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Math Help - Trapezoid Area problem

  1. #1
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    Trapezoid Area problem

    Trapezoid Area problem-img_0089.jpg

    ABCD Trapezoid

    BC = 4

    AD = 8

    BD = 6

    <ADB = 90
    <DBC = 90

    find the area of the darkened part of figure.

    please help
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  2. #2
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    Re: Trapezoid Area problem

    It can easily be shown that triangles ADO and CBO are similar with a 2:1 ratio, using an angle-angle-angle argument. Therefore, BO = \frac{1}{2}DO \Rightarrow BO = 2.

    Triangle AOB has a "base" of 2 and a height equal to AD, or 8 (since BDA is a right angle). Hence,

    [ABC] = \frac{1}{2}(2)(8) = 8
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  3. #3
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    Re: Trapezoid Area problem

    thanks!
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  4. #4
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    Re: Trapezoid Area problem

    Hello, Telo!

    Trapezoid ABCD\!:\;BC = 4,\;AD = 8,\;BD = 6,\;\angle ADB = 90^o,\;\angle DBC = 90^o

    Find the area of the shaded portion.

    Code:
                          B   4   C
                         .* - - - * 
                       .*:|    * *
                     .*:::| *   *
                   .*::::*|O   *
                 .*:::*  6|   *
               .*::*      |  *
             .*:*         | *
            **            |*
        A * - - - - - - - * D
                  8

    Since \Delta OBC \,\sim\,\Delta ODA\!:\:OB = 2,\;OD = 4.


    Refering to the areas of the triangles:

    . . \Delta BDA \:=\:\tfrac{1}{2}(8)(6) \:=\:24

    . . \Delta ODA \:=\:\tfrac{1}{2}(8)(4) \:=\:16

    Therefore: . \Delta AOB \:=\:\Delta BDA - \Delta ODA \:=\:24 - 16 \:=\:8
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