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Math Help - "A parallelogram is a rhombus if and only if its diagonals are at right angles"

  1. #1
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    "A parallelogram is a rhombus if and only if its diagonals are at right angles"

    I proved that ||a+b|| = ||a-b|| if and only if a.b = 0

    However the next part says using this explain why a paralleogram is a rhombus if and only if its diagonals are at right angles..

    I can't figure this out?
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  2. #2
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    This is a neat problem.
    If \vec{a}~\&~\vec{b} are consecutive sides of a parallelogram then \vec{a}+\vec{b}~\&~\vec{a}-\vec{b} are its diagonals.
    Now (\vec{a}+\vec{b}) \cdot (\vec{a}-\vec{b})=\vec{a}\cdot \vec{a}-\vec{b}\cdot\vec{b} .

    So if the parallelogram is a rhombus then \left\| \vec{a} \right\|=\left\| \vec{b} \right\| .
    What does that imply?

    What is the converse?
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  3. #3
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    Quote Originally Posted by Plato View Post
    This is a neat problem.
    If \vec{a}~\&~\vec{b} are consecutive sides of a parallelogram then \vec{a}+\vec{b}~\&~\vec{a}-\vec{b} are its diagonals.
    Now (\vec{a}+\vec{b}) \cdot (\vec{a}-\vec{b})=\vec{a}\cdot \vec{a}-\vec{b}\cdot\vec{b} .

    So if the parallelogram is a rhombus then \left\| \vec{a} \right\|=\left\| \vec{b} \right\| .
    What does that imply?

    What is the converse?
    I understand exactly what you've done so far, but still I don't know what it implies!!!

    If the paralleogram is a rhombus then it's diagonals must be at right angles. This means a.b = 0

    Now what?!
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  4. #4
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    I frankly do not know what to say to that.
    \left\| {\vec{a}} \right\|=\left\| {\vec{b}} \right\| if and only if  \vec{a} \cdot\vec{a}=\vec{b} \cdot\vec{b}
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