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Answers
Answer:
Converse of Basic proportionality Theorem
Statement : If a line divide any two sides of a triangle (Δ) in the same ration, then the line must be parallel (||) to third side.
If
DE
AD
=
EC
AE
then DE||BC.
Prove that : DE||BC.
Given in ΔABC, D and E are two points of AB and AC respectively, such that,
DB
AD
=
EC
AE
______ (1)
Let us assume that in ΔABC, the point F is an intersect on the side AC. So, we can apply the
Thales theorem,
DB
AD
=
FC
AF
_______ (2)
Simplify (1) and (2)
EC
AE
=
FC
AF
adding 1 on both sides
EC
AE
+1=
FC
AF
+1
⇒
EC
AE+EC
=
FC
AF+FC
⇒
EC
AC
=
FC
AF
⇒AC=FC
.
Converse of Basic Propostionality Theorm
statement :
If a line divided any two sides
of a triangle(⚠️) in the same ratio,
then the line must be parallel (||)
third side.
If AD/DE.AE/EC then DE || BC.
Prove that : DE || BC.
Given in ⚠️ABC.
D and E are two points of AB
and AC respectively, such that,
AD/DM.AE/EC-----(1)
Let us assume that in ⚠️ABC,
the point F is an intersect on
the side Ac.
So we Can apply the
Thales theorem,
AD/DB.AF/FC-----(2)
Simplify eq(1) and eq(2)
AE/CE.AF/FC
adding '1' on both sides
=> AE/CE+1=AF/FC+1
=>AE+CE/CE=AF+FC/FC
=>AC/CE=AF/FC
=>AC=FC
From the above we can say
that the points E and F are
coincide on Ac.i.e; DF Coincide
with DE. Since DF is parallel to
BC, DE is also parallel to BC .
Hence, the Converse of Basic proportionality Theorm is proved.