Question

prove that is a line is drawn paralled one
0f a triangle to intersect the other
sides in district points then other
two sides are divided in the
same ratio By using this theroem
prove that if triangle ABC De parallel to Be then
AD/ BD=AE/AC​

Answer 1

Answer: Thales Theorem.

Step-by-step explanation: Given: DE is Parallel to BC. DE Intersects AB And AC. To prove:AB/AD=AE/EC

Construction:Draw EN Perpendicular to AB and DM Perpendicular to AC, Join BE and CD.

Proof: Ar(ADE)=1/2 AD×EN. (Taking base AD)

Ar(BDE)=1/2 DB×EN.

This implies: Ar(ADE)/Ar(BDE)=1/2×AD×EN/1/2DB×EN=AD/DB (Eq. 1)

Ar(ADE=1/2 AE*DM(Taking AE Base)

Ar(DEC)=1/2×EC×DM

This implies Ar(ADE)/Ar(DEC)=1/2 AE×DM=1/2 EC×DM=AE/EC. Eq.2

Ar(BDE)=Ar(DEC(Triangles on the same base and the same parallels are equal in area). Eq.3

From Equations 1, 2,3 we get:AD/DB=AE/EC.

Hence the Theorem is Proved.*

* This is a very important Theorem in the chapter Triangles class 10.