State whether True or False.If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

Question

Accepted Answer

Consider a Δ ABC, in which a line DE parallel to BC intersects AB at D and AC at E, i.e., DE ∥ BCJoin BE and CD. We know that, Area of a triangle = (1/2) × b × h.So,Area of Δ ADE

=
    1
    2
   
   ×AD×EN
   Area of Δ BDE

=
    1
    2
   
   ×BD×EN
   So,

ar(ΔADE)
    
     ar(ΔBDE)
   
   =

1
      2
     
     ×AD×EN

1
      2
     
     ×BD×EN

ar(ΔADE)
      
       ar(ΔBDE)
     
     =

1
        2
       
       ×AD×EN

1
        2
       
       ×BD×EN

⇒
      
       ar(ΔADE)
      
       ar(ΔBDE)
     
     =
      
       AD
      
       BD
     
     ...............(1)

Similarly, find the area of ΔADE and ΔCDE. Area of Δ ADE

=
    1
    2
   
   ×AE×DM
   Area of Δ CDE

=
    1
    2
   
   ×AE×DM
   So,

ar(ΔADE)
        
         ar(ΔCDE)
       
       =

1
          2
         
         ×AE×DM

1
          2
         
         ×EC×DM

⇒
        
          Area of ΔADE
        
          Area of ΔCDE
       
       =
        
         AE
        
         EC
       
       ……(2)

We know the theorem, if two triangles are on the same base and between the same parallels, then they are equal in area. Since Δ BDE and ΔCDE stand on the same base DE and between the same parallel lines BC and DE,ar(Δ BDE) = ar(Δ CDE) … (3)Using eq (1), (2), and (3), we have,

AD
    
     DB
   
   =
    
     AE
    
     EC

Hence, it is proved that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.So, the correct option is (1).

State whether True or False.

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

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