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Math Help - Proving type of quadrilateral by its diagonal lengths

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    Thumbs up Proving type of quadrilateral by its diagonal lengths

    Hi Forum

    Is there an easy and objective way to find the type of quadrilateral just by its diagonal lengths? I mean, with no calculation.
    If we were to find the area of a quadrilateral with diagonals 12 and 8. Having the diagonals it's not enough to find the area.
    Only if the quadrilateral is a rhombus, but we can't just assume that it is a rhombus. Since there is no more information given, like bisection of diagonals and angles.

    Thanks!
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by Zellator View Post
    Hi Forum

    Is there an easy and objective way to find the type of quadrilateral just by its diagonal lengths? I mean, with no calculation.
    If we were to find the area of a quadrilateral with diagonals 12 and 8. Having the diagonals it's not enough to find the area.
    Only if the quadrilateral is a rhombus, but we can't just assume that it is a rhombus. Since there is no more information given, like bisection of diagonals and angles.

    Thanks!
    This assumption is wrong. There is \aleph different quadrilaterals with diagonals 12 and 8.
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    Quote Originally Posted by Also sprach Zarathustra View Post
    This assumption is wrong. There is \aleph different quadrilaterals with diagonals 12 and 8.
    Hi!
    Thanks for the reply!

    This is the question
    The diagonals of the quadrilateral LEAK have diagonals LA=12 and EK=8. Find the maximum area this quadrilateral can have under these conditions
    In the answer is just assumed that LEAK is a rhombus and the answer is 1/2xLAxEK

    If we are after the maximum area is it safe to simple use a rhombus?
    Is it because of its shape that the area will be maximized?
    Sorry I left that bit of information out. I see it was actually important. : )
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by Zellator View Post
    Hi!
    Thanks for the reply!

    This is the question
    The diagonals of the quadrilateral LEAK have diagonals LA=12 and EK=8. Find the maximum area this quadrilateral can have under these conditions
    In the answer is just assumed that LEAK is a rhombus and the answer is 1/2xLAxEK

    If we are after the maximum area is it safe to simple use a rhombus?
    Is it because of its shape that the area will be maximized?
    Sorry I left that bit of information out. I see it was actually important. : )
    You can prove using calculus tools that maximum area of quadrilateral is when the diagonals are perpendicular.
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    MHF Contributor Also sprach Zarathustra's Avatar
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    We can prove that maximum area of quadrilateral is when the diagonals are perpendicular, by proving that for triangles. If ABC is triangle and x is the angle between a and b then S, the area of ABC is give by: (1/2)*a*b*sin(x).
    max{S}=max{ (1/2)*a*b*sin(x) }=(1/2)*a*b*max{sin(x)}=(1/2)*a*b*1 (Why?), hence sin(x)=1 ==> x=/pi/2.
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    Quote Originally Posted by Also sprach Zarathustra View Post
    We can prove that maximum area of quadrilateral is when the diagonals are perpendicular, by proving that for triangles. If ABC is triangle and x is the angle between a and b then S, the area of ABC is give by: (1/2)*a*b*sin(x).
    max{S}=max{ (1/2)*a*b*sin(x) }=(1/2)*a*b*max{sin(x)}=(1/2)*a*b*1 (Why?), hence sin(x)=1 ==> x=/pi/2.
    Amazing! Yes I agree!
    Clever resolution for it. It is a calculus question really. So we should use calculus.
    Thanks for your time Also sprach Zarathustra!
    That really clears a lot of things up.
    All the best!
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by Zellator View Post
    Amazing! Yes I agree!
    Clever resolution for it. It is a calculus question really. So we should use calculus.
    Thanks for your time Also sprach Zarathustra!
    That really clears a lot of things up.
    All the best!
    Thank you!
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