Like us!

Have a Look!

WhatsApp Message

Share Link

Basic Proportionality Theorem and its Converse

Triangles I - Concepts

Similar Figures

Similarity of Triangles

Basic Proportionality Theorem and its Converse

Similarity of Triangles by AAA Criteria

Similarity of Triangles by AA Criteria

Similarity of Triangles by SSS Criteria

Similarity of Triangles by SAS Criteria

Class - RRB Technician Grade I Signal Subjects

Concept Explanation

Basic Proportionality Theorem and its Converse

Basic Proportionality Theorem ( Thales Theorem ) and its converse

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

Given: A triangle ABC and DE || BC

$large To Prove: frac{AD}{DE}=frac{AE}{EC}$

Construction: Join BE and CD and draw DM $large perp$ AC and EN $large perp$ AB

Proof:

$large area;of;(Delta ADE)= frac{1}{2} ;X ;AD ;X ;EN$ .......(1)

$large area;of;(Delta BDE)= frac{1}{2} ;X ;DB ;X ;EN$ ......(2)

$large Also;area;of;(Delta ADE)= frac{1}{2} ;X ;AE ;X ;DM$ .....(3)

$large area;of;(Delta DEC)= frac{1}{2} ;X ;EC ;X ;DM$ .....(4)

Using Eq 1, 2, 3 and 4

$large frac{ar(ADE)}{ar(BDE)}= frac{frac{1}{2}X AD X EN}{frac{1}{2}X DB X EN}=frac{AD}{DB}$ ........(5)

$large frac{ar(ADE)}{ar(DEC)}= frac{frac{1}{2}X AE X DM}{frac{1}{2}X EC X DM}=frac{AE}{EC}$ ........(6)

Now $large Delta$ BDE and $large Delta$ DEC are on the same base DE and between the same parallel BC || DE

So ar(BDE) = ar( DEC)

Using the relation in Eq 5 and 6 we get

$large frac{AD}{DB}=frac{AE}{EC}$

Hence Proved

Converse of the basic proportionality theorem is also true,

Theorem If a line divides any twosides of a triangle in the same ratio, the line is parallel to the third side.

Given : a triangle ABC and That is, if in fig. $large frac{AD}{DB}=frac{AE}{EC}$ ,

To Prove : $large DEparallel BC$

Construction: Draw DF || BC

Proof: If DF || BC

then $large frac{AD}{DB}= frac{AF}{FC}$ ...... (1)

But we are given $large frac{AD}{DB}=frac{AE}{EC}$ ...............(2)

Then from 1 and 2 we get

$large frac{AF}{FC}= frac{AE}{EC}$

This is possible only when F and E coincide.

This implies DE || BC

Example :.

$large dpi{120} large If ; DEparallel BC; and;$ $large AD=4x-3, AE=8x-7,BD=3x-1$ $large and;CE=5x-3,; Find ;x ;when; x>0$

Solution

$large As;DEparallel BC$ ,

$large frac{AD}{DB}=frac{AE}{EC}$ [ Basic Proportionality Theorem]

$large Rightarrow$ $large frac{4x-1}{3x-1}=frac{8x-7}{5x-3}$

$large Rightarrow$ $large (4x-3)(5x-3)=(3x-1)(8x-7)$

$large Rightarrow$ $large 20x^{2}-15x-12x+9=24x^{2}-8x-21x+7$

$large Rightarrow$ $large 4x^{2}-2x-2=0;;or;;2x^{2}-x-1=0$

$large Rightarrow$ $large 2x^{2}-2x+x-1=0$

$large Rightarrow$ $large 2x(x-1)+(x-1)=0$

$large Rightarrow$ $large (2x+1)(x-1)=0;;;Rightarrow ;;x=-frac{1}{2},1$

As $large x> 0,x=1$

Sample Questions

(More Questions for each concept available in Login)

Question : 1

Find AB, when in $dpi{110} Delta ABC ,DE; \|\|; BC$ so that AD = 2.4 CM, AE = 3.2 cm and EC = 4.8 cm.
A. 8 cm	B. 5 cm	C. 6 cm	D. 2 cm
Right Option : C
View Explanation

Question : 2

In a $bigtriangleup ABC$ , DE \|\| BC and $frac{AD}{DB}=frac{3}{5}$ . If AC = 5.6 cm, find AE.
A. 0.7	B. 2.1	C. 3.1	D. 3.7
Right Option : B
View Explanation

Question : 3

Find AB, when in $bigtriangleup ABC,DE\|\|BC$ so that AD = 2.4 cm, AE = 3.2 cm and EC = 4.8 cm.
A. 6	B. 3.6	C. 2.4	D. 1.2
Right Option : A
View Explanation

Video Link - Have a look !!!

Language - English

Students / Parents Reviews [10]

Abhyas is a complete education Institute. Here extreme care is taken by teacher with the help of regular exam. Extra classes also conducted by the institute, if the student is weak.

Om Umang

10th

It has a great methodology. Students here can get analysis to their test quickly.We can learn easily through PPTs and the testing methods are good. We know that where we have to practice

Barkha Arora

10th

Abhyas Methodology is very good. It is based on according to student and each child manages accordingly to its properly. Methodology has improved the abilities of students to shine them in future.

Manish Kumar

10th

It was good as the experience because as we had come here we had been improved in a such envirnment created here.Extra is taught which is beneficial for future.

Eshan Arora

8th

My experience with Abhyas academy is very good. I did not think that my every subject coming here will be so strong. The main thing is that the online tests had made me learn here more things.

Hiya Gupta

8th

One of the best institutes to develope a child interest in studies.Provides SST and English knowledge also unlike other institutes. Teachers are co operative and friendly online tests andPPT develo...

Aman Kumar Shrivastava

10th

It was a good experience with Abhyas Academy. I even faced problems in starting but slowly and steadily overcomed. Especially reasoning classes helped me a lot.

Cheshta

10th

My experience with Abhyas is very good. I have learnt many things here like vedic maths and reasoning also. Teachers here first take our doubts and then there are assignments to verify our weak poi...

Shivam Rana

7th

A marvelous experience with Abhyas. I am glad to share that my ward has achieved more than enough at the Ambala ABHYAS centre. Years have passed on and more and more he has gained. May the centre f...

Archit Segal

7th

About Abhyas metholodology the teachers are very nice and hardworking toward students.The Centre Head Mrs Anu Sethi is also a brilliant teacher.Abhyas has taught me how to overcome problems and has...

Shreya Shrivastava

8th

View All

Attention !!!!!

Basic Proportionality Theorem and its Converse

Basic Proportionality Theorem and its Converse

Students / Parents Reviews [10]

Om Umang

Barkha Arora

Manish Kumar

Eshan Arora

Hiya Gupta

Aman Kumar Shrivastava

Cheshta

Shivam Rana

Archit Segal

Shreya Shrivastava