CBSE Class 10 Answered
derive the converse of BPT without using BPT in the proof.
Asked by sahil.95rana | 09 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 | 15 Jul, 2010, 02:00: PM
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