Trapezoid question - please give correct solution for 10 pts *urgent*?

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.

preguntad... can you explain to me how to get from your second equation to your third? and i don't understand your steps after that; how does abef/efcd=13/25 imply ef/cd=13/25? the heights are also not the same.

Answer 1

I follow the previous response up to the point you questioned. I do not think the answer is right. How could EF be almost the same as AB, if the area ratio is so close to one? I have made my own analysis, which is difficult to show directly but it is available here for viewing.

http://img244.imageshack.us/img244/7461/trapezoidfr3.png

I get EF = 24

EDIT: After reviewing other responses, I think they have errors about the altitude ratios. I have expanded my explanation in case more detail will help. It is in two pages. Here is page 1: http://img172.imageshack.us/img172/3950/trapezoid1za9.png
and page 2: http://img246.imageshack.us/img246/4613/trapezoid2wu4.png

Answer 2

Let:
A1= Area ABEF
A2= Area CDEF
At= Area ABCD
h1= Height of ABEF Trapezoid
h2= Height of CDEF Trapezoid
Ht= Height of ABCD Trapezoid

Consider Equations below:
At= A1+A2 -->1st equation
Ht=h1+h2 -->2nd equation

Note: The ratio of ABEF and CDEF
is 13:12 is also the same ratio as it
height h1 and h2 is 13:12 therefore

h1/h2=13/12 or
h1=12/13h2--->3rd equation

Compute for Ht considering h1=12/13h2:
Ht=h1+h2
Ht=12/13h2 +h2
Ht=25/13h2 -->4rd equation

Using 2nd equation:
At=A1+A2

Using the Formula of Area for a Trapezoid is:
A=[(b1+b2)/2]h
So;
At=A1+A2

[(AB+CD)/2]Ht=[(AB+EF)/2]h1+
[(CD+EF)/2]h2-->5th equation

Input 3rd and 4th equation in the 5th equation:
[(AB+CD)/2]*[(25/13)h2]=
[(AB+EF)/2]*[(12/13)h2]
+[(CD+EF)/2]h2

And input the values of AB and CD:
[(15+30)/2]*[(25/13)h2]=
[(15+EF)/2]*[(12/13)h2]
+[30+EF])/2]h2

Elimanate h2 in the equation to show:
[(45/2)]*[(25/13)]={[(15/2)+(EF/2)]}
*[12/13]+[(30/2)+(EF/2)]

Elimanate 2 in the equation:
45*(25/13)=[(15+EF)*(12/13)]
+(30+EF)
Simplify:
(1125/13)=(180/13)+(12/13)EF+
30+EF

(1125/13)-(180/13)-30=(12/13)EF+EF

(945/13)-30=(12/13)EF+(13/13)EF

(945/13)-(390/13)=(25/13)EF

555=25EF

EF=555/25 or 22.2

Answer 3

We use the similarity of trapezoids.
given ABFE:EFCD=13:12
implies ABFE:ABCD=13:25
implies EF:CD=13:25
CD=30
therefore EF=13*30/25=15.6CM

Answer 4

Area(ABFE)/Area(EFCD) = 13/12, then
Area(EFCD)/Area(ABFE) = 12/13, so
Area(ABCD)/Area(ABFE) = 25/13

This implies CD/EF = 25/13. Since CD = 30, then EF = 30*13/25 = 390/25=78/5=15.6

The answer is 15.6

Answer 5

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