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Pythagoras Theorem

 The Pythagorean theorem takes its name from the ancient Greek mathematician Pythagoras (569 B.C. -500 B.C.), who was perhaps the first to offer a proof of the theorem. But people had noticed the special relationship between the sides of a right triangle long before Pythagoras.

The Pythagorean theorem states that the sum of the squares of the lengths of the two other sides of any right triangle will equal the square of the length of the hypotenuse, or, in mathematical terms, for the triangle shown at right, a2 + b2 = c2. Integers that satisfy the conditions a2 + b2 = c2 are called "Pythagorean triples."

Ancient clay tablets from Babylonia indicate that the Babylonians in the second millennium B.C., 1000 years before Pythagoras, had rules for generating Pythagorean triples, understood the relationship between the sides of a right triangle, and, in solving for the hypotenuse of an isosceles right triangle, came up with an approximation of accurate to five decimal places. [They needed to do so because the lengths would represent some multiple of the formula: 12 + 12 =(√2)2.]

A Chinese astronomical and mathematical treatise called Chou Pei Suan Ching (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, ca. 500-200 B.C.), possibly predating Pythagoras, gives a statement of and geometrical demonstration of the Pythagorean theorem.

Ancient Indian mathematicians also knew the Pythagorean theorem, and the Sulbasutras (of which the earliest date from ca. 800-600 B.C.) discuss it in the context of strict requirements for the orientation, shape, and area of altars for religious purposes. It has also been suggested that the ancient Mayas used variations of Pythagorean triples in their Long Count calendar.

We do not know for sure how Pythagoras himself proved the theorem that bears his name because he refused to allow his teachings to be recorded in writing. But probably, like most ancient proofs of the Pythagorean theorem, it was geometrical in nature.

That is, such proofs are demonstrations that the combined areas of squares with sides of length a and b will equal the area of a square with sides of length c, where a, b, and c represent the lengths of the two sides and hypotenuse of a right triangle.

Pythagoras himself was not simply a mathematician. He was an important philosopher who believed that the world was ruled by harmony and that numerical relationships could best express this harmony. He was the first, for example, to represent musical harmonies as simple ratios.

Pythagoras and his followers were also a bit eccentric. Pythagoras's followers were sworn to absolute secrecy, and their devotion to their master bordered on the cult-like. Pythagoreans followed a strict moral and ethical code, which included vegetarianism because of their belief in the reincarnation of souls. They also refused to eat beans!

    It was ancient Indians mathematicians who discovered Pythagoras theorem. This might come as a surprise to many, but it’s true that Pythagoras theorem was known much before Pythagoras and it was Indians who actually discovered it at least 1000 years before Pythagoras was born! It was Baudhāyana who discovered the Pythagoras theorem. Baudhāyana listed Pythagoras theorem in his book called Baudhāyana Śulbasûtra (800 BCE). Incidentally, Baudhāyana Śulbasûtra is also one of the oldest books on advanced Mathematics. The actual shloka (verse) in Baudhāyana Śulbasûtra that describes Pythagoras theorem is given below :

“dīrghasyākṣaṇayā rajjuH pārśvamānī, tiryaDaM mānī, cha yatpṛthagbhUte kurutastadubhayāṅ karoti.”

Interestingly, Baudhāyana used a rope as an example in the above shloka which can be translated as – A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together. As you see, it becomes clear that this is perhaps the most intuitive way of understanding and visualizing Pythagoras theorem (and geometry in general) and Baudhāyana seems to have simplified the process of learning by encapsulating the mathematical result in a simple shloka in a layman’s language.

Some people might say that this is not really an actual mathematical proof of Pythagoras theorem though and it is possible that Pythagoras provided that missing proof. But if we look in the same Śulbasûtra, we find that the proof of Pythagoras theorem has been provided by both Baudhāyana and Āpastamba in the Sulba Sutras! To elaborate, the shloka is to be translated as –

The diagonal of a rectangle produces by itself both (the areas) produced separately by its two sides.

The implications of the above statement are profound because it is directly translated into Pythagorean Theorem are geometrical in nature, the Sulba Sutra’s numerical proof was unfortunately ignored. 

Though, Baudhāyana was not the only Indian mathematician to have provided Pythagorean triplets and proof. Āpastamba also provided the proof for Pythagoras theorem, which again is numerical in nature but again unfortunately this vital contribution has been ignored and Pythagoras was wrongly credited by Cicero and early Greek mathematicians for this theorem. Baudhāyana also presented geometrical proof using isosceles triangles so, to be more accurate, we attribute the geometrical proof to Baudhāyana and numerical  proof to Āpastamba. Also, another ancient Indian mathematician called Bhaskara later provided a unique geometrical proof as well as numerical which is known for the fact that it’s truly generalized and works for all sorts of triangles and is not incongruent (not just isosceles as in some older proofs).

One thing that is really interesting is that Pythagoras was not credited for this theorem till at least three centuries after! It was much later when Cicero and other Greek philosophers/mathematicians/historians decided to tell the world that it was Pythagoras that came up with this theorem! How utterly ridiculous! In fact, later on many historians have tried to prove the relation between Pythagoras theorem and Pythagoras but have failed miserably. In fact, the only relation that the historians have been able to trace it to is with Euclid, who again came many centuries after Pythagoras!

This fact itself means that they just wanted to use some of their own to name this theorem after and discredit the much ancient Indian mathematicians without whose contribution it could’ve been impossible to create the very basis of algebra and geometry!

Many historians have also presented evidence for the fact that Pythagoras actually travelled to Egypt and then India and learned many important mathematical theories (including Pythagoras theorem) that western world didn’t know of back then! So, it’s very much possible that Pythagoras learned this theorem during his visit to India but hid his source of knowledge he went back to Greece! 

Pythagoras and his works

Pythagoras of samos was an lonian Greek philosopher, and has been credited as the founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after 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 may be India, and in 520 AD a 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 often revered as a great mathematician and scientist and is best known for the Pythagorean 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 accounts mention that the philosophy associated with the Pythagoras was related to mathematics and that numbers were important. It was said that he was the first man to call himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Aristotle, and Plato, and through him, all of western philosophy.

Pythagoras theorem – (1) geometric proof using squares

The Pythagoras Theorem states that, in right triangle, the square of a(a²) plus the square of b(b²) is equal to the square of c(c²).

a²+b²+c²

Proof of the Pythagorean Theorem using Algebra We can show that a2 + b2 = c2 using Algebra Take a look at this diagram ... it has that "abc"

triangle in it (four of them actually):

Area of Whole Square It is a big square, with each side having a length

of a+b, so the total area is: A = (a+b)(a+b) Area of The Pieces Now let's add up the areas of all the smaller

pieces: First, the smaller (tilted) square has an area of A

= c2   And there are four triangles, each one has an area of A =½abSo all four of them combined is A = 4(½ab) = 2ab   So, adding up the tilted square and the 4 triangles gives: A = c2+2ab.

Both Areas Must Be Equal The area of the large square is equal to the

area of the tilted square and the 4 triangles. This can be written as:

(a+b)(a+b) = c2+2ab Now, let us rearrange this to see if we can get

the pythagoras theorem: Start with: (a+b)(a+b)=c2 + 2ab     Expand

(a+b)(a+b): a2 + 2ab + b2=c2 + 2ab     Subtract "2ab" from both sides: a2 + b2=c2       DONE!   

Now we can see why the Pythagorean Theorem works ... and it is actually a proof of the Pythagorean Theorem.

Formulas for generating pythagorean triples

A Pythagorean triple is a triple of positive integers and  such that a right triangle exists with legs  and hypotenuse. By the Pythagorean theorem, this is equivalent to finding positive integers , , and  satisfying

(1) The smallest and best-known Pythagorean

triple is . The right triangle having these side lengths is sometimes called the 3, 4, 5 triangle.

Plots of points in the -plane such that  is a Pythagorean triple are shown above for successively larger bounds. These plots include negative values of  and , and are therefore symmetric about both the x- and y-axes.

Similarly, plots of points in the -plane such that  is a Pythagorean triple are shown above for successively larger bounds

It is usual to consider only primitive Pythagorean triples (also called "reduced"triples) in which  and  are relatively prime, since other solutions can be generated trivially from the primitive ones. The primitive triples are illustrated above, and it can be seen immediately that the radial lines corresponding to imprimitive triples in the original plot are absent in this figure. For primitive solutions, one of  or must be even, and the other odd (Shanks 1993, p. 141), with  always odd.

In addition, one side of every Pythagorean triple is divisible by 3, another by 4, and another by 5. One side may have two of these divisors, as in (8, 15, 17), (7, 24, 25), and (20, 21, 29), or even all three, as in (11, 60, 61).

Given a primitive triple , three new primitive triples are obtained from

where U

A

D

Hall (1970) and Roberts (1977) prove that  is a primitive Pythagorean triple if

(8) where  is a finite product of the matrices. It

therefore follows that every primitive Pythagorean triple must be a member of the infinite array

Pythagoras and the Babylonians gave a formula for generating (not necessarily primitive) triples as

for , which generates a set of distinct triples containing neither all primitive nor all imprimitive triples (and where in the special case, ).

The early Greeks gave

where  and  are relatively prime and of opposite parity (Shanks 1993, p. 141), which generates a set of distinct triples containing precisely the primitive triples (after appropriately sorting  and ).

Let  be a Fibonacci number. Then

generates distinct Pythagorean triples (Dujella 1995), although not exhaustively for either primitive or imprimitive triples. More generally, starting with positive integers and constructing the Fibonacci-like sequence with terms generates distinct Pythagorean triples

(Horadam 1961), where

where  is a Lucas number. For any Pythagorean triple, the product of the two

nonhypotenuse legs (i.e., the two smaller numbers) is always divisible by 12, and the product of all three sides is divisible by 60. It is not known if there are two distinct triples having the same product. The existence of two such triples corresponds to a nonzero solution to the Diophantine equation

(Guy 1994, p. 188). For a Pythagorean triple (a, b, c),

where  is the partition function P (Honsberger 1985). Every three-term progression of squares , ,  can be associated with a Pythagorean triple (X, Y, Z) by

r = X – Y s = Z t = X + Y

(Robertson 1996). The area of a triangle corresponding to the Pythagorean

triple ( u² - v², 2uv, u²+ v²) is A = ½ (u² - v²)(2uv) = uv (u² - v²) Fermat proved that a number of this form can never be

a square number. To find the number  of possible primitive triangles which

may have a leg (other than the hypotenuse) of length , factor  into the form

The number of such triangles is then

i.e., 0 for singly even  and 2 to the power one less than the number of distinct prime factors of  otherwise (Beiler 1966, pp. 115-116). The first few numbers for , 2, ..., are 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 2, ... (OEIS A024361). To find the number of ways L(s)  in which a number s can be the leg (other than the hypotenuse) of a primitive or nonprimitive right triangle, write the factorization of s as

Then

(Beiler 1966, p. 116). Note that L(s) iff  is prime or twice a prime. The first few numbers for s=1, 2, ... are 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, ...

To find the number of ways  in which a number  can be the hypotenuse of a primitive right triangle, write its factorization as

where the ps are of the form  and the s are of the form . The number of possible primitive right triangles is then

Uses & interesting problems

Uses

Real Life Applications Some real life applications to introduce the concept of Pythagoras's

theorem to your middle school students are given below: 1) Road Trip: Let’s say two friends are meeting at a playground.

Mary is already at the park but her friend Bob needs to get there taking the shortest path possible. Bob has two way he can go - he can follow the roads getting to the park - first heading south 3 miles, then heading west four miles. The total distance covered following the roads will be 7 miles. The other way he can get there is by cutting through some open fields and walk directly to the park. If we apply Pythagoras's theorem to calculate the distance you will get:

(3)2 + (4)2 = 9 + 16 = C2

√25 = C 5 Miles. = C

Walking through the field will be 2 miles shorter than walking along the roads. .

2) Painting on a Wall: Painters use ladders to paint on high buildings and often use the help of Pythagoras' theorem to complete their work. The painter needs to determine how tall a ladder needs to be in order to safely place the base away from the wall so it won't tip over. In this case the ladder itself will be the hypotenuse. Take for example a painter who has to paint a wall which is about 3 m high. The painter has to put the base of the ladder 2 m away from the wall to ensure it won't tip. What will be the length of the ladder required by the painter to complete his work? You can calculate it using Pythagoras' theorem:

(5)2 + (2)2 = 25 + 4 = C2

√100 = C 5.3 m. = C Thus, the painter will need a ladder about 5 meters

high.

3) Buying a Suitcase: Mr. Harry wants to purchase a suitcase. The shopkeeper tells Mr. Harry that he has a 30 inch of suitcase available at present and the height of the suitcase is 18 inches. Calculate the actual length of the suitcase for Mr. Harry using Pythagoras' theorem. It is calculated this way:

(18)2 + (b)2 = (30)2

324 + b2 = 900 B2 = 900 – 324 b= √576 = 24 inches 4) What Size TV Should You Buy? Mr. James saw an

advertisement of a T.V.in the newspaper where it is mentioned that the T.V. is 16 inches high and 14 inches wide. Calculate the diagonal length of its screen for Mr. James. By using Pythagoras' theorem it can be calculated as:

(16)2 + (14)2 = 256 + 196 = C2

√452 = C 21 inches approx. = C

5) Finding the Right Sized Computer: Mary wants to get a computer monitor for her desk which can hold a 22 inch monitor. She has found a monitor 16 inches wide and 10 inches high. Will the computer fit into Mary’s cabin? Use Pythagoras' theorem to find out:

(16)2 + (10)2 = 256 + 100 = C2

√356 = C 18 inches approx. = C

Interesting problems Example: Shane marched 3 m east and 6 m north. How far

is he from his starting point? Solution: First, sketch the scenario. The path taken by

Shane forms a right-angled triangle. The distance from the starting point forms the hypotenuse.

x = = 6.71 m Example: The rectangle PQRS represents the floor of a room. Ivan stands at point A. Calculate the distance of

Ivan from a) the corner R of the room b) the corner S of the room

Solution: a) AR = = 4.47 m Ivan is 4.47 m from the corner R of the room b) AS = = 10.77 m Ivan is 10.77m from the corner S of the room Example: In the following diagram of a circle, O is the centre

and the radius is 12 cm. AB and EFare straight lines.

Find the length of EF if the length of OP is 6 cm. Solution: OE is the radius of the circle, which is 12 cm OP 2 + PE 2 = OE 2 6 2 + PE 2 = 12 2 PE =  EF = 2 × PE = 20.78 cm

Extensions

Theorem 67: If a, b, and c represent the lengths of the sides of a triangle, and c is the longest length, then the triangle is obtuse if c2 > a2 + b2, and the triangle is acute if c2 < a2 + b2.

Figures 1 (a) through (c) show these different triangle situations and the sentences comparing their sides. In each case, c represents the longest side in the triangle.

Figure 1 The relationship of the square of the longest side to the sum of the squares of the other two sides of a right triangle, an obtuse triangle, and an acute triangle.

Example 1: Determine whether the following sets of three values could be the lengths of the sides of a triangle. If the values can be the sides of a triangle, then classify the triangle. (a) 16‐30‐34, (b) 5‐5‐8, (c) 5‐8‐15, (d) 4‐4‐5, (e) 9‐12‐16, (f) 1-1-√2

34 ? 16 + 30 34 < 46 (So these can be the sides of a triangle.) 1156 ? 256 + 900

1156 = 1156 This is a right triangle. Because its sides are of

different lengths, it is also a scalene triangle. 8 ? 5 + 5 8 < 10 (So these can be the sides of triangle.) 8² = 5² + 5² 64 = 25 + 25  64 > 25 This is an obtuse triangle. Because two of its sides

are of equal measure, it is also an isosceles triangle.

15 ? 5 + 8 15 > 13 (So these cannot be the sides of a

triangle.) 5 ? 4 + 4 5 < 8 (So these cannot be the sides of a triangle)

5² ? 4² + 4² 25² ? 16 +16   25 < 32 This is an acute triangle. Because two of its sides are of

equal measure, it is also an isosceles triangle. 16 ? 9 + 12 16 < 21 (So these can be the sides of triangle.) 16² ? 9² + 12² 256 ? 81 + 44 256 > 225 This is an obtuse triangle. Because all sides are of different

lengths, it is also a scalene triangle. √2 ? 1 + 1 √2 < 2 (So these can be the sides of a triangle.) (√2)² ? 1 +1 2 ? 1 +1 2 = 2 This is a right triangle. Because two of its sides are of equal

measure, it is also an isosceles triangle.

Converse of pythagoras theorem

    What is Pythagoras Theorem ?  As in the diagram, ABC is a right-angled triangle with right angle at C, then          a2 + b2 = c2

The converse of Pythagoras Theorem is:   If             a2 + b2 = c2        holds then           DABC is a right angled triangle with           right angle at C.

The converse of Pythagoras Theorem is:  If               a2 + b2 = c2        holds  then           DABC is a right angled triangle with           right angle at C.

How to prove The converse of Pythagoras Theorem ?  

Now construct another triangle as follows :        EF = BC = a ÐF is a right angle.    FD = CA = b

   In  DDEF,    By Pythagoras Theorem,           ……..(2)    By (1), the given,             Therefore,                 AB = DE    But by construction,       BC = EF   and                                 CA = FD                              D ABC @ D DEF (S.S.S.)

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MADE BY-NISHITA X - D