# Need help with CONGRUENCY!! (trapezium) @___@

• Nov 30th 2007, 12:25 AM
kapitanvicki
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:confused:

Attachment 4564
• Nov 30th 2007, 04:37 AM
Soroban
Hello, kapitanvicki!

Quote:

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 $\displaystyle \Delta AOD.$

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