FOR CLASS 8th

1. BASIC
PROPORTIONALITY
THEOREM
Class-X

2. STATEMENT
If a line drawn parallel to one
side of a triangle to intersect the
other two sides in distinct point,
the other two sides are divided in
the same ratio.

3. PROOF
 We are given a triangle ABC in which a line parallel to
side BC intersects other two sides AB and AC at D and E
respectively.
 We need to prove that AD/DB =AE/EC
 Let us join BE and CD and then draw DM ┴ AC and EN ┴
AB
 Now area of ∆ADE (1/2 x base x height) = 1/2 x AD x EN
 So, ar(ADE) = ½ AD x EN
 Similarly , ar(BDE) = ½ BD x EN
 ar(ADE) = ½ AE x DM
 ar(DEC) = ½ EC x DM

4. Therefore ,ar(ADE)/ar(BDE) = ½ AD x EN / ½ BD x EN
= AD/BD ………………………(1)
and ,ar(ADE)/ar(DEC) = ½ AE x DM / ½ EC x DM
= AE/EC …………………………….(2)
Note that ∆BDE and ∆DEC are on the same base DE and between the
same parallels BC and DE
So, ar(ADE) = ar(DEC) …………………………………..(3)
Therefore from (1),(2)and (3), we have ,
AD/BD = AE/EC
Hence proved.