Three vertices of the trapezoid are a(4d,4e), b(4f,4e), and c(4g,0). the fourth vertex lies on the origin. find the midpoint of the midsegment of the trapezoid.

We have that

point C and point D have y = 0> (the bottom of the trapezoid).

point A and point B have y = 4e > (the top of the trapezoid)

the y component of  midpoint would be halfway between these lines
 y = (4e+ 0)/2 = 2e. 

the x component of the midpoint of the midsegment would be halfway between the midpoint of AB and the midpoint of CD.

x component of midpoint of AB is (4d + 4f)/2.
x component of midpoint of CD is (4g + 0)/2 = 4g/2.
x component of a point between the two we just found is
[(4d + 4f)/2 + 4g/2]/2 = [(4d + 4f + 4g)/2]/2 = (4d + 4f + 4g)/4 = d + f + g. 

therefore

the midpoint of the midsegment is (d + f + g, 2e)