How many choices can we use to prove the Pythagorean theorem?

Question

I'm asking about the Theorem, not the triple.

To prove the Pythagorean theorem: a^2=b^2+c^2

And show them, please.

Answer 1

I know of two:

Translation and rotation:

1) translation :

Take a rectangle and cut a diagonal and extended the lines from all three side of the triangle at right angle to it from the side that is made from cutting the rectangle until those lines equal the lenght of the side of the triangle drawn from and it form three boxes around all three sides of the triangle. Now let take the big side an calulate the square of it ,then calulate the squares of the other two sides and add themup, it should equal the square of the big side and that proves the pythagorean theorem.

Rotation:

See the show : Project Mathematics by Cal tech : Entitled "The Pythagorean Theorem" it explain better than I can, It worth watching. Very good show on Math & Geometry

Answer 2

Pythagorean Triples are sets of 3 numbers that work with the pythagorean theorem.

There are infinite choices.

Answer 3

We can use 3 choices to prove the pythagorean theorem

Answer 4

If you are making choices, you are probably not proving the Pythagorean theorem, which is a general result. It works for *all* right triangles.

If you are asking how many different proofs there are, then at last count there are hundreds.

Answer 5

The proof of the theorem that I am aware of uses calculus and is reasonably straightforward.

You start by considering a pseudo right triangle where the hypoteneus is a "stair" rather than a line.

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For this figure, the pseudo-hypoteneuse equals the sum of the of the base and the height. (the horizontal pieces of the steps is equal to the horizontal base and the vertical pieces of the steps are equal to the vertical height).

The more smaller steps that are used, the closer it gets to a "real" hypoteneuse rather than a pseudo one. If we calculate the sum of all the steps at the point in the function that represents the limet as the step size becomes zero, then we will derive the function and form the proof that x^2 + y^2 = z^2

I started to try to show the calculus here but its too hard to type.

I leave the rest as an excersize for the interested student (as the books say).

Answer 6

Infinite-any right triangle can prove the Pythagorean theorem

Answer 7

There are INFINITE choices can be used to prove the theorem.

where a is perpendicular to b

place 4 of these triangles so that you have (a + b) on each side of a square

the hypotenuse will join the places where a & b are joined on each side.. forming a square in the middle

The area of the BIG square is (a+b)(a+b)

or a^2 +2ab +b^2

the area of the SMALL square is c^2

there are 4 equal triangles and one small square that make up the BIG square.

area of each triangle is (1/2)ab

times 4... gives 4*(1/2)ab = 2ab for just the triangles

now... area of BIG square minus area of the triangles equals the area of the small square.. or...

a^2 + 2ab +b^2 - 2ab = c^2

combining.

a^2 + b^2 +2ab - 2ab = c^2

or

a^2 + b^2 = c^2




ok.. for the small square in the middle... a and b are in a straight line... so their associated angles add up to 90 degrees

so you have angle A + angle B + unknown angle between the c's is equal to 180 ... so the unknown angle has to be 90 degrees.. so it is a square in the middle.

sorry i can't draw it out ... it keeps getting re-formatted... but if you do it on paper..you will see it.

Answer 8

the one method i know is using the similar triangle method..