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Math Help - Quadrilateral Construction

  1. #1
    Junior Member
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    Quadrilateral Construction

    Construct a convex quadrilateral and then a quadrilateral similar to it whose area is three-fourths of the area of the original quadrilateral.
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  2. #2
    Super Member

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    Hello, thamathkid1729!

    I already solved this at another site . . .

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  3. #3
    Super Member

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    Hello, everyone!

    This is the long and clunky solution I posted elsewhere . . .


    Construct a convex quadrilateral and then a quadrilateral similar to it
    whose area is three-fourths of the area of the original quadrilateral.

    Suppose the original quadrilateral has sides a,b,c,d.
    The smaller quadratilateral has sides multiplied by a factor of \frac{\sqrt{3}}{2}.

    Construct an equilateral triangle with side 1.
    Contruct an altitude; its length is \frac{\sqrt{3}}{2}
    (I'll leave the proof to you.)


    On a line, mark off points A,\,B,\,C,\,\text{ where }AB \,=\,1,\:BC \,=\,\frac{\sqrt{3}}{2}


    Code:
          o-----o----o
          A     B    C



    From \,A, draw a line to the upper-right.


    Code:
                                *
                              *
                            *
                          *
                        *
                      *
                    *
                  *
                *
              *
            *
          o-----o----o
          A     B    C



    On that line, mark off AP \,=\,a


    Code:
                                *
                              *
                            *
                          *
                     P  *
                      o
                    *
               a  *
                *
              *
            *
          o-----o----o
          A     B    C



    Draw line segment BP.


    Code:
                                *
                              *
                            *
                          *
                     P  *
                      o
                    */
               a  * /
                *  /
              *   /
            *    /
          o-----o----o
          A     B    C



    From \,C construct CQ\,\parallel\, BP


    Code:
                                o Q
                              */
                            * /
                          *  /
                     P  *   /
                      o    /
                 a  */    /
                  * /    /
                *  /    /
              *   /    /
            *    /    /
          o-----o----o
          A     B    C

    Then: PQ \:=\:\frac{\sqrt{3}}{2}\,a \:=\:a'

    Repeat the process with side \,b and find b'.



    With that information, you can complete the construction.


    Code:
                        a
                  * * * * * * *
                 *  a'    *     *
                *           *     *
               * b'        d' *     * d
           b  *                 *     *
             *          c'        *     *
            * * * * * * * * * * * * *     *
           *                                *
          * * * * * * * * * * * * * * * * * * *
                             c

    Note that: . c'\parallel c\,\text{ and }\,d'\parallel d.

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