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Math Help - Bow Tie Shapes. Please help

  1. #1
    Newbie Schuyla's Avatar
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    Bow Tie Shapes. Please help

    Chuck used a dynamic geometry utility and kept drawing bow tie shapes. To do this, he used two parallel lines where A, and B are on one line, C and D are on the other. He moved only point D to change the shape. Elizabeth noted that no matter where he moved D, one thing remained true about the triangles making the bow tie. What did she observe about triangles ACE and DBE?
    Bow Tie Shapes. Please help-untitled.png
    I don't know how to go about finding the solution. Can someone help me out please.
    Attached Thumbnails Attached Thumbnails Bow Tie Shapes. Please help-untitled.png  
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  2. #2
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    Re: Bow Tie Shapes. Please help

    I can't imagine how she could observe anything about "triangles ACE and DBE" because there is NO point "E" in your picture or in your description of the problem.
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  3. #3
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    Re: Bow Tie Shapes. Please help

    Hello, Schuyla!

    Chuck used a dynamic geometry utility and kept drawing bowtie shapes.
    To do this, he used two parallel lines where A and B are on one line, C and D are on the other.
    He moved only point D to change the shape.
    Elizabeth noted that no matter where he moved D, one thing remained true about the triangles making the bowtie.
    What did she observe about triangles ACE and DBE?

    Code:
                  C                 D
        - - - - - o - - - - - - - - o - -
                 *|:*           *::*|
                *:|:::* E   *:::::* |
               *::|h::::o::::::::*  |h
              *:::| *     *:::::*   |
             *::* |         *::*    |
        - - o - - + - - - - - o - - + - -
            A     F           B     G
    Draw altitudes CF and DG.
    Since AB \parallel CD,\;CF = DG = h.

    \Delta ACB has base AB and height h.
    \Delta ADB has base AB and height h.
    . . Hence, referring to areas: . \Delta ACB \,=\,\Delta ADB.

    They both have \Delta AEB in common.

    Hence: . \Delta ACB - \Delta AEB \;=\;\Delta ADB - \Delta AEB

    Therefore: . . . . . \Delta ACE \;=\;\Delta DBE

    The two halves of the bowtie have equal areas.
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