2. 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."

3.  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.

4.  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.

5.  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!

6. Baudhayan’s Sutra
 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 :

7.  “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.

8.  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.

9.  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).

10.  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!

11.  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!

12. 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.

13.  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.

14. 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):

15.  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.

16.  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.

17. 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.

18.  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

19.  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

20.  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

21.  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

22.  (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

23.  (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

24.  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

25. 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

26.  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.

27.  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

28.  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

29. 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

30.  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

31. 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

32. 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)

33.  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.

34. 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.

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

36. 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.)