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Math Help - need a urgent and complete solution for this

  1. #1
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    Post need a urgent and complete solution for this

    prove thatA.M, H.M
    ,AND G.M of two positive nos. 'a' and 'b' can be lengths of right angled triangle if and only a=b(G^3)
    where G=0.5*(1+(5^0.5))
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  2. #2
    MHF Contributor red_dog's Avatar
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    We have A.M.\geq G.M.\geq H.M.
    So A.M., G.M., H.M. can be lengths of a right angled triangle if and only if
    \displaystyle\left(\frac{a+b}{2}\right)^2=(\sqrt{a  b})^2+\left(\frac{2ab}{a+b}\right)^2
    This is equivalent to a^4-18a^2b^2+b^4=9
    Divide the equation by b^4 and note \displaystyle\left(\frac{a}{b}\right)^2=t
    t^2-18t+1=0\Rightarrow t_{1,2}=9\pm 4\sqrt{5}=(\sqrt{5}\pm 2)^2
    We can suppose that a\geq b\Rightarrow\frac{a}{b}\geq 1
    So, t=(2+\sqrt{5})^2.
    \displaystyle\left(\frac{a}{b}\right)^2=(2+\sqrt{5  })^2\Rightarrow\frac{a}{b}=2+\sqrt{5}\Rightarrow a=b(2+\sqrt{5})
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