Given: klmn is a trapezoid, kf =10 mf ║ lk aklmf = afmn find: kn

You can see the trapezoid in the attached picture.

The data given in the problem are:
KF = 10
LK // MF
A(KLMF) = A(FMN)

We know that KLMF is a parallelogram because:
LM // KF (bases of the trapezoid) 
LK // MF (hypothesis).
The area of a parallelogram is given by the formula: 
A(KLMF) = b × h
               = KF × h
               = 10 × h

The area of a triangle is given by the formula:
A(FMN) = (b × h) / 2
             = (FN × h) / 2

The problem states that the two areas are congruent, therefore:
A(KLMF) = A(FMN)
10 × h = FN × h / 2
10 = FN / 2
FN = 20

Therefore we can calculate:
KN = KF + FN = 10 + 20 = 30

Hence, KN is 30 units long.