Basic Proportionality Theorem (BPT), Thales Theorem

Basic Proportionality Theorem Class 10th 

(OR)  B.P.T.   or Thales Theorem
 Converse of basic proportionality theorem, thales theorem 10th standard, theorem 6.2 class 10

Statement:-  

If a line is drawn parallel to one side of the triangle to intersect the other two sides in two distinct points, the other two sides are divided in the same ratio.

Given:- 

A  Δ ABC in which line l ॥ BC, intersect side AB and AC at point D and E

To Prove :-  

Construction :- 

Draw  EM 丄 AB and DN 丄 AC. Also join BE and CD

Proof :- Area of triangle =  ✕ Base ✕ Height

Area of △ ADE =  ✕ AD ✕ EM ..............(1)

Area of △ BDE =  ✕ BD ✕ EM ..............(2)

Divide equation (1) by equation (2) we get

Similarly

ΔBED and ΔCDE are two triangles on the same base and lie between the same parallel DE and BC

 ∴ Ar(ΔBED) = Ar(ΔCDE)   .................. (5)

From equation  (3), (4), (5) we get

Hence prove the Basic Proportionality Theorem.

Note:- For the examination point of view students should study the basic proportionality theorem only its converse is only a motivational theorem. 

Converse of Basic Proportionality Theorem

Statement:- 

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.  

Given:-  In Triangle ABC,   

To Prove :-  Line DE॥ BC

Construction:-

If DE is not parallel to BC, then let us take another line DE'॥ BC

Proof:-

In ΔABC,  DE'॥ BC Therefore by B.P.T

Therefore  

Adding  1 on both side

This is possible only if E and E' coincide with each other

⇒ E and E' represent the same point on the side of the triangle.

    Hence DE is parallel to the side BC

Converse of BPT is proved

Important result based on BPT

If a line intersects side AB and AC of a ΔABC at D and E respectively and is parallel to BC, then prove that 

          

Solution :-  

It is given that DE॥BC, therefore by BPT

  

     .........  (∵  By Invertendo

Adding 1 on both side we get

     ..........  ( ∵  By Invertendo

THANKS FOR YOUR VISIT

PLEASE COMMENT BELOW