Exercise 5.4.16* (The area of a right triangle with integer sides cannot be a square)

Question

Consider a right triangle the length of whose sides are integers. Prove that the area cannot be a perfect square.

Richard Ganaye · Accepted Answer

\begin{proof} 
Assume for the sake of contradiction that there is a right triangle with positive integer sides $x,y,z$ (where $z$ is the length of the hypotenuse) and integer area $t$. Then
\begin{align}
\left\{
\begin{array}{rl}
x^2 + y^2 &= z^2\
\frac{1}{2}\, xy &= t^2.
\end{array}
ight.
\end{align}
Then 
\begin{align*}
(x+y)^2 = x^2 + y^2 + 2xy = z^2 + 4t^2,\
(x-y)^2 = x^2 + y^2 - 2xy = z^2 - 4t^2.
\end{align*}
If $w = 2t$, then $z^2 + w^2$ and $z^2 - w^2$ are both perfect squares, which is impossible by Problem 15.
\end{proof}
(The difficulty is in Problem 15, and Problem 14, not in Problem 16.)

Exercise 5.4.16* (The area of a right triangle with integer sides cannot be a square)

Answers

Comments

Exercise 5.4.16* (The area of a right triangle with integer sides cannot be a square)

Answers

Comments

Add answer