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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 $EF.$

 

 Dec 29, 2023
 #1
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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

 

cool cool cool

 Dec 29, 2023
edited by CPhill  Dec 29, 2023

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