+1418 Trapezoid $ABCD$ has bases $\overline{AB}$ and $\overline{CD}$. Line segment $\overline{EF},$ which is parallel to the bases, divides trapezoid $ABCD$ into two trapezoids of equal area. Find the length $EF.$
+129771 Let the height of the upper trapezoid = x and the height of the lower trapezoid = y
Since the areas are equal
(1/2) (x) ( 1 + EF) = (1/2)y (2 + EF)
x ( 1 + EF) = y(2 + EF)
x/y = (2 + EF) / ( 1 + EF)
height = x + y = (2 + EF) + (1 + EF) = 3 + 2EF
And twice the area of the top trapezoid = the area of the whole trapezoid
2 ( 1/2) x (1 + EF) = (1/2)(x + y)(1 + 2)
x (1 + EF) = (3/2)(x + y)
(2 + EF) ( 1 + EF) = (3/2)(3 + 2EF)
2 + 3EF + EF^2 = (3/2)(3 + 2EF)
EF^2 + 3EF + 2 = 3EF + 9/2
EF^2 = 9/2 - 2
EF^2 = 5/2
EF = sqrt (5/2) ≈ 1.58
