In a triangle, ABC, D is a point on AB such that AB = 4AD and E is a point on AC such that AC = 4AE. Prove that BC = 4ED.


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Answer:


Step by Step Explanation:
  1. It is given in the question that D and E are the points on side AB and BC of ΔABC respectively. Join DE.
  2. Given:- AB = 4AD
    or, AD =  
    1
    4
     AB
    AC = 4AE
    or, AE =  
    1
    4
     AC
  3. We need to prove that the ΔADC and ΔABC are similar.
    Where, A is the common angle in ΔADE and ΔABC.
    Therefore,  
    AD
    AB
      =  
    AE
    AC
        [By BPT theorem.]
    Then,  
    AD
    AB
      =  
    AE
    AC
      =  
    ED
    BC
      ------(1)
    So, ΔABC ∼ ΔADE   [By SAS criteria.]
  4. As,  
    AD
    AB
      =  
    1
    4
      ------(2)
    and  
    AE
    AC
      =  
    1
    4
      ------(3)
  5. On comparing (1), (2) and (3), we get:  
    ED
    BC
      =  
    1
    4
     
  6. Hence, BC = 4ED.

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