Area (FGE) = Area (CDE) - AREA(CDFG)
Now length FG = 12cm which is 9 + half the difference between 9 and 15 (As FG is halfway down AD and BC).
So area of trapezoid CDFG is half sum of parallel sides times height = [(15 + 12) / 2] * 2 = 27 sq cm
Now we need area of CDE. We can easilty see that a formula for the width of this triangle as a function of the height from the bottom of the triangle must be linear, in fact it must be W = 15 - (3/2)*H.
This is constructed from realising that the foumula must give 15 when height from bottom (H) = 0, and 9 when H = 4 (also 12 when H = 2). So we can imagine a graph of H (x-axis) versus W (y-axis) and we must have a straight line passing through (4,1) and (2,0.5) and we can therefore find its equation easily as y = x/4 so our formula is W = 15 - (H/4)*(15 - 9) = 15 - (3/2)*H (as above)
So rearrange this to make H the subject and insert W = 0 (as it does at the top of the triangle) and we get H = 10. So the height of CDE is 10 and the base is 15 therefore area CDE is 0.5 * 10 * 15 = 75 sq cm.
Therefore area of FGE = 75 - 27 = 48 sq cm