PROJECT ON :PYTHAGORAS THEOREMPRESENTED BY GROUP 5

CREATED BY SOURAV NETI

SUBJECT TEACHER – D.K. BHUI Sir

GROUP MEMBERS :

SWATI SAGARIKA ROUT ( L )MITALI GOUDA SOURAV NETI ( A.L. )SATYA PRAKASH RAY RAKESH KUMAR BEHERA ABHIJEET LENKA

Pythagoras (~580-500 B.C.)

He was a Greek philosopher responsible for important developments in mathematics,

astronomy and the theory of music.

Pythagoras of Samos was an lonian Greek philosopher, mathematician , and has been credited as the founder of the movement called Pythagoreanism . Most of the information about Pythagoras was written down centuries after he lived , so very little reliable information is known about him . He was born on the island of Samos and travelled visiting Egypt and Greece and maybe India and in 520 BC returned to Samos . Around 530 BC he moved to Croton in Magna Graecia , and there established some kind of school or guild .

Pythagoras made influential contributions to philosophy and religion in the late 6th century BC . He is also reversed as a great mathematician and scientist and is best known for Pythagoras theorem which bears his name . However , because legend and obfuscation cloud his work even more than that of the other pre-Socratic philosophers , one can give only a tentative account of his teachings , and some have questioned whether he contributed much to mathematics or natural philosophy . Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors .

Some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important . It was said that he was the first man to call himself as philosopher , or lover of wisdom , and Pythagorean ideas exercised a marked influence on Plato , and through him , all of western philosophy .

SCULPTOR OF PYTHAGORAS

STATEMENTIn a right triangle the square of hypotenuse is equal to the sum of the square of other two sides .

MATERIALS REQUIRED1. Colour paper2. Pair of scissors3.Glue4. Geometry Box

PROCEDURE1) LET US DRAW A TRIANGLE ABC OF SIDES OF 3CM , 4CM AND 5CM RESPECTIVELY .2)TAKE OR DRAW ANOTHER TRIANGLE OF SAME MEASUREMENT AND DRAW OR CUT SQUARES OF SIDES 3CM , 4CM AND 5CM AND PASTE THEM ON SIDES OF TRIANGLE .3)TAKE ANOTHER TWO SQUARES OF SIDES 3CM , 4CM AND 5CM AND PASTE FOUR TRIANGLES ACCORDING .4)TAKE ANOTHER SQUARE OF SIDE 5 CM AND PASTE YOUR PIECES OF THE TRIANGLE , SO THAT THIS FORM A SQUARE OF SIDE (A+B) .

1. cut a triangle with base 4 cm and height 3 cm

0 1 2 3 4 5

4 cm

01

23

45

3 cm

2. measure the length of the hypotenuse .

0

1

2

3

4

5

Here we have the first figure :

5 cmA

B C

a

b

c FIG 1

a

a

a

a

b

b

b

b

b

a cc

cc

c

FIG 2

Consider a square PQRS with sides a + b

a

a

a

a

b

b

b

b cc

cc

4 congruNow, the square is cut into

- 4 congruent right-angled triangles and

- n- 1 smaller square with sides c

t right-angled triangles .

- 1 smaller square with sides c .

P Q

R S

FIG 3

a + b

a + b

A B

CD

FIG 4

a + b

a + b

A B

CD

Area of square ABCD

= (a + b) 2

b

b

a b

b

a

a

a

cc

cc

P Q

RS

Area of square PQRS

= 4 + c 2

ab

2

a 2 + 2ab + b 2 = 2ab + c 2

a2 + b2 = c2

FIG 5

Theorem states that:"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."

The Pythagorean Theorem asserts that for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: a2 + b2 = c2

The figure above at the right is a visual display of the theorem's conclusion. The figure at the left contains a proof of the theorem, because the area of the big, outer, green square is equal to the sum of the areas of the four red triangles and the little, inner white square: c2 = 4(ab/2) + (a - b)2 = 2ab + (a2 - 2ab + b2) = a2 +

b2

ANIMATED PROOF OF PYTHAGORAS THEOREM

WE HAVE PROVED PYTHAGORAS THEOREM BUT THERE IS ONE MORE WAY TO PROVE AND THAT IS SHOWN FROM THE NEXT SLIDE .

STATEMENTIn a right triangle the square of hypotenuse is equal to the sum of the square of other two sides .

MATERIALS REQUIRED1. Colour paper2. Pair of scissors3.Glue4. Geometry Box

PROCEDURE1) Let us draw a triangle ABC of sides of 3cm , 4cm and 5cm respectively .

2)Take or draw another triangle of same measurement and draw or cut squares of sides 3cm , 4cm and 5cm and paste them on sides of triangle .

3)Take another two squares of sides 3cm , 4cm and 5cm and paste four triangles according .4)Take another square of side 5 cm and paste your pieces of the triangle , so that this form a square of side (a+b) .

a

a

a

a

b

b

b

b

a

a

a a

b

b

bbc

c

ccFIG 6

5) The area of fig.6 and fig.3 are (a+b)^2 respectively . Hence the areas of 5 and 3 are equal 6) Now , we remove four triangles from fig. 5 and 3 .So the remaining figures must have the same area . Hence , areas of fig.6 and fig.7 are equal . b

b

b

ba

a

a

a

FIG 7

By removing four triangles from fig.6 we get this figure .

Area of square = (side)^2 = a^2Similarly,Area of second square = (side)^2 =>a^2=a^2+b^2

By removing four triangles from fig.3 we get the following figure . a

a

aaFIG 8

Thus , by comparing fig. 7 and 8 we get that [ a^2 + b ^2 ] .

HENCE PROVED

a

ANIMATION OF PYTHAGORAS THEOREM

THANK

YOU