In trapezium ABCD, AB||DC and

DC=2AB.EF
is drawn parallel to AB cuts AD in F and BC in E such that

BE:EC=3:4
. Diagonal D B intersects EF at G. State if the given statement is condition true or false, “

7EF=10AB
”.

Question

In trapezium ABCD, AB||DC and

DC=2AB.EF
   is drawn parallel to AB cuts AD in F and BC in E such that

BE:EC=3:4
   . Diagonal D B intersects EF at G. State if the given statement is condition true or false, “

7EF=10AB
  ”.

Accepted Answer

It is given that:ABCD is a trapezium. where AB||DC And, DC = 2 AB .......(1) and BE: EC = 3:4.............(2)From equation we can write that:

BE
      
       EC
     
     =
      3
      4

⇒
      
       BE
      
       EC
     
     +1=
      3
      4
     
     +1

⇒
      
       BE+EC
      
       EC
     
     =
      7
      4

⇒
      
       BC
      
       EC
     
     =
      3
      4

Since, the ΔCGE and triangles ΔCAB are similar triangles the ratio of corresponding sides is equal.

EG
    
     AB
   
   =
    
     EC
    
     BC
   
   =
    4
    7

→7EG=4AB

Therefore, 
    
     AF
    
     FD
   
   =
    
     EB
    
     EC
   
   =
    3
    4

FD
      
       AF
     
     =
      4
      3

⇒
      
       FD
      
       AF
     
     +1=
      4
      3
     
     +1

⇒
    
     FD+AF
    
     AF
   
   =
    7
    3

⇒
    
     AD
    
     AF
   
   =
    7
    3

Since, the triangles AFG and ADC are similar triangles.

FG
      
       CD
     
     =
      
       AF
      
       AD
     
     =
      3
      7

⇒7FG=3CD

From equation (1)

7FG=6AB….(4)

Adding (3) and (4)

7(EG+FG)=10AB

⇒7EF=10AB

From the given conditions, it is derived that 7EF=10AB.The given statement is true.Hence, the correct option is 2.

In trapezium ABCD, AB||DC and $D C = 2 A B . E F$ is drawn parallel to AB cuts AD in F and BC in E such that $B E : E C = 3 : 4$ . Diagonal D B intersects EF at G. State if the given statement is condition true or false, “ $7 E F = 10 A B$ ”.