such triplets of integers are called Pythagorean triples.)

Answer to Problem 56AYU

$c^2$

Explanation of Solution

Given:

Suppose that $m$ and $n$ are positive integers with $m>n$. If $a=m^2-n^2$,$b=2mn$, and $c=m^2+n^2$.

Calculation:

By Pythagorean Theorem $c^2=a^2+b^2$.

Here $a=m^2-n^2$,$b=2mn$, and $c=m^2+n^2$.

$a^2+b^2= {\left(m^2-n^2\right)}^2+{\left(2mn\right)}^2$

$=m^4+n^4-2m^2{n}^2+4m^2{n}^2$

$=m^4+n^4+2m^2{n}^2$

$={\left(m^2+n^2\right)}^2$

$=c^2$