Pythagors Theorem : Geometrical interpretation

Question

If x,y,z are the 3 sides of a right angled triangle, where say, x = hypo, y=vertical, z=base then I learnt that the area of a square on hypo = area of square on vertical + area of square on base i.e $$x^2 = y^2 + z^2$$ That's the usual picture students are taught at elementary math. I never thought that this also means : area of a circle(x=dia) on hypo = area of circle(z=dia) on base + area of circle(y=dia) on base

Question :

1. Is it true for any other regular polygons ?
2. Am I right when I say x,y,z can be rational or irrational?
3. Is there any connection between Pythagoras Theorem and FLT ?

Answer 1

First a right triangle is NOT a "regular polygon" itself so asking if it is true for "other regular polygons" makes no sense. Second, yes, that is true whether the side lengths are rational or irrational. For example $3^2+ 4^2= 25= 5^2$ where 3, 4, and 5 are rational (in fact integer) while $\pi^2+ \pi^2= (\sqrt{2}\pi)^2$ where $\pi$ and $\sqrt{2}\pi$ are irrational.

It is easy to see that the equation $x^2+ y^2= z^2$ has integer solutions, x= 3, y= 4, z= 5, above. "FTL" says that the equation $x^n+ y^n= z^n$, with n a positive integer larger than 2, has NO integer solutions.

Answer 2

For your first question, Pythagoras theorem works for any object as long as their side lengths are similar

Reference here:

https://betterexplained.com/articles/surprising-uses-of-the-pythagorean-theorem/#:~:text=The%20Pythagorean%20Theorem%20can%20be,formula%20that%20squares%20a%20number.

For your second question, Yes you are right a simple example of this is:

$x = \sqrt{\sqrt{2}}$, $y = \sqrt{\sqrt{3}}$

$\sqrt{2} + \sqrt{3} = c^2$

$c = \sqrt{\sqrt{2} + \sqrt{3}}$

All numbers are irrational.

For your last question on FLT there is a connection with Pythagorean triplets:

https://www.math.mun.ca/~drideout/pytrip06.pdf