trapezoid question?

Question

In trapezoid ABCD, bases AB = 15 and CD = 30. Points E and F are on AD and BC with EF || AB. If the ratio ofIn trapezoid ABCD, bases AB = 15 and CD = 30. Points E and F are on AD and BC with EF || AB. If the ratio of the area of ABFE to that of EFCD is 13:12, compute EF.

10 points to the first person who can give me a complete solution.

Answer 1

Draw a line from point A to line CD.(perpendicular to CD)
Name the intersection of this line wiyh line EF, G and with line CD, H.
AG=h1
GH=h2
EF=x
For trapezoid ABFE
S1=(15+x)*h1/2
For trapezoid EFCD
S2=(30+x)*h2/2
S1/S2=13/12
==>
(15+x)*h1/((30+x)*h2=13/12
==>
h2/h1=12(15+x)/(13(30+x))
equation #1
For trapezoid ABCD
S=S1+S2
=(15+30)*(h1+h2)/2
==>
(15+x)*h1/2+(30+x)*h2/2=
45*(h1+h2)/2
==>
(x-15)*h2=(30-x)*h1
==>
h2/h1=(30-x)/(x-15)
equation #2
Now consider equations #1 and #2
==>
(30-x)/(x-15)=
12*(15+x)/((13*(30+x))
==>
13*(900-x^2)=
12(x^2-225)
==>
x^2=576
==>
x=EF=24
I believe you owe me 10 pts.
Sorry for my English. If you correct this solution grammatically and send me my errors I will appreciate that. Thanks.

Answer 2

draw a line from point A to line perpendicular to CD
call the intersection of this line with line EF, G and with line CD, H.
AG=h1
GH=h2
EF=x
for trapezoid ABFE:
S1=(15+x)h1/2
For trapezoid EFCD:
S2=(30+x)h2/2
S1/S2=13/12

(15+x)h1/((30+x)h2=13/12

h2/h1=12(15+x)/(13(30+x))
equation 1
For trapezoid ABCD
S=S1+S2
=(15+30)(h1+h2)/2

(15+x)h1/2+(30+x)*h2/2=
45(h1+h2)/2

(x-15)h2=(30-x)h1

h2/h1=(30-x)/(x-15)