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Math Help - Pythagoras relationships between perimeter and area

  1. #1
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    Pythagoras relationships between perimeter and area

    Wasn't sure where to put this so I put it here
    Is there any relationship between the perimeter of a pythagorean triple and the area of one. I am really stuck on this help? thanks
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  2. #2
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    Hello, SeaN187!

    Is there any relationship between the perimeter of a Pythagorean trangle and its area?
    No wonder you're stuck . . . There is no "neat" answer.


    A Pythagorean Triple is of the form: . (a,b,c) .where c\:=\:\sqrt{a^2+b^2}

    . . Its perimeter is: . P\;=\;a + b + \sqrt{a^2+b^2}

    . . Its area is: . A\;=\;\frac{1}{2}ab

    And there is no "neat" relationship between P and A.




    Another approach . . .

    A Pythagorean Triple can be generated by: \begin{Bmatrix}a\:=\:m^2-n^2 \\ b\:=\:2mn \\ c\:=\:m^2+n^2\end{Bmatrix}
    . . for any positive integers m > n.


    So there are (and always will be) two parameters (variables) in the problem.


    The perimeter is: . P \;= \;(m^2-n^2) + 2mn + (m^2+n^2)\:= 2m^2 + 2mn \:=\:2m(m+n)

    The area is: . A\;=\;\frac{1}{2}bh\;=\;\frac{1}{2}(m^2-n^2)(2mn) \;=\;mn(m^2-n^2)


    Again, no "neat" relationship between P and A.


    The best we can do is a ratio: . \frac{A}{P}\;=\;\frac{m(m^2-n^2)}{2m(m+n)}\;=\;\frac{n(m-n)}{2}

    Not much better . . .
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  3. #3
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    Quote Originally Posted by Soroban
    Hello, SeaN187!


    No wonder you're stuck . . . There is no "neat" answer.

    .
    There is alwats Heron's formula, but that is a relation between the area,
    perimiter and the triple

    RonL
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  4. #4
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    yeh i searched on wikipedia and got the stuff on m>n and all the formulas but really i need sum1 to talk me through it all and explain it
    although it was an extention for my maths coursework so im just gunna leave it

    just out of intrest what is heron's formula?
    Last edited by SeaN187; June 10th 2006 at 01:46 PM.
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  5. #5
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    Quote Originally Posted by SeaN187
    just out of intrest what is heron's formula?
    A formula they stopped teaching in American schools because it is too complicated

    Given a triangle with sides,
    a,b,c
    Then its area is,
    A=\sqrt{s(s-a)(s-b)(s-c)}
    Where s=\frac{a+b+c}{2}=\mbox{semiperimeter}
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  6. #6
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    thts not exactly super hard is it and its easy to remember too
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  7. #7
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    Quote Originally Posted by SeaN187
    thts not exactly super hard is it and its easy to remember too
    But remember we are talking about american high schools students
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