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Thread: 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 $\displaystyle ABCD$ with diagonals $\displaystyle AC$ and $\displaystyle BD$,
    . . intersecting at $\displaystyle O.$

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

    They share $\displaystyle \Delta AOD.$


    $\displaystyle \begin{array}{cccc}
    \text{We have:} & \Delta ABD & = & \Delta ACD \\
    \text{Subtract }\Delta AOD\!: & \text{-}\Delta AOD & = & \text{-}\Delta AOD \\
    \text{Therefore:}& \Delta AOB & = & \Delta COD \end{array}$

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