I have a trapezoid. I know it's height and bases. How can I split it in 2 parts of given area by line parallel to bases. For example my trapezoid has area of S. And I want to get 2 trapezoids with areas aS and (1-a)S.
Assuming that the values of $base_1$, $base_2$, $S$, $h$ and $a$ are known then you can set up a system of equations to find the two unknowns. We will call the unknown height of the bottom trapezoid $x$ and the unknown line parallel to the bases $base_3$.
$(1-a)S$ = $(1/2)(base_1 + base_3)$ $*$ $(h-x)$
$aS$ = $(1/2)(base_2 + base_3)$ $*$ $(x)$
Solving the system would give you the value of $x$ which determines the height of the two trapezoids and the value of $base_3$ which determines their shared side.