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.
