A trapezoid $ABGE$ is partitioned into three smaller shapes: a trapezoid $ABCD$, a parallelogram $CDEF$, and a triangle $CFG$. The sides $AB || CD$, and $2AB = CD$. The number in each colored region indicates the area of each shape.
What is the area of the blue parallelogram?
The question basically states that the answer will be the same no matter what trapezium (trapezoid in your country) you use so long as it fits the description.
So I took the liberty of making angle AEG a right angle.
Here is the pic I used
I worked out that k=3/2 and then you can work out the blue area.
Angle AEG = 90 deg. angle BGE = 45 deg.
[CDEF] = 8 square units
AB = AD = 1.632993161
DC = EF = 2*AB = 3.265986322
DE = CF = FG = 2.449489743
[CDEF] = 3.265986322 * 2.449489743 = 8
Diagram not to scale
Where do all your added numbers come from?
I know what the answer is. I answered first.
But neither of you 2 has said where any of your numbers come from.
Oh I get it.
You both arbitrarily chose angle AEG to be 90 degrees AND angle EGB=45 degrees.
And you arbitrarily chose AB to equal AD
Only you didn't bother to explain that is what you were doing.
This does make it easier,
You do not need to approximate anything though.
If you let AB=x units and CF=h units then
it is easy to work out that h=sqrt6, and x=sqrt(8/3)
And then that blue area = 8 units squared.