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Thread: help proving midpoints of quadrilaterals

  1. #1
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    [solved] help proving midpoints of quadrilaterals

    help proving midpoints of quadrilaterals-proving-midpoints.png
    Last edited by djo4567; Feb 18th 2013 at 07:10 PM.
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  2. #2
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    Re: help proving midpoints of quadrilaterals

    For both questions, look for congruent triangles. Once you've identified the congruent triangles, study the angles.
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  3. #3
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    Re: help proving midpoints of quadrilaterals

    .
    Last edited by djo4567; Feb 18th 2013 at 05:38 PM.
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    Re: help proving midpoints of quadrilaterals

    Hello, djo4567!

    I'll do the first one.


    Connect the midpoint of the sides of a rhombus.
    Prove that the midpoints form a rectangle.

    $\displaystyle \text{We have rhombus }ABCD,\,\text{ and midpoints }E, F, G, H.$

    Code:
                          E
                A o * * * o * * * o B
                 *     *   *     *
                *   *       *   *
               * *           * *
            H o               o F
             * *           * *
            *   *       *   *
           *     *   *     *
        D o * * * o * * * o C
                  G
    $\displaystyle \text{Draw diagonal }BD.$

    $\displaystyle \text{In }\Delta ABD,\,E\text{ and }H\text{ are midoints of }AB\text{ and }AD.}$
    $\displaystyle \text{Hence: }\,EH = \tfrac{1}{2}BD\text{ and }EH \parallel BD.$
    $\displaystyle \text{(Theorem: the segment joining the midpoints of two sides of a triangle}$
    . . $\displaystyle \text{is parallel to and one-half the length of the third side.)}$

    $\displaystyle \text{In }\Delta CBD,\,F\text{ and }G\text{ are midpoints of }CB\text{ and }CD.$
    $\displaystyle \text{Hence: }\,FG = \tfrac{1}{2}BD\text{ and }FG \parallel BD.$

    $\displaystyle \text{Hence, }EFGH\text{ is a parallelogram.}$
    $\displaystyle \text{(Theorem: if two sides of a quadrilateral are equal and parallel,}$
    . . $\displaystyle \text{the quadrilateral is a parallelogram.)}$


    $\displaystyle \text{Draw diagonal }AC.$
    $\displaystyle \text{Then: }\,AC \perp BD.$
    $\displaystyle \text{(Diagonals of a rhombus are perpendicular..)}$

    $\displaystyle \text{In }\Delta ABC,\,EF = \tfrac{1}{2} AC\text{ and }EF \parallel AC.$

    $\displaystyle \text{Since }EH \parallel BD,\,EF \perp EH.$


    $\displaystyle \text{Therefore, }EFGH\text{ is a rectangle.}$
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  5. #5
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    Re: help proving midpoints of quadrilaterals

    Quote Originally Posted by Soroban View Post
    Hello, djo4567!

    I'll do the first one.



    $\displaystyle \text{We have rhombus }ABCD,\,\text{ and midpoints }E, F, G, H.$

    Code:
                          E
                A o * * * o * * * o B
                 *     *   *     *
                *   *       *   *
               * *           * *
            H o               o F
             * *           * *
            *   *       *   *
           *     *   *     *
        D o * * * o * * * o C
                  G
    $\displaystyle \text{Draw diagonal }BD.$

    $\displaystyle \text{In }\Delta ABD,\,E\text{ and }H\text{ are midoints of }AB\text{ and }AD.}$
    $\displaystyle \text{Hence: }\,EH = \tfrac{1}{2}BD\text{ and }EH \parallel BD.$
    $\displaystyle \text{(Theorem: the segment joining the midpoints of two sides of a triangle}$
    . . $\displaystyle \text{is parallel to and one-half the length of the third side.)}$

    $\displaystyle \text{In }\Delta CBD,\,F\text{ and }G\text{ are midpoints of }CB\text{ and }CD.$
    $\displaystyle \text{Hence: }\,FG = \tfrac{1}{2}BD\text{ and }FG \parallel BD.$

    $\displaystyle \text{Hence, }EFGH\text{ is a parallelogram.}$
    $\displaystyle \text{(Theorem: if two sides of a quadrilateral are equal and parallel,}$
    . . $\displaystyle \text{the quadrilateral is a parallelogram.)}$


    $\displaystyle \text{Draw diagonal }AC.$
    $\displaystyle \text{Then: }\,AC \perp BD.$
    $\displaystyle \text{(Diagonals of a rhombus are perpendicular..)}$

    $\displaystyle \text{In }\Delta ABC,\,EF = \tfrac{1}{2} AC\text{ and }EF \parallel AC.$

    $\displaystyle \text{Since }EH \parallel BD,\,EF \perp EH.$


    $\displaystyle \text{Therefore, }EFGH\text{ is a rectangle.}$
    THANK YOU!!!! That helped soo much.
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  6. #6
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    Re: help proving midpoints of quadrilaterals

    Here's a second proof obtained just by considering angles; I think it's more elementary than the above solution:

    help proving midpoints of quadrilaterals-mhfgeometry10.png
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