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estudier

  • 2 years ago

Show that area of Pythagorean triangle x,y,z cannot be square (U can use that a^4-b^4 = c^2 has no positive solution)

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  1. AbhimanyuPudi
    • 2 years ago
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    the base and height of pythagorean triangle are 3d and 4d where d is a constant Area = 1/2 x 3d x 4d = 3d x 2d = 6d^2 this is never a square

  2. AbhimanyuPudi
    • 2 years ago
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    Its not generalised though!!

  3. estudier
    • 2 years ago
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    It's a start... Let's say we have x^2 + y^2 = z^2 and xy = 2n^2 (triangle area)

  4. AbhimanyuPudi
    • 2 years ago
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    Okay so we take it as hypothesis and prove it a contradiction..I'll try

  5. experimentX
    • 2 years ago
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    |dw:1350135906515:dw|

  6. experimentX
    • 2 years ago
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    |dw:1350136054246:dw|

  7. estudier
    • 2 years ago
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    Promising...how will you get a quartic so as to use the given?

  8. estudier
    • 2 years ago
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    You need to multiply the first line of your first post with something...

  9. estudier
    • 2 years ago
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    Writing your first line in full: (x+y)^2 = x^2 + y^2 +2xy = z^2 + 4n^2

  10. experimentX
    • 2 years ago
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    |dw:1350136543480:dw|

  11. estudier
    • 2 years ago
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    (x+y)^2 = x^2 + y^2 +2xy = z^2 + 4n^2 Your first line (x-y)^2 = x^2 + y^2 -2xy = z^2 -4n^2

  12. experimentX
    • 2 years ago
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    oh!!

  13. estudier
    • 2 years ago
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    :-)

  14. experimentX
    • 2 years ago
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    that would yield extra non dependent solution.

  15. experimentX
    • 2 years ago
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    *equation (solution)

  16. estudier
    • 2 years ago
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    We, are just trying to get a contradiction....

  17. estudier
    • 2 years ago
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    We are assuming a triangle that is square....

  18. experimentX
    • 2 years ago
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    yeah ... three variables 3 non dependent equations ... it's possible to deduce from that.

  19. estudier
    • 2 years ago
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    If u multiply those two above, u get (x^2-y^2)^2 = z^4 - (2n)^4 which contradicts the given so no Pythagorean triangle area is square

  20. experimentX
    • 2 years ago
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    yeah i see the trick!!

  21. estudier
    • 2 years ago
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    U almost had it, a bit more time, u would have had it, I think....

  22. experimentX
    • 2 years ago
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    well .. I'm not too good with algebraic manipulations.

  23. estudier
    • 2 years ago
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    I shall put up a few more while it's stil quiet....

  24. experimentX
    • 2 years ago
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    I usually takes quite lot more than necessary steps to do things.

  25. experimentX
    • 2 years ago
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    I might go offline for 2hrs ... well I would enjoy if you tag me in few interesting problems if you have.

  26. estudier
    • 2 years ago
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    <k

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