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Math Help - Need help with CONGRUENCY!! (trapezium) @___@

  1. #1
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    Question Need help with CONGRUENCY!! (trapezium) @___@

    A trapezium is divided into 4 triangles by its diagonals. Prove the ares of the coloured triangles are equal. Thank you

    Need help with CONGRUENCY!! (trapezium) @___@-trapezium.jpg
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  2. #2
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    Hello, kapitanvicki!

    A trapezium is divided into 4 triangles by its diagonals.
    Prove the areas of the coloured triangles are equal.
    Code:
               B                   C
                *  *  *  *  *  *  *
               *|:::*         *:::|:*
              *:|:::::::::*:::::::|:::::*
             *::|:::::*   O     *:|:::::::*
            *:::|:*               |   *:::::::*
           *::* |                 |         *:::*
          *::*  *  *  *  *  *  *  *  *  *  *  *  *  *
         A      E                 F                  D

    We have trapezoid ABCD with diagonals AC and BD,
    . . intersecting at O.

    \Delta ABD and \Delta ACD have a common base, AD, and equal altitudes, BE = CF.
    . . Hence: . \text{area }\Delta ABD \:=\: \text{area }\Delta ACD

    They share \Delta AOD.


    \begin{array}{cccc}<br />
\text{We have:} & \Delta ABD & = & \Delta ACD \\<br />
\text{Subtract }\Delta AOD\!:  & \text{-}\Delta AOD & = & \text{-}\Delta AOD \\<br />
\text{Therefore:}& \Delta AOB & = & \Delta COD \end{array}

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