Pythagoras theorem

 

• In mathematics, the Pythagoreantheorem or Pythagoras' theorem is arelation in Euclidean geometry amongthe three sides of a right triangle(right-angled triangle ). In terms ofareas, it states:

• In any right triangle, the area of thesquare whose side is the hypotenuse(the side opposite the right angle) isequal to the sum of the areas of thesquares whose sides are the two legs(the two sides that meet at a rightangle).

 

  The theorem can be written as anequation relating the lengths ofthe sides a , b  and c , often calledthe Pythagorean equation :

where c  represents the length ofthe hypotenuse, and a  and b  represent the lengths of the othertwo sides.

 

Right-angledTriangle

 

The Pythagorean Theorem

a

b

c

a2 + b2 = c 2

Which side is

the

hypotenuse? 

a2 = c 2  – b2

The right angle points to the hypotenuse.

It’s the side labelled “c”. 

OR

 

6

8

c

Calculate side

c.

c 2 = 82 + 62 

c 2 = 64 +

36c 2 = 100

100c 

c  = 10

When calculating the hypotenuse,

we add the area of the squares of

the other two sides.

c 2 = a2 + b2 

 

 x 2 = 122  – 72 

 x 2 = 144  – 

49 x 2 = 95

95 x 

 x  = 9.7

Calculate the

length of side x .

When calculating a side, we

determine the difference of the area

of the square of the hypotenuse

and the area of the square of the

known side.

b2 = c 2  – a2 

a  

c  

b   x  

12

7

 

The length and width of a

rectangle are 12 cm and 15 cm.

Calculate the length of the

diagonal.

15 cm

12 cmdd 2 = 152 +

122 d 2 = 225 +

144d 2 =

369 369d  

d  = 19.2 cm 

c 2 = a2  + b2 

 

Tanya is making a party hat

using a cone made out of paper.

Determine the height of the

cone.b2 = c 2 – a2 

h2 = 144

h = 12 cm

h2 = 132 – 52 

h2 = 169 – 25

144h

h

5 cm

13 cm

 

 History

 

• The Pythagorean theorem is namedafter the Greek mathematicianPythagoras(569 B.C.?-500 B.C.?), whoby tradition is credited with itsdiscovery and proof, although it is oftenargued that knowledge of the theorempredates him. There is evidence thatBabylonian mathematicians understoodthe formula, although there is littlesurviving evidence that they fitted itinto a mathematical framework.

 

• The theorem has numerous proofs, possiblythe most of any mathematical theorem. Theseare very diverse, including both geometricproofs and algebraic proofs, with some datingback thousands of years. The theorem can begeneralized in various ways, including higher-dimensional spaces, to spaces that are notEuclidean, to objects that are not righttriangles, and indeed, to objects that are nottriangles at all, but n -dimensional solids. ThePythagorean theorem has attracted interestoutside mathematics as a symbol ofmathematical abstruseness, mystique, orintellectual power; popular references inliterature, plays, musicals, songs, stamps andcartoons abound.

 

 Biography

 

Biography• Pythagoras was born in 572BC on the island of Samos, Greece.

• In about 530BC Pythagoras left Samos in hatred for its ruler

Polycrates and settled in Cretona, Italy.

• He joined a religious group known as the Pythagoreans.

• He formed a philosophical and religious school where they

studied mathematics, science and music. This attracted many

followers.

• When involved with this group he discovered what is now known

as Pythagoras’ Theorem. 

•  Also during his time there he out the mathematics of octaves

and harmony.

• Because of the secrecy in the group there is nothing of

Pythagoras’ writings or books. 

• Pythagoras was murdered at the age of 77, in 495BC and the

religious school was separated.

 

Contemporary importance

Pythagoras’ theorem has been used over thousands of years for many differentaspects of human life.

Today his theorem is used mainly in building, architecture, carpenter, navigation,astronomy and many other fields of work that involve mathematicalcalculations.

Each of these fields uses his theorem to try and decide either the hypotenuse orthe two other sides in a right angled triangle.

-Builders use Pythagoras’ theorem to work out dimensions of different aspects oftheir constructions. This allows them to work out the exact requirements ofbuilding materials needed.

-Architects use his theorem to work out designs for the builders to use. Histheory may be used to work out exact lengths of roofs, also the framework ofa house just to name a few.

-Carpenter’s uses his theorem to interprete the size of sides of their timberstructures eg: corner furniture.

-Navigators and astronomers use his theorem to establish distances betweenplanets, towns, countries and stars.

 

RelevancePythagoras is often described as one of the pure mathematicians of his time

and an extremely important figure in the expansion of mathematics.

Pythagoras’ theorem is studied from Year 8 to Year 12 in NSW schools. Students today often wonder why geometry is so important. It allows people

to think more logically and as I have shown in ContemporaryImportance  his theorem is used in numerous jobs and work areas.

Students pursuing technical majors in college are expected to understandand extend this knowledge on geometry.

Geometry proofs are also an important way to increase disciplined .

 As you can see geometry is still just as important now as it was inPythagoras’ time.

 

Mathematical achievements

• Pythagoras has contributed various theories to geometry, algebra,number .etc.

• All of these theories were discovered during Pythagoras‟ time withthe Pythagoreans.

• Pythagoras‟ theory is : The square on the hypotenuse in any right-angled triangle is equal to the sum of the squares on the other twosides.

So for example:

 A

BCa

bc 4

3

c

Using the formula a²+b²=c² find side “c” on the triangleDEF D

EF

4² + 3² = c²

16 + 9 = c²

25 = c²

√25 = c² c = 5

 

• Pythagoras believed:1. All things are numbers. Mathematics

is the basis for everything, andgeometry is the highest form ofmathematical studies. The physicalworld can understood throughmathematics.

2. Certain symbols have a mysticalsignificance.

3. All members of the society shouldobserve strict loyalty and secrecy.

 

4. The world depends upon the interactionof opposites, such as male and female,lightness and darkness, warm and cold,dry and moist, light and heavy, fast andslow.

5. The soul resides in the brain, and isimmortal. It moves from one being toanother, sometimes from a human into ananimal, through a series of reincarnationscalled transmigration until it becomespure. Pythagoras believed that bothmathematics and music could purify.

 

• Some of the students of Pythagoraseventually wrote down the theories,teachings and discoveries of thegroup, but the Pythagoreans alwaysgave credit to Pythagoras as theMaster for:

1. The five regular solids(tetrahedron, cube, octahedron,icosahedrons, dodecahedron). It isbelieved that Pythagoras knew howto construct the first three but notlast two.

 

2. The sum of the angles of a triangle isequal to two right angles.

3. Pythagoras taught that Earth was asphere in the center of the Kosmos(Universe), that the planets, stars, andthe universe were spherical becausethe sphere was the most perfect solidfigure. He also taught that the pathsof the planets were circular.Pythagoras recognized that themorning star was the same as theevening star, Venus.

 

4.  Pythagoras studied odd and evennumbers, triangular numbers, andperfect numbers. Pythagoreanscontributed to our understanding ofangles, triangles, areas, proportion,polygons, and polyhedra.

5. Pythagoras also related music tomathematics. He had long played theseven string lyre, and learned howharmonious the vibrating stringssounded when the lengths of the stringswere proportional to whole numbers,such as 2:1, 3:2, 4:3. Pythagoreans alsorealized that this knowledge could beapplied to other musical instruments.

 

• The reports of Pythagoras' death are varied. Heis said to have been killed by an angry mob, tohave been caught up in a war between theAgrigentum and the Syracusans and killed by theSyracusans, or been burned out of his school inCrotona and then went to Metapontum where hestarved himself to death. At least two of thestories include a scene where Pythagoras refusesto trample a crop of bean plants in order toescape, and because of this, he is caught.

• The Pythagorean Theorem is a cornerstone ofmathematics, and continues to be so interestingto mathematicians that there are more than 400different proofs of the theorem, including anoriginal proof by President Garfield.

 

Statementof the

Theorem

 

• It is believed that the statement ofPythagorean's Theorem was discovered ona Babylonian tablet circa 1900-1600 B.C.The Pythagorean Theorem relates to thethree sides of a right triangle. It statesthat c2=a2+b2, C is the side that isopposite the right angle which is referredto as the hypoteneuse. a and b are thesides that are adjacent to the right angle.In essence, the theorem simply stated is:the sum of the areas of two small squaresequals the area of the large one. 

 

The theorem statesthat: 

•  For any right triangle, thesquare of the hypotenuseis equal to the sum of thesquares of the other twosides. 

 

Verificationof

Theorem

 

1. cut a triangle with

base 4 cm and

height 3 cm

0 1 2 3 4 5

4 cm

 0  

1  

2  

 3  

4  

 5  

 3   c m 2. measure the length

of the hypotenuse

Now take out a square paper and a

ruler.

 

Consider a square PQRS with sides a + ba

a

a

a

b

b

b

bc

c

cc

Now the square is cut into

- 4 congruent right-angled triangles and

- 1 smaller square with sides c

Proof of Pythagoras’ Theorem  

P Q

R S

 

a + b

a + b

 A B

CD

 Area of square

 ABCD= (a  + b ) 2  

b

b

a b

b

a

a

a

c

c

cc

P Q

RS

 Area of square

PQRS

= 4 + c  2ab 

2

  

   

a 2  + 2ab + b 2 = 2ab + c2

a 2  + b 

2  = c 

2

 

Theorem states that:

" The area of the square bu i l t upon the hypo tenuse of a r ight

t r iang le 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 the Pythagorean TheoremBelow is an animated proof of the Pythagorean Theorem. Starting with a right triangle and squares

on each side, the middle size square is cut into congruent quadrilaterals (the cuts through the

center and parallel to the sides of the biggest square). Then the quadrilaterals are hinged and

rotated and shifted to the big square. Finally the smallest square is translated to cover the

remaining middle part of the biggest square. A perfect fit! Thus the sum of the squares on the

smaller two sides equals the square on the biggest side.

 Afterward, the small square is translated back and the four

quadrilaterals are directly translated back to their original position.

The process is repeated forever.

 

PythagoreanTriplets

 

• The sides of a right triangle follows thePythagorean Theorem,

•   a2 + b2 = c2• where a and b are the lengths of the legs

of the right triangle while c is the lengthof the hypothenuse.

• A right triangle with sides of lengths 3, 4and 5 is a special right triangle in that allthe sides have whole number lengths. Thethree numbers 3, 4 and 5 forms aPythagorean triplet or Pythagorean triple.

 

• A Pythagorean triplet is a set ofthree whole numbers where the sumof the squares of the first two isequal to the square of the thirdnumber.  Below are examples ofPythagorean triplets:

3 4 5

5 12 13

7 24 25

9 40 41

11 60 61

 

• One equation satisfying a PythagoreanTriplet A, B, C is

•   Given A is odd, then

B = (A2 - 1)/2

C = (A2 + 1)/2

• Another equation derived by Plato was

(m2+1)2 = (m2-1)2 + (2m)2

where m is a natural number. The aboveequation is called Plato's Formula.

 

Why MemorizePythagorean Triples?

• Remember how much time it took to figureout 8 x 8 before you memorized it?(8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 64)

• Think of all the work involved to solve thisproblem:

a2 + b2 = c2 

32 + 42 = x2 

9 + 16 = x2

25 = x2

5 = x 

 

Wouldn‟t itbe nice

to just knowthis is 5?

 

Good PythagoreanTriples to Memorize:

a b c a2 + b

2 = c

2

3 4 5 9 + 16 = 25

5 12 13 25 + 144 = 169

8 15 17 64 + 225 = 289

7 24 25 49 + 576 = 625

 

And multiplesof each, like:3x2, 4x2, 5x2

6 8 10

 

ApplicationOf

PythagorasTheorem

 

Applications• The Pythagorean theorem has far-reaching ramifications

in other fields (such as the arts), as well as practical

applications.

• The theorem is invaluable when computing distances

between two points, such as in navigation and land

surveying.

• Another important application is in the design of ramps.

Ramp designs for handicap-accessible sites and for

skateboard parks are very much in demand.

 

Baseball ProblemA baseball “diamond” is really a square. 

You can use the Pythagorean theoremto find distances around a baseballdiamond.

 

Baseball ProblemThe distance between

consecutive bases is 90

feet. How far does a

catcher have to throw

the ball from home

plate to second base?

 

Baseball ProblemTo use the Pythagorean

theorem to solve forx, find the right angle.

Which side is thehypotenuse?

Which sides are thelegs?

Now use: a2 + b2 = c2 

 

Baseball ProblemSolution

• The hypotenuse is thedistance from home tosecond, or side x in thepicture.

• The legs are fromhome to first and fromfirst to second.

• Solution:

x2 =  902 + 902 = 16,200

x = 127.28 ft

 

Ladder ProblemA ladder leans againsta second-story windowof a house.If the ladder is 25meters long,and the base of theladder is 7 metersfrom the house,how high is thewindow? 

 

Ladder ProblemSolution

• First draw a diagramthat shows the sidesof the right triangle.

• Label the sides: – Ladder is 25 m 

– Distance from houseis 7 m

• Use a2 + b2 = c2 tosolve for the missingside. Distance from house: 7 meters

 

Ladder ProblemSolution

72 + b2 = 252

49 + b2 = 625

b2 = 576

b = 24 m

How did you do?A = 7 m

 

  Indirect Measurement

• Support Beam: Theskyscrapers shown on page535 are connected by askywalk with support beams.You can use the PythagoreanTheorem to find theapproximate length of eachsupport beam.

 

• Each support beam forms thehypotenuse of a right triangle. Theright triangles are congruent, so thesupport beams are the same length.Use the Pythagorean Theorem toshow the length of each supportbeam (x).

 

(hypotenuse)2 = (leg)2 + (leg)2

x2 = (23.26)2 + (47.57)2 

x2 = √ (23.26)2 + (47.57)2 

x ≈ 13

Pythagorean Theorem

Substitute values.

Multiply and find the

positive square root.

Use a calculator to

approximate.

Solution:

 

Lets learnwith Fun

 

Pythagoras Board Game

Rules:• To begin, roll 2 dice. The person with the highest sum goes

first.

• To move on the board, roll both dice. Substitute thenumbers on the dice into the Pythagorean Theorem for thelengths of the legs to find the value of the length of thehypotenuse.

• Using the Pythagorean Theorem a²+b²=c², a player movesaround the board a distance that is the integral part of c.

• For example, if a 1 and a 2 were rolled, 1²+2²=c²; 1+4=c²;5=c²; Since c = √5 or approximately 2.236, the play movestwo spaces. Always round the value down.

• When the player lands on a „?‟ space, a question card isdrawn. If the player answers the question correctly, he orshe can roll one die and advance the resulting number ofplaces.

• Each player must go around the board twice to completethe game. A play must answer a „?‟ card correctly tocomplete the game and become a Pythagorean

 

Pythagoras Board GameWhat are the

lengths of the legs

of a 30-60-90

degree triangle

with a hypotenuse

of length 10?

Answer: 5 and 5√3 

If you hiked 3

km west and the

4 km north, how

far are you from

your starting

point?

Answer: 5 km 

The square of the

 ______ of a right

triangle equals

the sum of the

squared of the

lengths of the two

legs.

Answer: 

hypotenuse 

Find the missing

member of the

Pythagorean triple

(7, __,, 25).

Answer: 24 

What is the length

of the legs in a 45-

45-90 degree right

triangle with

hypotenuse of

length √2? 

Answer: 1 

Using a²+b²=c,

find b if c = 10

and a = 6

Answer: b=8²

True or false?

Pythagoras lives

circa A.D. 500

Answer: false (500

B.C.) 

Have the person to

your left pick two

numbers for the legs

of a right triangle.

Compute the

hypotenuse

Can an isosceles

triangle be a right

triangle?

Answer: yes 

Pythagoras was

of what

nationality?

Answer: Greek  

Is (7, 8, 11) a

Pythagorean

triple?

Answer: no 

How do you spell

Pythagoras?

The Pythagorean

Theorem is

applicable for what

type of triangle?

Answer: a right

triangle 

What is the

name of the

school that

Pythagoras

founded?

Answer: The

Pythagorean

School 

True or false?

Pythagoras

considered

number to be the

basis of creation?

Answer: true 

True or false?

Pythagoras

formulated the only

proof of the

Pythagorean

Theorem?

Answer: false (there

are about 400

possible proofs) 

 

THANK YOU

By-

Rashmi Sharma

VIII-A