similarity at Line parallel to another line to make Z letter

Question

$AB = BC =CD =5$ cm. AB // CD. DE // BC. DE = 2 cm. What is the length of $BF$?

Answer 1

Assuming $DE\parallel BC$ (otherwise we haven't enough data) one gets: $$ AB:BF=CG:FC=DG:DE, \quad\hbox{that is:}\quad 5:BF=(5-DG):(5-BF)=DG:2. $$ Solve for $BF$ and $DG$.

Answer 2

Complete the figure to get the point $H$ such that $BCDH$ is a parallelogram.

• It is immediate that $HE+ED=BC=5$ so $HE=3$
• By construction we also have $AC=CH$ so by applying Thales $CF=\frac 12 HE=\frac 32$
• Finally $CF+BF=BC=5$ so $BF=\frac 72$