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Math Help - Proving parallelogram (vectors)

  1. #1
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    Proving parallelogram (vectors)

    The midpoints of the adjacent sides of a quadrilateral are joined. Show that the resulting figure is a parallelogram.

    I know that you suppose to somehow use vectors to prove this, but I dont know how you are suppose to model this?
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  2. #2
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    Suppose that A,~B,~C,~D are the consecutive vertices of the quadrilateral.
    Let midpoints be P \in \overrightarrow {AB},~ Q \in \overrightarrow {BC},~ R \in \overrightarrow {CD},~ S \in \overrightarrow {DA} .
    We know that \overrightarrow {AB}+\overrightarrow {BC}+\overrightarrow {CD}+\overrightarrow {DA}=0 .
    Moreover, 0.5(\overrightarrow {AB})=\overrightarrow {PB}~\&~ 0.5(\overrightarrow {BC})=\overrightarrow {BQ}
    So 0.5(\overrightarrow {AB}+\overrightarrow {BC})=\overrightarrow {PQ}
    Do the same thing for  \overrightarrow {RS}
    Prove that  \overrightarrow {PQ}=-\overrightarrow {RS}
    You will be done.
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    Quote Originally Posted by Plato View Post
    We know that \overrightarrow {AB}+\overrightarrow {BC}+\overrightarrow {CD}+\overrightarrow {DA}=0 .
    What if the quadrilateral you were doing this proof in was a trapezium then how would this be true? (since all the sides are unequal). How would the vectors of the sides add up 0?

    Quote Originally Posted by Plato View Post
    So 0.5(\overrightarrow {AB}+\overrightarrow {BC})=\overrightarrow {PQ} .
    Sorry but I dont understand why the inner side of a parallelogram = 2 x 1/2 of the sides of a quadrilateral.
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    Quote Originally Posted by SyNtHeSiS View Post
    What if the quadrilateral you were doing this proof in was a trapezium then how would this be true? (since all the sides are unequal). How would the vectors of the sides add up 0?
    Sorry but I dont understand why the inner side of a parallelogram = 2 x 1/2 of the sides of a quadrilateral.
    This is a standard question. It is done in any geometry course.
    I have given you the standard vector proof.

    BTW. This theorem is true for any quadrilateral, even non-convex quadrilaterals.
    And the proof is the same.

    I suspect that you do not understand vector addition.
    If that is the case, you no chance proving this.
    As I said, the proof I gave is the standard.
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