Kinetic Energy Pythagorean Theorem Proof! (Math w/ Physics #2)

Forget proofs by rearrangement. This proof is one that every physics student should know, by proving it with kinetic energy. Here's a basic rundown:

A particle starts at the origin of the x-y plane. It first starts with a velocity a in the positive y-direction. This has a kinetic energy of 0.5ma^2. Then while still keeping this y-velocity, it adds a velocity b in the positive x direction. This has a kinetic energy of 0.5mb^2. Together, this will give the combined final velocity c of the particle which has a kinetic energy of 0.5mc^2. Setting these kinetic energies equal to each other and simplifying will give us the Pythagorean theorem.

[dancl]

Thank you for the video. Please, sir. what software program did you use to illustrate with?

[Dekoherence-iipw]

I don't think this works. Because... actually, the fact that the kinetic energies of the x and y components are independent of each other IS ITSELF something that follows from Pythagoras's theorem.

So it's actually circular logic to prove Pythagoras's theorem from this, without first justifying the independence of the x and y components.