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Thread: Application of the Pythagorean Theorem

  1. #1
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    Question Application of the Pythagorean Theorem

    Greetings,

    I was hoping you could help me prove the following:

    For the case of all 3, 4, 5 special right triangles, the relationship of:

    a/3+b=c

    is always true.

    Or can you find a counterexample? I believe it to be true always but am looking for a formal proof.

    It is much easier to work with than:

    a^2+b^2=c^2

    Please prove me right

    Thanks much,

    Andy
    Last edited by andyimm; May 8th 2019 at 05:33 PM.
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  2. #2
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    Re: Application of the Pythagorean Theorem

    Quote Originally Posted by andyimm View Post
    Greetings,
    I was hoping you could help me prove the following:
    For the case of all 3, 4, 5 special right triangles, the relationship of:
    a/3+b=c
    is always true.
    Or can you find a counterexample? I believe it to be true always but am looking for a formal proof.
    It is much easier to work with than:
    a^2+b^2=c^2
    Please prove me right
    Well it all depends upon what you mean by 3, 4, 5 special right triangles.
    A $\displaystyle 5,~12,~13 $ is clearly a right triangle that lacks that property
    BUT does $\displaystyle 3k,~4k,~5k,~k\in\mathbb{Z}^+$ form right triangle with that property?
    Last edited by Plato; May 8th 2019 at 08:44 PM.
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  3. #3
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    Re: Application of the Pythagorean Theorem

    I think what the OP is asking is to show that for any triplet a = 3k, b=4k, c = 5k it is true that a/3 + b = c.

    Simple enough: Start by adding a/3 + b:

    a/3+b = 3k/3 + 4k = 5k = c


    Hence a/3+b =c.

    As Plato points out this does not work for other examples of Pythagorean triples, such as (5, 12, 13). You might be tempted to think (a, c,b,c)= (5, 12, 13) has the property a/5+b = c, and hence the denominator of the 'a' term is always the value of 'a' for the base Pythagorean triplet - but that only works if the base triple has the property c = b+1, such as for the triplets (5, 12, 13), or (7, 24, 25), etc. It does not work if the base Pythagorean triplet has the property c = b+2, such as for the triplets (8, 15, 17), or (12, 35, 37), etc.
    Thanks from topsquark
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