Like us!

Have a Look!

WhatsApp Message

Share Link

Pythagoras Theorem

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 IMO Subjects

Concept Explanation

Pythagoras Theorem

Pythagoras Theorem:

Pythagoras, the famous Greek philosopher, lived about 572 B.C. to 501 B.C. He proved the relation between the lengths of the sides of a triangle, Although, this theorem was known to the Babylonians 1000 years ealier, but, Pythagoras is believed to have the first to discovered a proof of this theorem. However, long ago ( 800 B.C.), the Indian mathe,atician Baudhaya had stated and proved this property of a right angled triangle.

Let us have a look at the fig. $large Delta ABC$ is a right triangle, right angled at C, so that AB is the hypotenuse and AC and BC are the sides of the right triangle,

Then, $large (AB)^{2}=(BC)^{2}+(CA)^{2}$

i.e., $large (Hypotenuse)^{2}=(Base)^{2}+(perpendicular)^{2}$

Given: A triangle ABC right angled at B

To Prove: $large AB^2+BC^2=AC^2$

Construction: From B draw BD $large perp$ AC

Proof: Since BD $large perp$ AC

$large Delta ADBsimDelta ABC$ [By the above theorem]

$large Rightarrow frac{AD}{AB}=frac{AB}{AC}$

$large Rightarrow AB^2= AD;X;AC$ ............(1)

Now $large Delta BDCsimDelta ABC$ [By the above theorem]

$large Rightarrow frac{BC}{AC}=frac{DC}{BC}$

$large Rightarrow BC^2= AC;X;DC$ ............(2)

Adding Eq (1) and (2)

$large AB^2 +BC^2= AC;X;AD + AC;X;DC$

$large AB^2 + BC^2= AC;(AD + ;DC)$

$large AB^2 +BC^2= AC;X;AC$

$large AB^2 +BC^2= AC^2$

Hence Proved

Example 1: The lengths of two sides of a right triangle are 5 cm and 12 cm. Find the length of the hypotenuse.

Solution: Suppose BC = 5cm and AC = 12 cm.

$large therefore$ By Pythagoras theorem,

$large (AB)^{2}=(BC)^{2}+(AC)^{2}$

$large (AB)^{2}=(5)^{2}+(12)^{2}$

$large =25+144$

$large =169$

$large (AB)^{2}=169$

$large ABtimes AB=13times 13$

$large Rightarrow$ $large or;;AB=13 cm$

$large therefore$ Length of the hypotenuse = 13 cm.

Converse of Pythagoras Theorem:

THeorem: In a triangle if the square of one side is equal to the sum of squares of the other two sides, then the angle opposite to the first side is a right angle.

Given: A triangle ABC such that $large AB^2+ BC^2=AC^2$

To Prove : $large angle B =90^0$

Construction : Construct a triangle DEF such that DE= AB, EF= BC and $large angle E =90^0$

Proof: Since $large Delta DEF$ is a right angled triangle with right angle at E

$large therefore DF^2=DE^2+EF^2$

Since DE = AB and EF = BC replacing them in the above equation

$large therefore DF^2=AB^2+BC^2$

But $large AB^2+ BC^2=AC^2$ [Given]

$large therefore DF^2=AC^2$

DF = AC

Now in $large Delta$ ABC and $large Delta$ DEF

AB = DE [By Construction]

BC= EF [By Construction ]

DF = AC [Proved Above]

$large therefore Delta ABCcongDelta DEF$ [By SSS Congruence Criterion]

$large Rightarrow angle B = angle E=90^0$

Hence $large Delta$ ABC is right angled at B

Illustration: ABC is an isosceles triangle with Ac = BC. If $large AB^2=2AC^2$ . Prove that ABC is aright triangle.

In $large Delta$ ABC, we are given that

AC = BC and $large AB^2=2AC^2$

Now $large AC^2= BC^2$

Adding $large AC^2$ To both sides we get

$large AC^2+AC^2=BC^2+AC^2$

$large BC^2+AC^2=2AC^2$

But $large AB^2=2AC^2$

$large therefore BC^2+AC^2=AB^2$

From Converse of Pythagoras Theorem we can say that

Triangle ABC is a right angled at C

Sample Questions

(More Questions for each concept available in Login)

Question : 1

In a right angle triangle ABC, right angled at B , AB = 6cm , BC = 8cm , then AC = _________________ .
A. 10	B. 12	C. 21	D. 14
Right Option : A
View Explanation

Question : 2

A man goes 150 m due east and then 200 m due north. How far is he from the starting point?
A. 150 m	B. 250 m	C. 130 m	D. 130 m
Right Option : B
View Explanation

Question : 3

A 13-m long ladder reaches a window of building 12 m above the ground. Determine the distance of the foot of the ladder from the building.
A. 7	B. 15	C. 5	D. 9
Right Option : C
View Explanation

Video Link - Have a look !!!

Language - English

Chapters

Real Numbers I

Real Numbers II

Pair of Linear Equations I

Pair of Linear Equations II

Polynomial I

Polynomials II

Quadratic Equations I

Quadratic Equations II

Introduction to Trigonometry I

Introduction to Trigonometry II

Arithmetic Progression I

Arithmetic Progression II

Triangles I

Triangles II

Some Applications of Trigonometry

Coordinate Geometry I

Coordinate Geometry II

Introduction to Trigonometry

Direct and Inverse Variation

Surface Area and Volume I

Surface Area and Volume II

Areas Related to Circles

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]

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 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

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

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

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

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 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

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

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

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

View All

Attention !!!!!

Pythagoras Theorem

Pythagoras Theorem

Students / Parents Reviews [10]

Aman Kumar Shrivastava

Shivam Rana

Ayan Ghosh

Barkha Arora

Archit Segal

Om Umang

Eshan Arora

Cheshta

Shreya Shrivastava

Prisha Gupta