Like us!

Have a Look!

WhatsApp Message

Share Link

Area of Similar Triangles

Triangles II - Concepts

Calculating Perimeter of Triangle

Area of Similar Triangles

Similarity Related To Right Triangle

Pythagoras Theorem

Results Deduced From Pythagoras Theorem

Class - 10th CBSE Subjects

Concept Explanation

Area of Similar Triangles

Theorem: Ratio of the areas of two similar triangles is equal to the ratio of the square of their corresponding sides.

Given: $large Delta ABC sim Delta PQR$

To Prove:

$large frac{area(Delta ABC)}{area(Delta PQR)}=frac{AB^2}{PQ^2}=frac{BC^2}{QR^2}=frac{AC^2}{PR^2}$

Construction: Draw AD $large perp$ BC and PS $large perp$ QR

Proof: In $large Delta$ ABD and $large Delta$ PQS

$large angle B = angle Q$ $large [;Delta ABC sim Delta PQR;]$

$large angle D = angle S$ $large [ ;Each ;90^0;]$

$large Rightarrow Delta ABD sim Delta PQS$ [ By AA Similarity Criterion ]

Ratio of corresponding sides of similar triangle

$large frac{AB}{PQ}=frac{AD}{PS}$ ..........................(1)

But we are given that $large Delta ABC sim Delta PQR$

$large frac{AB}{PQ}=frac{AC}{PR}=frac{BC}{QR}$ ..........................(2)

From Eq (1) and (2)

$large frac{BC}{QR}=frac{AD}{PS}$

Now

$large frac{area(Delta ABC)}{area(Delta PQR)}=frac{frac{1}{2}X BC;X;AD}{frac{1}{2}X QR;X;PS}$ $large [;area ;of ;Delta = frac{1}{2};X;base;X;height;]$

$large frac{area(Delta ABC)}{area(Delta PQR)}=frac{BC;X;AD}{QR;X;PS}=frac{BC^2}{QR^2}$

As $large frac{AB}{PQ}=frac{AC}{PR}=frac{BC}{QR}$

Hence $large frac{area(Delta ABC)}{area(Delta PQR)}=frac{AB^2}{PQ^2}=frac{BC^2}{QR^2}=frac{AC^2}{PR^2}$

Illustration: If the area of two similar triangles are equal, prove that they are congruent.

Let $large Delta ABC sim Delta PQR$

$large Rightarrow large frac{AB}{PQ}=frac{AC}{PR}=frac{BC}{QR}$

Given

$large frac{area(Delta ABC)}{area(Delta PQR)}=1$

We know that

$large frac{area(Delta ABC)}{area(Delta PQR)}=frac{AB^2}{PQ^2}=frac{BC^2}{QR^2}=frac{AC^2}{PR^2}$

$large Rightarrow frac{AB^2}{PQ^2}=frac{BC^2}{QR^2}=frac{AC^2}{PR^2}=1$

$large Rightarrow$ AB = PQ , BC= QR and AC = PR

Hence

$large Delta ABC cong Delta PQR$ [By SSS Criterion of Congruence]

Sample Questions

(More Questions for each concept available in Login)

Question : 1

In the following figure, $bigtriangleup ABCsim bigtriangleup bigtriangleup DEF$ . If AB = 2DE and area of $bigtriangleup ABC=56;cm^{2}$ , find the area of $bigtriangleup DEF.$
A. 16	B. 18	C. 14	D. 20
Right Option : C
View Explanation

Question : 2

$large Delta AMBsim Delta CMD$ . Also 2ar ( $large Delta AMB$ ) = ar ( $large Delta CMD$ ) the length of MD is ___________________
A. $large sqrt2 ;MB$	B. $large sqrt2 ;MD$	C. $large frac{sqrt2}{MB}$	D. $large frac{sqrt2}{MD}$
Right Option : A
View Explanation

Question : 3

Two triangles ABC and DEF are similar to each other in which AB = 10 cm and DE = 8 cm. Find the ratio of the areas of $bigtriangleup ABC;and;bigtriangleup DEF$ .
A. 25:16	B. 16:25	C. 5:4	D. 4:5
Right Option : A
View Explanation

Video Link - Have a look !!!

Language - English

Chapters

Polynomial I

Polynomials II

Pair of Linear Equations I

Pair of Linear Equations II

Real Numbers I

Real Numbers II

Coordinate Geometry I

Coordinate Geometry II

Quadratic Equations I

Quadratic Equations II

Arithmetic Progression I

Arithmetic Progression II

Introduction to Trigonometry I

Introduction to Trigonometry II

Surface Area and Volume I

Surface Area and Volume II

Areas Related to Circles

Some Applications of Trigonometry

Content / Category

IMO-Mathematics Olympiad

Central Government Service

ISO - Science Olympiads

State Government Services

Banking And Insurance

Railways & Metro Services

MBA Entrance

Law Entrance

English Proficiency

Reasoning Proficiency

Computer Proficiency

Navodaya Entrance 6th & 9th

Sainik School Entrance

Class / Course

Students / Parents Reviews [10]

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

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

Being a parent, I saw my daughter improvement in her studies by seeing a good result in all day to day compititive exam TMO, NSO, IEO etc and as well as studies. I have got a fruitful result from m...

Prisha Gupta

8th

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

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

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

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

I have spent a wonderful time in Abhyas academy. It has made my reasoning more apt, English more stronger and Maths an interesting subject for me. It has given me a habbit of self studying

Yatharthi Sharma

10th

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

My experience was very good with Abhyas academy. I am studying here from 6th class and I am satisfied by its results in my life. I improved a lot here ahead of school syllabus.

Ayan Ghosh

8th

View All

Attention !!!!!

Area of Similar Triangles

Area of Similar Triangles

Students / Parents Reviews [10]

Manish Kumar

Shivam Rana

Prisha Gupta

Shreya Shrivastava

Cheshta

Eshan Arora

Om Umang

Yatharthi Sharma

Aman Kumar Shrivastava

Ayan Ghosh