Prove Pythagorean Theorem

Discussion in 'Algebra' started by nycmathguy, Jul 6, 2022.

  1. nycmathguy

    nycmathguy

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    College Algebra
    Chapter 1/Section 3

    Prove the Pythagorean Theorem. Explain each step along the way.
     
    nycmathguy, Jul 6, 2022
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  2. nycmathguy

    MathLover1

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    The Pythagorean Theorem says that, in a right triangle, the

    a^2 + b^2 = c^2

    Proof:

    We can show that a^2 + b^2 = c^2 using Algebra

    Take a look at this diagram:
    [​IMG]

    it has that "abc" triangle in it (four of them actually):

    It is a big square, with each side having a length of a+b, so the total area is:

    A = (a+b)(a+b)

    Now let's add up the areas of all the smaller pieces:

    First, the smaller (tilted) square has an area of: c^2

    Each of the four triangles has an area of: ab/2

    So all four of them together is: 4ab/2 = 2ab

    Adding up the tilted square and the 4 triangles gives: A = c^2 + 2ab

    The area of the large square is equal to the area of the tilted square and the 4 triangles. This can be written as:

    (a+b)(a+b) = c^2 + 2ab

    NOW, let us rearrange this to see if we can get the Pythagoras theorem:

    Start with: (a+b)(a+b) = c2 + 2ab
    Expand (a+b)(a+b): a^2 + 2ab + b^2 = c^2 + 2ab

    Subtract "2ab" from both sides:

    a^2 + b^2 = c^2 DONE!
     
    MathLover1, Jul 6, 2022
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    nycmathguy likes this.
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  3. nycmathguy

    nycmathguy

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    Wow! You went beyond with this reply.
     
    nycmathguy, Jul 6, 2022
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