note that the area of rectangle ABCD is given by AD*AB

now, ABED is a trapezium, thus its area is given by half the sum of the two parallel sides times the distance between them. that is (1/2)(BE + AD)*AB, and this is 2/3, so

(1/2)(BE + AD)*AB = 2/3

=> (BE + AD)*AB = 4/3

but, BE = (1/2)AD (this should be pretty obvious from a diagram if you drew or were given one), so that

=> (3/2)AD*AB = 4/3

=> AD*AB = 8/9