Disproving Pythagorean Theorem

In summary, the conversation discusses the idea of the Pythagorean theorem being wrong and the speaker's attempts to disprove it. They share a proof they made that appears to contradict the theorem, but ultimately it is revealed that the proof is based on a misunderstanding of the properties of right triangles.
  • #1
Big Gus
2
0
Ever since I was in grade school I have been fascinated with the idea that the Pythagorean theorem, or any other universally respected theorem, could be wrong. When I was younger I found a little proof I made to disprove it, and I came across it in an old notebook of mine. Now after taking calculus and other more advanced maths I see that Pythagoras could not be wrong, however, I am having trouble actually disproving the proof I made to disprove Pythagoras's theorem, if that makes sense. I may just be having a serious brain fart so don't kill me.

The proof I made was this:

Make 4 congruent right triangles with side lengths of "a" and "b" and a hypotenuse with a length "c" and put the triangles together to make a square where the hypotenuses of the triangles are on the outside of the square. it should look like this: http://4.bp.blogspot.com/-YF-2E8vTRLs/TmoSmBD65wI/AAAAAAAABX8/BPFkCMM0vGE/s1600/QST.png

Obviously the area of the triangle is A=c^2, the Area could also be the area of each triangle A=1/2ab, there are 4 of them so it becomes A=2ab. Through substitution we get c^2=2ab.

c^2=2ab doesn't agree with the Pythagorean theorem, the problem I have having is explaining why this is so... I can't find an error in my logic. So can you guys help me out and tell me where I went wrong?
 
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  • #2
"c^2=2ab doesn't agree with the Pythagorean theorem"

I agree with you up til here. So you created a square, right? Meaning that each triangle is a 45 45 90, right? So then a=b. Then c^2=a^2+b^2=a^2+a^2=2a^2=2ab. I see no reason why this contradicts the Pythagorean Theorem.
 
  • #3
hmm... well I do you see they don't have to be 45 45 90 triangles, it could be a rectangle too, sorry for saying square. Try an example let's say a=4 b=5 and by the Pythagorean theorem c=5 but by the formula c^2=2ab c=sqrt20..
 
  • #4
This site is not the place to discuss your misunderstandings.

Thread closed.
 
  • #5
Big Gus said:
hmm... well I do you see they don't have to be 45 45 90 triangles, it could be a rectangle too, sorry for saying square.

In reference to your drawing, and working backwards, if you start with a rectangle and draw two diagonal lines, the four angles at the center will not be congruent. That means that none of the angles at the center can be 90°.

The only way that the central angles can all be congruent (and thus 90° each) is when the rectangle is actually a square. In this case the triangles are all 45° - 45° - 90° right triangles, and the two legs are equal in length. If the square is a units on each side, each leg of the triangle that makes up a quarter of the square will be a/√2.

Big Gus said:
Try an example let's say a=4 b=5 and by the Pythagorean theorem c=5 but by the formula c^2=2ab c=sqrt20..
No. If the legs of a right triangle are a = 4 and b = 5, then c can't be 5. This would be too short. By the Pythagorean Theorem, c = √(42 + 52) = √41 ≈ 6.4.
 

FAQ: Disproving Pythagorean Theorem

1. How do you disprove the Pythagorean Theorem?

To disprove the Pythagorean Theorem, you would need to find a counterexample where the theorem does not hold true. This means finding a set of three numbers that do not follow the equation a^2 + b^2 = c^2, where a, b, and c represent the sides of a right triangle. However, despite numerous attempts, the Pythagorean Theorem has not been successfully disproved and is still considered a fundamental and universally applicable mathematical concept.

2. What would happen if the Pythagorean Theorem was disproven?

If the Pythagorean Theorem was disproven, it would have a significant impact on mathematics and many other fields that rely on its principles. It would mean that the relationship between the sides of a right triangle would not follow a consistent pattern, and the theorem could not be used to solve problems involving right triangles. It could potentially require a complete overhaul of mathematical principles and formulas.

3. Can a single counterexample disprove the Pythagorean Theorem?

Yes, a single counterexample can disprove the Pythagorean Theorem. However, since the theorem has been extensively tested and proven to be true in countless scenarios, it is highly unlikely that a single counterexample would be found. This is why the Pythagorean Theorem is considered a law rather than a theory.

4. Has anyone ever successfully disproved the Pythagorean Theorem?

No, the Pythagorean Theorem has never been successfully disproved. It has been proven to be true in numerous scenarios and is widely accepted as a fundamental mathematical principle. However, there have been attempts to disprove the theorem, but none have been successful.

5. Why is it important to test and potentially disprove mathematical theories?

Testing and potentially disproving mathematical theories is important because it helps to refine and improve our understanding of mathematics. It allows us to identify any flaws or limitations in current theories and develop new and more accurate concepts. It also encourages critical thinking and creativity in the field of mathematics.

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