Question

Fig. 1.25
2) Converse of basic proportionality
theorem
PS PT
In APOR, if
SỞ TR
then seg ST || seg OR
Fig. 1.26​

Answer 1

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  AD /DE  = AE​ /EC  then DE||BC.

Prove that : DE||BC.

Given in ΔABC, D and E are two points of AB and AC respectively, such that,

AD/DB =  AE/EC   ______ (1)

Let us assume that in ΔABC, the point F is an intersect on the side AC. So, we can apply the  

Thales theorem,

AD/DB  = AF /FC_______ (2)

Simplify (1) and (2)

​AE/EC  =  AF/FC

adding 1 on both sides

AE/EC +1=  AF/FC +1

⇒ AE+EC /EC  = AF+FC /FC

⇒ AC /EC  = AF /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.

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

Step-by-step explanation:

HOPE IT HELPS YOU ^_^