Abcd is a trapezoid with sides ab parallel to cd, where ab = 50, cd = 20. e is a point on the side ab with the property, that the segment de divides the given trapezoid into two parts of equal area (see figure). calculate the length ae.

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.