ABCD is a trapezium. AB is parallel to DC, BD is a diagonal and E is the midpoint of AD. A line is drawn through E parallel to AB intersecting BC at F. F is the midpoint of BA. Check whether the given statement is true or false.

Question

ABCD is a trapezium. AB is parallel to DC, BD is a diagonal and E is the midpoint of AD. A line is drawn through E parallel to AB intersecting BC at F. F is the midpoint of BA. Check whether the given statement is true or false.

Accepted Answer

Now let's say the intersection of BD and EF is O.So we have a figure likeNow we know that ABCD is a trapezoid.Hence AB is parallel to CD.Now it is given that a line parallel to AB is drawn through E.EF is therefore parallel to AB.Now we can say that Now consider the triangle ABD.In this triangle ED = EA ………………….. (1)Also we have EO parallel to AB.Using the fundamental theorem of proportionality, we thus get,

ED
    
     EA
   
   =
    
     DO
    
     OB

But from equation (1) we have ED = EA.Hence we get,

EA
    
     EA
   
   =
    
     DO
    
     OB

.Hence we can say that

DO
    
     OB
   
   =1
  By cross multiplication we get, DO = OB …… (2)Therefore O is the midpoint of DB.Now consider the triangle BCD.In this triangle, we have OF parallel to CD.Now using the Angle Axis Theorem we can say that