converse of BPT theorem?
Answers
Answer:
Converse of Basic Proportionality theorem:
If a line divides the two sides of a triangle in equal proportion then it is parallel to the third side.
Given : In Triangle ABC, Line i intersects sides AB and AC of triangle ABC on points D and E respectively. such that,
AD/DB = AE/EC
To prove: DE || BC
Proof:
If possible suppose DE is not parallel to BC.
Then there must be a line parallel to BC through D.
So, Let DF || BC
therefore,By Basic proportionality theorem,
=> AD/DB= AF/FC
But, AD/DB = AE/EC (given)
thus, AF/FC = AE/ EC
=> AF/FC+1= AE/EC +1 (Adding 1 on
both the sides) =>AF+FC/FC= AE+EC/ EC
=> AC/FC = AC/ EC
=> 1/FC = 1/EC
=> EC = FC
This is possible only if F & E coincides. i.e., DF is the line i itself .
Hence, i || BC
=> DE || BC
----- Proved-----
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Step-by-step explanation:
