# trapezoid 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.

Update:

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

Relevance

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.

• 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)

Source(s): h