assgmnt geometri

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ESB 4144 MODERN GEOMETRICS

INTRODUCTION

GEOMETRY

Geometry is a branch of mathematics that deals with the measurement, properties, and

relationships of points, lines, angles, and two- and three-dimensional figures. "Geometry,"

meaning "measuring the earth," is the branch of math that has to do with spatial relationships.

In other words, geometry is a type of math used to measure things that are impossible to

measure with devices.

The word geometry Originates from the Greek words (geo meaning world, metri meaning

measure) and means, literally, to measure the earth. It is an ancient branch of mathematics,

but its modern meaning depends largely on context; however, geometry largely encompasses

forms of non-numeric mathematics, such as those involving measurement, area and perimeter

calculation, and work involving angles and position. It was one of the two fields of pre-

modern mathematics, the other being the study of numbers.

In modern times, geometric concepts have been generalized to a high level of abstraction and

complexity, and have been subjected to the methods of calculus and abstract algebra, so that

many modern branches of the field are barely recognizable as the descendants of early

geometry. For example, no one has been able take a tape measure around the earth, yet we are

pretty confident that the circumference of the planet at the equator is 40,075.036 kilometers

(24,901.473 miles) . How do we know that? The first known case of calculating the distance

around the earth was done by Eratosthenes around 240 BCE. What tools do you think current

scientists might use to measure the size of planets? The answer is geometry.

THE GEOMETERS SKETCHPAD

The Geometer's Sketchpad is a popular commercial interactive geometry software program

for exploring Euclidean geometry, algebra, calculus, and other areas ofmathematics. It was

created by Nicholas Jackiw. It is designed to run on Windows 95 orWindows NT 4.0 or later

and Mac OS 8.6 or later (including Mac OS X). It also runs on Linux underWine with few

bugs.

Geometer's Sketchpad includes the traditional Euclidean tools of classical Geometric

constructions; that is, if a figure (such as the pentadecagon) can be constructed with compass

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and straight-edge, it can also be constructed using this program. However, the program also

allows users to employ transformations to "cheat," creating figures impossible to construct

under the traditional compass-and-straight-edge rules (such as the regularnonagon). You can

animate objects. Also, you are able to find the midpoint and mid segments of objects.

Geometer's Sketchpad also allows to measure lengths ofsegments, measures

ofangles, area,perimeter, etc. Some of the tools one can use include; construct function,

which allows the user to create objects in relation to selected objects. The transform function

allows the user to create points in relation to objects, which include distance, angle, ratio, and

others. With these tools, one can create numerous different objects, measure them, and

potentially figure out hard-to-solve math problems.

PYTHAGOREAN THEOREM

The Pythagorean Theorem states that, in a right triangle, the square of a (a) plus the square

of b (b) is equal to the square of c (c): a2 + b2 = c2. In a right angled triangle the square of

the long side (the "hypotenuse") is equal to the sum of the squares of the other two sides. It is

stated in this formula: a2 + b2 = c2.

Years ago, a man named Pythagoras found an amazing fact about triangles: If the triangle

had a right angle (90) ...... and you made a square on each of the three sides, then ...... the

biggest square had the exact same area as the other two squares put together! The longest

side of the triangle is called the "hypotenuse". In a right angled triangle the square of the

hypotenuse is equal to the sum of the squares of the other two sides.So, the square of a (a)

plus the square of b (b) is equal to the square of c (c): a2+ b2= c2. The theorem that the

sum of the squares of the lengths of the sides of a right triangle is equal to the square of the

length of the hypotenuse.

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HISTORY OF PYTHAGORAS

Pythagoras of Samos is often described as the first pure mathematician. He is an extremely

important figure in the development of mathematics yet we know relatively little about his

mathematical achievements. Unlike many later Greek mathematicians, where at least we have

some of the books which they wrote, we have nothing of Pythagoras's writings. The society

which he led, half religious and half scientific, followed a code of secrecy which certainly

means that today Pythagoras is a mysterious figure. Pythagoras lived in the 500s BC, and

was one of the first Greek mathematical thinkers. Pythagoreans were interested in

Philosophy, especially in Music and Mathematics. The statement of the Theorem was

discovered on a Babylonian tablet circa 1900 1600 B.C. Professor R. Smullyan in his book

5000 B.C. and Other Philosophical Fantasies tells of an experiment he ran in one of his

geometry classes. He drew a right triangle on the board with squares on the hypotenuse and

legs and observed the fact the the square on the hypotenuse had a larger area than either of

the other two squares. Then he asked, Suppose these three squares were made of beaten

gold, and you were offered either the one large square or the two small squares. Which would

you choose? Interestingly enough, about half the class opted for the one large square and

half for the two small squares. Both groups were equally amazed when told that it would

make no difference.

We do have details of Pythagoras's life from early biographies which use important original

sources yet are written by authors who attribute divine powers to him, and whose aim was to

present him as a god-like figure. What we present below is an attempt to collect together the

most reliable sources to reconstruct an account of Pythagoras's life. There is fairly good

agreement on the main events of his life but most of the dates are disputed with different

scholars giving dates which differ by 20 years. Some historians treat all this information as

merely legends but, even if the reader treats it in this way, being such an early record it is of

historical importance.

Pythagoras's father was Mnesarchus, while his mother was Pythais and she was a native of

Samos. Mnesarchus was a merchant who came from Tyre, and there is a story that he brought

corn to Samos at a time of famine and was granted citizenship of Samos as a mark of

gratitude. As a child Pythagoras spent his early years in Samos but travelled widely with his

father. There are accounts of Mnesarchus returning to Tyre with Pythagoras and that he was

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taught there by the Chaldaeans and the learned men of Syria. It seems that he also visited

Italy with his father.

Little is known of Pythagoras's childhood. All accounts of his physical appearance are likely

to be fictitious except the description of a striking birthmark which Pythagoras had on his

thigh. It is probable that he had two brothers although some sources say that he had three.

Certainly he was well educated, learning to play the lyre, learning poetry and to recite Homer.

There were, among his teachers, three philosophers who were to influence Pythagoras while

he was a young man. One of the most important was Pherekydes who many describe as the

teacher of Pythagoras.

The other two philosophers who were to influence Pythagoras, and to introduce him tomathematical ideas, were Thales and his pupil Anaximanderwho both lived on Miletus. In it

is said that Pythagoras visitedThales in Miletus when he was between 18 and 20 years old.

By this time Thales was an old man and, although he created a strong impression on

Pythagoras, he probably did not teach him a great deal. However he did contribute to

Pythagoras's interest in mathematics and astronomy, and advised him to travel to Egypt to

learn more of these subjects.Thales's pupil, Anaximander, lectured on Miletus and

Pythagoras attended these lectures. Anaximander certainly was interested in geometry

andcosmology and many of his ideas would influence Pythagoras's own views.

In about 535 BC Pythagoras went to Egypt. This happened a few years after the tyrant

Polycrates seized control of the city of Samos. There is some evidence to suggest that

Pythagoras and Polycrates were friendly at first and it is claimed that Pythagoras went to

Egypt with a letter of introduction written by Polycrates. In fact Polycrates had an alliance

with Egypt and there were therefore strong links between Samos and Egypt at this time. The

accounts of Pythagoras's time in Egypt suggest that he visited many of the temples and took

part in many discussions with the priests. According toPorphyryPythagoras was refused

admission to all the temples except the one at Diospolis where he was accepted into the

priesthood after completing the rites necessary for admission.

It is not difficult to relate many of Pythagoras's beliefs, ones he would later impose on the

society that he set up in Italy, to the customs that he came across in Egypt. For example the

secrecy of the Egyptian priests, their refusal to eat beans, their refusal to wear even cloths

made from animal skins, and their striving for purity were all customs that Pythagoras would

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later adopt. Porphyry in says that Pythagoras learnt geometry from the Egyptians but it is

likely that he was already acquainted with geometry, certainly after teachings

fromThales and Anaximander.

In 525 BC Cambyses II, the king of Persia, invaded Egypt. Polycrates abandoned his alliance

with Egypt and sent 40 ships to join the Persian fleet against the Egyptians. After Cambyses

had won the Battle of Pelusium in the Nile Delta and had captured Heliopolis and Memphis,

Egyptian resistance collapsed. Pythagoras was taken prisoner and taken to

Babylon. Iamblichus writes that Pythagoras :-

... was transported by the followers of Cambyses as a prisoner of war. Whilst he

was there he gladly associated with the Magoi ... and was instructed in their

sacred rites and learnt about a very mystical worship of the gods. He also

reached the acme of perfection in arithmetic and music and the other

mathematical sciences taught by the Babylonians...

In about 520 BC Pythagoras left Babylon and returned to Samos. Polycrates had been killed

in about 522 BC and Cambyses died in the summer of 522 BC, either by committing suicide

or as the result of an accident. The deaths of these rulers may have been a factor in

Pythagoras's return to Samos but it is nowhere explained how Pythagoras obtained his

freedom. Darius of Persia had taken control of Samos after Polycrates' death and he would

have controlled the island on Pythagoras's return. This conflicts with the accounts

ofPorphyry and Diogenes Laertius who state that Polycrates was still in control of Samos

when Pythagoras returned there.

Pythagoras made a journey to Crete shortly after his return to Samos to study the system of

laws there. Back in Samos he founded a school which was called the semicircle. Iamblichus

writes in the third century AD that:-

... he formed a school in the city [of Samos], the 'semicircle' of Pythagoras,

which is known by that name even today, in which the Samians hold political

meetings. They do this because they think one should discuss questions about

goodness, justice and expediency in this place which was founded by the man

who made all these subjects his business. Outside the city he made a cave the

private site of his own philosophical teaching, spending most of the night and

daytime there and doing research into the uses of mathematics...

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Pythagoras left Samos and went to southern Italy in about 518 BC (some say much earlier).

Iamblichus gives some reasons for him leaving. First he comments on the Samian response to

his teaching methods:-

... he tried to use his symbolic method of teaching which was similar in all

respects to the lessons he had learnt in Egypt. The Samians were not very keen

on this method and treated him in a rude and improper manner.

This was, according to Iamblichus, used in part as an excuse for Pythagoras to leave Samos:-

... Pythagoras was dragged into all sorts of diplomatic missions by his fellow

citizens and forced to participate in public affairs. ... He knew that all the

philosophers before him had ended their days on foreign soil so he decided to

escape all political responsibility, alleging as his excuse, according to some

sources, the contempt the Samians had for his teaching method.

Pythagoras founded a philosophical and religious school in Croton (now Crotone, on the east

of the heel of southern Italy) that had many followers. Pythagoras was the head of the society

with an inner circle of followers known as mathematikoi. The mathematikoi lived

permanently with the Society, had no personal possessions and were vegetarians. They were

taught by Pythagoras himself and obeyed strict rules. The beliefs that Pythagoras held were:

(1) that at its deepest level, reality is mathematical in nature,

(2) that philosophy can be used for spiritual purification,

(3) that the soul can rise to union with the divine,

(4) that certain symbols have a mystical significance, and

(5) that all brothers of the order should observe strict loyalty and secrecy.

Both men and women were permitted to become members of the Society, in fact several later

women Pythagoreans became famous philosophers. The outer circle of the Society was

known as the akousmatics and they lived in their own houses, only coming to the Society

during the day. They were allowed their own possessions and were not required to be

vegetarians.

Of Pythagoras's actual work nothing is known. His school practised secrecy and

communalism making it hard to distinguish between the work of Pythagoras and that of his

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followers. Certainly his school made outstanding contributions to mathematics, and it is

possible to be fairly certain about some of Pythagoras's mathematical contributions. First we

should be clear in what sense Pythagoras and the mathematikoi were studying mathematics.

They were not acting as a mathematics research group does in a modern university or other

institution. There were no 'open problems' for them to solve, and they were not in any sense

interested in trying to formulate or solve mathematical problems.

Rather Pythagoras was interested in the principles of mathematics, the concept of number, the

concept of a triangle or other mathematical figure and the abstract idea of a proof. As

Brumbaugh writes in :-

It is hard for us today, familiar as we are with pure mathematical abstractionand with the mental act of generalization, to appreciate the originality of this

Pythagorean contribution.

In fact today we have become so mathematically sophisticated that we fail even to recognise

2 as an abstract quantity. There is a remarkable step from 2 ships + 2 ships = 4 ships, to the

abstract result 2 + 2 = 4, which applies not only to ships but to pens, people, houses etc.

There is another step to see that the abstract notion of 2 is itself a thing, in some sense every

bit as real as a ship or a house.

Pythagoras believed that all relations could be reduced to number relations.

AsAristotle wrote:

The Pythagorean ... having been brought up in the study of mathematics, thought

that things are numbers ... and that the whole cosmos is a scale and a number.

This generalization stemmed from Pythagoras's observations in music, mathematics andastronomy. Pythagoras noticed that vibrating strings produce harmonious tones when the

ratios of the lengths of the strings are whole numbers, and that these ratios could be extended

to other instruments. In fact Pythagoras made remarkable contributions to the mathematical

theory of music. He was a fine musician, playing the lyre, and he used music as a means to

help those who were ill.

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Pythagoras studied properties of numbers which would be familiar to mathematicians today,

such as even and odd numbers,triangular numbers,perfect numbers etc. However to

Pythagoras numbers had personalities which we hardly recognize as mathematics today:

Each number had its own personality - masculine or feminine, perfect or

incomplete, beautiful or ugly. This feeling modern mathematics has deliberately

eliminated, but we still find overtones of it in fiction and poetry. Ten was the very

best number: it contained in itself the first four integers - one, two, three, and

four[1+ 2 + 3 + 4 = 10] - and these written in dot notation formed a perfect

triangle.

Of course today we particularly remember Pythagoras for his famous geometry theorem.

Although the theorem, now known as Pythagoras's theorem, was known to the Babylonians

1000 years earlier he may have been the first to prove it. Proclus, the last major Greek

philosopher, who lived around 450 AD wrote:

After[Thales, etc.] Pythagoras transformed the study of geometry into a liberal

education, examining the principles of the science from the beginning and

probing the theorems in an immaterial and intellectual manner: he it was who

discovered the theory ofirrationaland the construction of the cosmic figures.

AgainProclus, writing of geometry, said:-

I emulate the Pythagoreans who even had a conventional phrase to express what

I mean "a figure and a platform, not a figure and a sixpence", by which they

implied that the geometry which is deserving of study is that which, at each new

theorem, sets up a platform to ascend by, and lifts the soul on high instead of

allowing it to go down among the sensible objects and so become subservient to

the common needs of this mortal life.

Heath gives a list of theorems attributed to Pythagoras, or rather more generally to the

Pythagoreans.

1. The sum of the angles of a triangle is equal to two right angles. Also the Pythagoreans

knew the generalization which states that a polygon with n sides has sum of interior

angles 2n - 4 right angles and sum of exterior angles equal to four right angles.

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2. The theorem of Pythagoras - for a right angled triangle the square on

the hypotenuseis equal to the sum of the squares on the other two sides. We should note

here that to Pythagoras the square on the hypotenuse would certainly not be thought of as

a number multiplied by itself, but rather as a geometrical square constructed on the side.

To say that the sum of two squares is equal to a third square meant that the two squares

could be cut up and reassembled to form a square identical to the third square.

3. Constructing figures of a given area and geometrical algebra. For example they solved

equations such as a (a -x) =x2 by geometrical means.

4. The discovery of irrationals. This is certainly attributed to the Pythagoreans but it

does seem unlikely to have been due to Pythagoras himself. This went against

Pythagoras's philosophy the all things are numbers, since by a number he meant the ratio

of two whole numbers. However, because of his belief that all things are numbers it

would be a natural task to try to prove that the hypotenuse of an isosceles right angled

triangle had a length corresponding to a number.

5. The five regular solids. It is thought that Pythagoras himself knew how to construct

the first three but it is unlikely that he would have known how to construct the other two.

6. In astronomy Pythagoras taught that the Earth was a sphere at the centre of the

Universe. He also recognized that the orbit of the Moon was inclined to the equator of the

Earth and he was one of the first to realize that Venus as an evening star was the same

planet as Venus as a morning star.

Primarily, however, Pythagoras was a philosopher. In addition to his beliefs about numbers,

geometry and astronomy described above, he held:

... the following philosophical and ethical teachings: ... the dependence of the

dynamics of world structure on the interaction of contraries, or pairs of

opposites; the viewing of the soul as a self-moving number experiencing a form

of metempsychosis, or successive reincarnation in different species until its

eventual purification (particularly through the intellectual life of the ethically

rigorous Pythagoreans); and the understanding ...that all existing objects were

fundamentally composed of form and not of material substance. Further

Pythagorean doctrine ... identified the brain as the locus of the soul; and

prescribed certain secret cultic practices.

In their practical ethicsare also described:-

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In their ethical practices, the Pythagorean were famous for their mutual

friendship, unselfishness, and honesty.

Pythagoras's Society at Croton was not unaffected by political events despite his desire to

stay out of politics. Pythagoras went to Delos in 513 BC to nurse his old teacher Pherekydes

who was dying. He remained there for a few months until the death of his friend and teacher

and then returned to Croton. In 510 BC Croton attacked and defeated its neighbour Sybaris

and there is certainly some suggestions that Pythagoras became involved in the dispute. Then

in around 508 BC the Pythagorean Society at Croton was attacked by Cylon, a noble from

Croton itself. Pythagoras escaped to Metapontium and the most authors say he died there,

some claiming that he committed suicide because of the attack on his Society. Iamblichus in

quotes one version of events:-

Cylon, a Crotoniate and leading citizen by birth, fame and riches, but otherwise

a difficult, violent, disturbing and tyrannically disposed man, eagerly desired to

participate in the Pythagorean way of life. He approached Pythagoras, then an

old man, but was rejected because of the character defects just described. When

this happened Cylon and his friends vowed to make a strong attack on

Pythagoras and his followers. Thus a powerfully aggressive zeal activated Cylon

and his followers to persecute the Pythagoreans to the very last man. Because of

this Pythagoras left for Metapontium and there is said to have ended his days.

This seems accepted by most but Iamblichus himself does not accept this version and argues

that the attack by Cylon was a minor affair and that Pythagoras returned to Croton. Certainly

the Pythagorean Society thrived for many years after this and spread from Croton to many

other Italian cities. Gorman argues that this is a strong reason to believe that Pythagoras

returned to Croton and quotes other evidence such as the widely reported age of Pythagoras

as around 100 at the time of his death and the fact that many sources say that Pythagoras

taught Empedokles to claim that he must have lived well after 480 BC.

The evidence is unclear as to when and where the death of Pythagoras occurred. Certainly the

Pythagorean Society expanded rapidly after 500 BC, became political in nature and also spilt

into a number of factions. In 460 BC the Society :-

... was violently suppressed. Its meeting houses were everywhere sacked and

burned; mention is made in particular of "the house of Milo" in Croton,

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where 50 or60 Pythagoreans were surprised and slain. Those who survived took

refuge at Thebes and other places

Pythagorean Theorem Theory

A

B

C

The Pythagorean Theorem: The sum of the areas of the two squares on the legs (a and b)

equals the area of the square on the hypotenuse (c).

In mathematics, the Pythagorean Theorem orPythagoras' Theorem is a relation

inEuclidean geometry among the three sides of a right triangle (right-angled triangle). In

terms of areas, it states:

In any right triangle, the area of the square whose side is the hypotenuse(the side opposite

the right angle) is equal to the sum of the areas of the squares whose sides are the two legs

(the two sides that meet at a right angle).

The theorem can be written as an equation relating the lengths of the sides a, b and c, often

called thePythagorean equation: where c represents the length of the hypotenuse,

and a and b represent the lengths of the other two sides.

The Pythagorean Theorem is named after the GreekmathematicianPythagoras, who by

tradition is credited with its discovery andproof, although it is often argued that knowledge

of the theorem predates him. There is evidence that Babylonian mathematicians understood

the formula, although there is little surviving evidence that they used it in a mathematical

framework.

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The theorem has numerousproofs, possibly the most of any mathematical theorem. These are

very diverse, including both geometric proofs and algebraic proofs, with some dating back

thousands of years. The theorem can be generalized in various ways, including higher-

dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles,

and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean

theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness,

mystique, or intellectual power; popular references in literature, plays, musicals, songs,

stamps and cartoons abound.

As pointed out in the introduction, ifc denotes the lengthof the hypotenuse

and a and b denotes the lengths of the other two sides, the Pythagorean Theorem can be

expressed as the Pythagorean equation:

If the length of both a and b are known, then c can be calculated as follows:

If the length of hypotenuse c and one leg (a orb) are known, then the length of the other leg

can be calculated with the following equations:

or

The Pythagorean equation relates the sides of a right triangle in a simple way, so that if the

lengths of any two sides are known the length of the third side can be found. Another

corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of

the legs, but less than the sum of them.

A generalization of this theorem is the law of cosines, which allows the computation of the

length of the third side of any triangle, given the lengths of two sides and the size of the angle

between them. If the angle between the sides is a right angle, the law of cosines reduces to the

Pythagorean equation.

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PROOF

A. Proof using similar triangles

H

A

B C

This proof is based on theproportionality of the sides of two similartriangles, that is, upon

the fact that theratio of any two corresponding sides of similar triangles is the same

regardless of the size of the triangles. LetABCrepresent a right triangle, with the right angle

located at C, as shown on the figure. We draw the altitude from point C, and callHits

intersection with the sideAB. PointHdivides the length of the hypotenuse c into

parts dand e. The new triangleACHis similarto triangleABC, because they both have a

right angle (by definition of the altitude), and they share the angle atA, meaning that the third

angle will be the same in both triangles as well, marked as in the figure. By a similar

reasoning, the triangle CBHis also similar toABC. The proof of similarity of the triangles

requires the Triangle postulate: the sum of the angles in a triangle is two right angles, and is

equivalent to theparallel postulate. Similarity of the triangles leads to the equality of ratios of

corresponding sides:

The first result equates the cosine of each angle and the second result equates thesines.

These ratios can be written as:

Summing these two equalities, we obtain

Which, tidying up, is the Pythagorean Theorem:

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The role of this proof in history is the subject of much speculation. The underlying question

is why Euclid did not use this proof, but invented another. One conjecture is that the proof by

similar triangles involved a theory of proportions, a topic not discussed until later in

theElements, and that the theory of proportions needed further development at that time.

B. Euclids Proof

B C

A

In outline, here is how the proof inEuclid'sElements proceeds. The large square is divided

into a left and right rectangle. A triangle is constructed that has half the area of the left

rectangle. Then another triangle is constructed that has half the area of the square on the left-

most side. These two triangles are shown to be congruent, proving this square has the same

area as the left rectangle. This argument is followed by a similar version for the right

rectangle and the remaining square. Putting the two rectangles together to reform the square

on the hypotenuse, its area is the same as the sum of the area of the other two squares. The

details are next.

LetA,B, Cbe the vertices of a right triangle, with a right angle atA. Drop a perpendicular

fromA to the side opposite the hypotenuse in the square on the hypotenuse. That line divides

the square on the hypotenuse into two rectangles, each having the same area as one of the two

squares on the legs.

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For the formal proof, we require four elementarylemmata:

1. If two triangles have two sides of the one equal to two sides of the other, each to each,

and the angles included by those sides equal, then the triangles are congruent (side-

angle-side).

2. The area of a triangle is half the area of any parallelogram on the same base and

having the same altitude.

3. The area of a rectangle is equal to the product of two adjacent sides.

4. The area of a square is equal to the product of two of its sides (follows from 3).

Next, each top square is related to a triangle congruent with another triangle related in turn to

one of two rectangles making up the lower square.

K

G

F

D E

I

H

A

CB

L

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K

BC

A

D

F

G

L

The proof is as follows:

1. Let ACB be a right-angled triangle with right angle CAB.

2. On each of the sides BC, AB, and CA, squares are drawn, CBDE, BAGF, and ACIH, in

that order. The construction of squares requires the immediately preceding theorems in

Euclid, and depends upon the parallel postulate

3. From A, draw a line parallel to BD and CE. It will perpendicularly intersect BC and DE

at K and L, respectively.

4. Join CF and AD, to form the triangles BCF and BDA.

5. Angles CAB and BAG are both right angles; therefore C, A, and G are collinear.

Similarly for B, A, and H.

6. Angles CBD and FBA are both right angles; therefore angle ABD equals angle FBC,

since both are the sum of a right angle and angle ABC.

7. Since AB is equal to FB and BD is equal to BC, triangle ABD must be congruent to

triangle FBC.

8. Since A-K-L is a straight line, parallel to BD, then parallelogram BDLK has twice the

area of triangle ABD because they share the base BD and have the same altitude BK,

i.e., a line normal to their common base, connecting the parallel lines BD and AL.

(lemma 2)

9. Since C is collinear with A and G, square BAGF must be twice in area to triangle FBC.

10. Therefore rectangle BDLK must have the same area as square BAGF = AB 2.

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11. Similarly, it can be shown that rectangle CKLE must have the same area as square

ACIH = AC2.

12. Adding these two results, AB2 + AC2 = BD BK + KL KC

13. Since BD = KL, BD BK + KL KC = BD(BK + KC) = BD BC14. Therefore AB2 + AC2 = BC2, since CBDE is a square.

This proof, which appears in Euclid'sElements as that of Proposition 47 in Book 1,

demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other

two squares. This is quite distinct from the proof by similarity of triangles, which is

conjectured to be the proof that Pythagoras used.

Algebraic proofs

The theorem can be proved algebraically using four copies of a right triangle with

sides a, b and c, arranged inside a square with side c as in the top half of the diagram.[16] The

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triangles are similar with area , while the small square has side b a and area (b a)2.

The area of the large square is therefore

But this is a square with side c and area c2, so

A similar proof uses four copies of the same triangle arranged symmetrically around a square

with side c, as shown in the lower part of the diagram. This results in a larger square, with

side a + b and area (a + b)2. The four triangles and the square side c must have the same area

as the larger square,

Giving,

A related proof was published by former U.S. President James A. Garfield. Instead of a

square it uses a trapezoid, which can be constructed from the square in the second of the

above proofs by bisecting along a diagonal of the inner square, to give the trapezoid as shown

in the diagram. The area of the trapezoid can be calculated to be half the area of the square,

that is

The inner square is similarly halved, and there are only two triangles so the proof proceeds as

above except for a factor of which is removed by multiplying by two to give theresult.

Proof using differentials

One can arrive at the Pythagorean theorem by studying how changes in a side produce a

change in the hypotenuse and employing calculus.

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The triangleABCis a right triangle, as shown in the upper part of the diagram, withBCthe

hypotenuse. At the same time the triangle lengths are measured as shown, with the

hypotenuse of lengthy, the sideACof lengthx and the sideAB of length a, as seen in the

lower diagram part.

Ifx is increased by a small amount dx by extending the sideACslightly toD, theny also

increases by dy. These form two sides of a triangle, CDE, which (withEchosen so CEis

perpendicular to the hypotenuse) is a right triangle approximately similar toABC. Therefore

the ratios of their sides must be the same, that is:

This can be rewritten as follows:

This is a differential equation which is solved to give

And the constant can be deduced fromx = 0,y = a to give the equation

This is more of an intuitive proof than a formal one: it can be made more rigorous if proper

limits are used in place ofdx and dy.

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APPLICATION

The Pythagorean theorem has far-reaching ramification in other fields (such as the arts), as

well as practical application. 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.

The most widely quoted "practical" application of the Pythagorean theorem is actually anapplication of its converse. The theorem of Pythagoras says that if a triangle has sides of

length a, b and c and the angle between the sides of length a and b is a right angle, then a^2 +

b^2 = c^2. The converse says that if a triangle has sides of length a, b and c and a^2 + b^2 =

c^2 then the angle between the sides of length a and b is a right angle. Such a triple of

numbers is called a Pythagorean triple, so 3,4,5 is a Pythagorean triple and so are 6, 8, 10 and

5, 12, 13.

The application is in construction. It is very important when starting a building to have a

square corner, and a Pythagorean triple provides an easy and inexpensive way to get one.

Drive a stake at the desired corner point and another stake 3 meters from the corner along the

line where you want one wall of the building. Then position a third stake so that its distance

from the corner is 4 meters and the third side of the triangle formed by the three stakes is 5

meters. Since 3, 4, 5 is a Pythagorean triple the angle at the corner is a right angle.

The Pythagorean theorem is a starting place for trigonometry, which leads to methods, forexample, for calculating the heights of mountains. The Pythagorean theorem is also an

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example of the somewhat rare situation where both the theorem and its converse are true. It is

also useful in calculating distances.

EASE AND PROBLEM OF USE GEOMETRY SKETCHPAD

Geometer's Sketchpad is a wonderful mathematical program that combines the use of

technology with geometry. This program has several benefits to its use. First of all, students

are able to explore and learn on their own the meaning of several important definitions in

geometry. Students, by simply clicking on the mouse, learn the many parts that make up

geometry. The students are learning for them, and therefore are likely to retain far much

more information than they would in a traditional classroom. In addition to learning

definitions, all of the student's work is "saved." Therefore, students can compare differenttrails with one another, instead of simply loosing the information. Also, because information

is preserved, students have the benefit to examine and compare several similar cases in

seconds, rather than hours it might take to draw the figures accurately. This, of course, will

lead to generalizations and patterns far quicker than in a traditional geometric classroom.

Frequent problems experienced in a traditional geometric classroom are often obsolete with

the use of Geometer's Sketchpad. Students will be learning at a quicker pace and therefore

will able to accomplish more. Instead of "leaving things out," which happens far too often in

traditional classrooms, students are encouraged to learn as much as possible. Also, students

who might often find it difficult to stay on task and understand what is being discussed will

have the option of the "help menu." This tool is used by so many people today. Again, with

this tool those students can also move beyond the Euclidean plane (2 D) into a far more

complex geometrical world. The program also contains polar coordinate capabilities, often

discussed in calculus classes. In fact, Geometer's Sketchpad is not only for math, it has been

used in Art, Science, and a variety of other subjects across the curriculum. The program's

versatility is amazing. We truly believe it is one of the best innovations for the mathematical

world.

Another great benefit to this program is its ability to graph data collected while using it. As

anyone whose studied geometry in detail knows, there is quite a bit of trial and error before

the correct answer is reached. By using Geometer's Sketchpad, students have the benefit of

recalling information collected by simply clicking a button instead of fumbling through heaps

of papers. These graphs could also become a way of completing assignments because they

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can be labeled, include captions, and perform many of the other techniques used in other

computer based programs like excel.

The lists of the possibilities of how this program could be used in a classroom are endless.

Teachers and students can work together through the use of an overhead, having different

members of the class come up and "try" certain things. Or, if computers are available in the

classroom, students could simply be given a problem and asked to solve. If that is not

possible, this program would be an excellent source for an enrichment activity to those

students who find it interesting.

With Geometer's Sketchpad, students are able to take their learning into their own hands.

Students will not only be forced to understand the material in order to work the program, butthey will most likely discover new things to them, they will want to go beyond what is

expected of them. The Key Curriculum Press feels that this program could be the program

that would help to make "award-winning mathematical discoveries.

With all advantages, there are disadvantages as well. Geometer's Sketchpad is a very

complicated tool to master. There are many steps that need to be followed, as well as, an

understanding of what all the "buttons" do and why they do it.

Beside that the disadvantage of geometers sketchpad is we need to be familiar and

comfortable with the use of the computer per secondly familiar with whatever software that is

supposed to be the treatment for the Pythagoras theorem.

We also have the opportunity to explore the software, the subject will be overly anxious into

wanting to concentrate on too many new things at the same time.

We need to focus and concentrate is supposed to be learned. We also need to focus and

concentrate on the content that is supposed to be learned. The geometers sketchpads need

using the computer software how to use it. It means the students must be having the computer

while using the sketchpads.

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Documents

assgmnt geometri