The trapezoid is shown in the picture attached.
We know:
LK // MF
KF = 10
A(KLMF) = A(FMN)
Since LM // KF (because they are bases of the trapezoid) and LK // MF because is given in the hypothesis, KLMF is a parallelogram.
The area of a parallelogram is given by the base times the height.
A(KLMF) = b × h
= 10 × h
The area of a triangle is given by the formula:
A(FMN) = (b × h) / 2
= (FN × h) / 2
We know that the two areas are congruent, therefore:
A(KLMF) = A(FMN)
10 × h = FN × h / 2
The two "h" cancel out because they are the same and we can solve for FN:
10 = FN / 2
FN = 20
Now we can calculate:
KN = KF + FN = 10 + 20 = 30
Hence, KN is 30 units long.
