derive the converse of BPT without using BPT in the proof.

Asked by sahil.95rana | 9th Jul, 2010, 05:55: AM

Expert Answer:

Dear Student,

The converse of BPT says:
If a line divides any two sides of a triangle in the same ratio, Then the line must be parallel to the third side.
Given: Triangle ABC in which a line ‘l’ intersects AB in D and AC in E. Such that
  AD         AE
 -----  =  -----
 DB         EC

To prove: DE parallel to BC

Proof: Let line ‘l’ is not parallel to BC. Then, there must be another line through D,  which is parallel to BC. Let DF parallel to BC.
Using Basic proportionality theorem, we have

    AD        AF
  ------  =  ------
   DB         FC

             AD          AF
 But     ------  =  ------       (given)
             DB          EC     

                         AF           AE
 Therefore        ------- =  --------     
                         FC           EC

Adding ‘1’ to both the sides, we get

        AF                AE
      ------ + 1  =  ------  + 1 
        FC                 EC

    AF + FC         AE + EC
  ----------- =  -----------
        FC                  EC

         AC           AC 
      ------  =   -------   Or   FC = EC
         FC           EC

But this is true only if F and E coincide, that is, DF coincides with DE.

Hence DE || BC (Basic proportionality theorem converse is proved)
 
Regards Topperlearning.

Answered by  | 15th Jul, 2010, 02:00: PM

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