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B Basic geometry problem with triangle

  1. Mar 13, 2017 #1
    This is a problem I thought of, and I was wondering how to mathematically solve it with an equation.

    I tried calling one leg x.
    So the other leg, because of Pitagora's theorem, is: √(52 - x2)

    The area is equal to the product of the legs divided by two, so:

    6 = (x * √(52 - x2))/2
    12 = x * √(52 - x2)

    Problem is, I don't know how to solve the equation beyond the first basic step lol

    Btw, I know it can be solved with Euclide's second theorem, but I was wondering how to solve it with Pitagora, if it's possible to solve that equation I wrote. So basically I'm asking how to solve that equation.
     
    Last edited: Mar 13, 2017
  2. jcsd
  3. Mar 13, 2017 #2
    Well, you know a2 + b2 = 25 and ab/2 = 6. So you can work out a2 + 2ab + b2 and a2 - 2ab + b2; hence a + b and a - b, hence a and b.
     
  4. Mar 13, 2017 #3

    mfb

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    You can solve the last equation by squaring it, afterwards it is a quadratic equation in x2.
    The approach described by mjc123 is easier, however.
     
  5. Mar 13, 2017 #4
    That's actually nice lol Thanks for the input.
    I think that's what I was looking forward to doing, but I don't know how that would work. So, could you elaborate on the passages required? I only know about the property where you can sum/subtract, multiply/divide something left and right in an equation, but I don't know how to proceed with "squaring".
     
  6. Mar 13, 2017 #5

    mfb

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    If c=d, then c2=d2. If you know that both c and d are positive (which is the case here), the reverse direction is true as well.
     
  7. Mar 13, 2017 #6
    Wouldn't I get 144 = x2 * (25-x2) from 12 = x * √(52 - x2) then?
    It would continue as 144 = 25 x2 - x4, which would be a grade-four equation.
     
  8. Mar 13, 2017 #7

    mfb

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    It is a quartic equation in x, but a quadratic equation in x2.

    144 = 25 (x2) - (x2)2

    You can solve for x2 with the usual formula for quadratic equations. Afterwards you can take the square root to find x.
     
  9. Mar 13, 2017 #8
    Aaaah I see. Neat, thanks a lot :D
     
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