AE is 35 units in length.
One of the two shapes that DE splits the trapezoid into is a triangle. Since the two sections of the trapezoid have an equal area, this means that the area of the triangle is 1/2 of the area of the trapezoid. Using the formulas for the area of a triangle and the area of a trapezoid we get:
1/2bh = 1/2(1/2(B+b)(h))
The base of the triangle, AE, is unknown. We do know that AE + EB = 50; let x be EB. That means that AE = 50-x.
B, the "big base" of the trapezoid, is 50. b in the trapezoid, the little base, is 20. Using all of this we now have:
1/2(50-x)h = 1/2(1/2(50+20)(h))
1/2(50-x)h = 1/2(1/2(70)h)
1/2(50-x)h=1/2(35)h
Since we have multiplied by 1/2 and h on both sides, we can divide by both of them at the same time to cancel. This will give us:
(50-x)=35
Subtract 50 from both sides:
50-x=50 = 35-50
-x = -15
x = 15
This means that EB is 15; thus AE = 50-15 =35.
