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

You can see the trapezoid in the picture I have attached.
The trapezoid is shown in the picture attached.

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

We can infer that KLMF is a parallelogram. Indeed:
LK // MF, and  LM // KF.

The area of a parallelogram is given by the formula:
A(KLMF) = b × h
               = KF × h
               = 10 × h

We know that the two areas are congruent, therefore:
A(KLMF) = A(FMN)
FN × h / 2 = 10 × h
The two heights are the same, and then cancel out
FN / 2 = 10
FN = 20

Now we can calculate KN by simply adding KF and FN:
KN = KF + FN = 10 + 20 = 30

Hence, KN is 30 units long.