state and prove BPT THEOREM
Answers
Answer:
prove of BPT :-
Step-by-step explanation:
Converse of Basic proportionality Theorem
Statement : If a line divide any two sides of a triangle (\Delta )(Δ) in the same ration, then the line must be parallel (||) to third side.
If \frac{AD}{DE}=\frac{AE}{EC}
DE
AD
=
EC
AE
then DE||BC.
Prove that : DE||BC.
Given in \Delta ABCΔABC, D and E are two points of AB and AC respectively, such that,
\frac{AD}{DB}=\frac{AE}{EC}
DB
AD
=
EC
AE
______ (1)
Let us assume that in \Delta ABCΔABC, the point F is an intersect on the side AC. So, we can apply the
Thales theorem,
\frac{AD}{DB}=\frac{AF}{FC}
DB
AD
=
FC
AF
_______ (2)
Simplify (1) and (2)
\frac{AE}{EC}=\frac{AF}{FC}
EC
AE
=
FC
AF
adding 1 on both sides
\frac{AE}{EC}+1= \frac{AF}{FC}+1
EC
AE
+1=
FC
AF
+1
\Rightarrow \frac{AE+EC}{EC}= \frac{AF+FC}{FC}⇒
EC
AE+EC
=
FC
AF+FC
\Rightarrow \frac{AC}{EC}=\frac{AF}{FC}⇒
EC
AC
=
FC
AF
\Rightarrow AC=FC⇒AC=FC
From the above we can sat that the points E and F are coincide on AC, i.e., DF coincides with DE. Since DF is parallel to BC, DE is also parallel to BC.
\therefore∴ Hence, the converse of Basic proportionality Theorem is proved.
Answer:
Refer to this attachment...
