Is there a formula/method for finding a vertical line that divides a trapezoid into 2 parts with equal areas when all the vertices are known?

Answer 1

Of course there is, though its form will depend on how the trapezoid is described. For simplicity, let's say that the bottom base goes from (0,0) to (a,0) and that the top base goes from (b,h) to (c,h). (Usually, this will mean that 0<=b<=c<=a)

The area of the trapezoid is (c-b + a-0)/2 * h

So, you want to find x such that

(x-b + x-0)/2 * h = ((c-b+a-0)/2 * h)/2

(2x-b)/2 = (c-b+a)/4
x = (c-b+a)/4 + (a+b)/2 = (a+b+c)/4

Note that if the trapezoid is a rectangle, so that b=0 and c=a, that gives

x = (a+0+a)/4 = a/2
or, halfway across the base.