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CBSE Assignments class 09 Mathematics

  Mathematics Assignments & Worksheets  For  Class IX Chapter-wise mathematics assignment for class 09. Important and useful extra questions strictly according to the CBSE syllabus and pattern with answer key CBSE Mathematics is a very good platform for the students and is contain the assignments for the students from 9 th  to 12 th  standard.  Here students can find very useful content which is very helpful to handle final examinations effectively.  For better understanding of the topic students should revise NCERT book with all examples and then start solving the chapter-wise assignments.  These assignments cover all the topics and are strictly according to the CBSE syllabus.  With the help of these assignments students can easily achieve the examination level and  can reach at the maximum height. Class 09 Mathematics    Assignment Case Study Based Questions Class IX 

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
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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  BC, intersect side AB and AC at point D and E

To Prove :-  
Construction :- 
Draw  EM AB and DN 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
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