Question

state converse of thales theorem

Answer 1

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,

DE / AD = EC / AA ______ (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/AA+1

EC/AE+EC= FC/AF+FC

EC/AC= FC/AF

⇒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.

∴ Hence, the converse of Basic proportionality Theorem is proved.