Area of triangle - Area of trapezoid

[sang-gye]

Not sure how to illustrate the problem ( I have a drawing in front of me I added a jpg -picture to my 360 degree yahoo page ... I don't really use it .. but it might be helful for this purpose);

http://ca.360.yahoo.com/edit/yphotos.ht ... 9sMgzMTuJ3

The tasK:
In the trapezoid ABCD below, BC= 2 and AD= 8. If the area of ABCD is 30, what is the area of triangle BCE?

Looks like this:

B C
________________




________________________________
A E D

You would have to imagine the lines between AB BE and CD. The task doesn't mention anything about right triangle or alike.

What I did:

Area of Trapezoid: A=1/2h(x+y) thus

30=1/2h(2+8)
30= 1/2h(10)
30= 5 h
6=h

But that doesn't seem to help me when trying to determine the height of the triangle or any of it's sides. Up until now, I only know that one side is 2(BC) and that BE cannot be < 8 ....

How would I tackle this?

Thanks you for your consideration.

sang-gye


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[soroban]

Hello, sang-gye!

You have everything you need . . . What's stopping you?

In the trapezoid ABCD below: BC= 2 and AD= 8.
If the area of ABCD is 30, what is the area of triangle BCE?

            B     2     C
            *-----------*
           * *    :    *  *
          *   *   :h  *     *
         *     *  :  *        *
        *       * : *           *
       *         *:*              *
      *-----------*-----------------*
      A           E                 D
      : - - - - - - 8 - - - - - - - :

What I did:
. . Area of Trapezoid: \(\displaystyle \:A\:=\:\frac{1}{2}h(x\,+\,y)\)

Thus: \(\displaystyle \:30 \:=\:\frac{1}{2}h(2\,+\,8)\;\;\Rightarrow\;\;30\:=\:\frac{1}{2}h(10) \;\;\Rightarrow\;\;30\:=\:5h\;\;\Rightarrow\;\;h \,=\,6\;\) Good!

Triangle BCE has base \(\displaystyle BC\,=\,2\) and height \(\displaystyle h\,=\,6\).

Well?

 

[sang-gye]

Soroban,

thanks a lot. I assumed that the height of the triangle would only equal height of trapezoid, if the line between BE would form a right angle to side BC, but that's right: Any point on BC has the same distance to line AD, so the length of the sides of the triangle changes, but it's height does not.

Sometimes, I just get stuck in a concept.

Thanks again.
sang-gye