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Math Help - Lenght of the Diagonal of a Quadrilateral given its 4 Side Lengths.

  1. #1
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    Question Lenght of the Diagonal of a Quadrilateral given its 4 Side Lengths.

    I was browsing the AMC 2010 Brochure, and came across problem
    10A-12, 12A10. It it possible to solve this problem with the given information? If so, how? Thanks in advance.
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  2. #2
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    Hello, asyed94!

    It's much easier thank you imagine . . .


    In quadrilateral ABCD\!:\;\;AB = 5,\;BC = 17,\;CD = 5,\;DA = 9.
    And the length of diagonal BD is an integer.
    Find the length of BD.
    Code:
                            A
                            o
                   9    *    *
                    *         *
           D    *              *
            o                   * 5
           *      *              *
        5 *             *         *
         *                    *    *
      C o   *   *   *   *   *   *   o B
                       17

    In \Delta ABD, the Triangle Inequality says: . AB + AD \:> \:BD

    That is: . 5 + 9 \:>\:BD \quad\Rightarrow\quad BD \:<\:14


    In \Delta BCD, the Triangle Inequality says: . BD + CD \:>\:17

    That is: . BD + 5 \:>\:17 \quad\Rightarrow\quad BD \:>\:12


    Therefore: BD is an integer between 12 and 14 . . . BD \,=\, 13.

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  3. #3
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    Many thanks for the quick and concise reply. This problem had me on the ropes all day . How were you able to determine the method necessary to solve it?
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