Pythagoras was an influential 6th century BCE Greek mathematician who developed a secret society called the Pythagoreans. They discovered the Pythagorean theorem, which states that in a right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. This allowed them to correctly construct buildings by ensuring columns had straight bases. The theorem revolutionized architecture and measurement.
Pythagoras was a Greek mathematician who contributed much to the mathematical world, mainly because of Pythagorean Theorem. The following PPT contains all the necessary information about Pythagoras's early and later life, as well as about his works and explanations.(If you find the fonts a little weird, its not my fault as Slideshare doesn't supports many fonts)
This was originally presented by Sachin Motwani (the creator of the PPT) in Goodley Public School (his Alma mater) on the occasion of the Indian Mathematics Day in 2016 in an audience of 12th Class Students.
"Application of 3D and 2D geometry" explains the importance of geometry in our lives. Geometry is found everywhere from nature to human made machines. I have tried to inculcate all
its applications.
I hope it helps in providing guidance to those who are aspiring to understand geometry. I have taken help from internet and some books to acquire knowledge.
thank you for clicking my slide.
Pythagoras was a Greek mathematician who contributed much to the mathematical world, mainly because of Pythagorean Theorem. The following PPT contains all the necessary information about Pythagoras's early and later life, as well as about his works and explanations.(If you find the fonts a little weird, its not my fault as Slideshare doesn't supports many fonts)
This was originally presented by Sachin Motwani (the creator of the PPT) in Goodley Public School (his Alma mater) on the occasion of the Indian Mathematics Day in 2016 in an audience of 12th Class Students.
"Application of 3D and 2D geometry" explains the importance of geometry in our lives. Geometry is found everywhere from nature to human made machines. I have tried to inculcate all
its applications.
I hope it helps in providing guidance to those who are aspiring to understand geometry. I have taken help from internet and some books to acquire knowledge.
thank you for clicking my slide.
What impact did Pythagoras have on EuclidSolutionPythagorasP.pdfformaxekochi
What impact did Pythagoras have on Euclid?
Solution
Pythagoras
Probably the most famous name during the development of Greek geometry is Pythagoras, even
if only for the famous law concerning right angled triangles. This mathematician lived in a secret
society which took on a semi-religious mission. From this, the Pythagoreans developed a number
of ideas and began to develop trigonometry. The Pythagoreans added a few new axioms to the
store of geometrical knowledge.
1)The sum of the internal angles of a triangle equals two right angles 180*.
2)The sum of the external angles of a triangle equals four right angles 360*
3)The sum of the interior angles of any polygon equals 2n-4 right angles, where n is the number
of sides.
4)The sum of the exterior angles of a polygon equals four right angles, however many sides.
5)The three polygons, the triangle, hexagon, and square completely fill the space around a point
on a plane - six triangles, four squares and three hexagons. In other words, you can tile an area
with these three shapes, without leaving gaps or having overlaps.
6)For a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of
the other two sides.
Most of these rules are instantly familiar to most students, as basic principles of geometry and
trigonometry. One of his pupils, Hippocrates, took the development of geometry further. He was
the first to start using geometrical techniques in other areas of maths, such as solving quadratic
equations, and he even began to study the process of integration. He solved the problem of
Squaring a Lune and showed that the ratio of the areas of two circles equalled the ratio between
the squares of the radii of the circles.
Euclid
Alongside Pythagoras, Euclid is a very famous name in the history of Greek geometry. He
gathered the work of all of the earlier mathematicians and created his landmark work, \'The
Elements,\' surely one of the most published books of all time. In this work, Euclid set out the
approach for geometry and pure mathematics generally, proposing that all mathematical
statements should be proved through reasoning and that no empirical measurements were
needed. This idea of proof still dominates pure mathematics in the modern world.
The reason that Euclid was so influential is that his work is more than just an explanation of
geometry or even of mathematics. The way in which he used logic and demanded proof for every
theorem shaped the ideas of western philosophers right up until the present day. Great
philosopher mathematicians such as Descartes and Newton presented their philosophical works
using Euclid\'s structure and format, moving from simple first principles to complicated
concepts. Abraham Lincoln was a fan, and the US Declaration of Independence used Euclid\'s
axiomatic system.
Apart from the Elements, Euclid also wrote works about astronomy, mirrors, optics, perspective
and music theory, although many of his works are lost to posterity. Certainl.
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Notes on Pythagoras!
1. Pythagoras and his Theorem
Pythagoras of Samos
a short history
Pythagoras is arguably one of
the most important
mathematicians of his time.
He was born in 569 BCE on
the small Greek island,
Samos. During his life, he
perfected his method of
traveling education, where he
taught in Middle-Eastern
cities.
!
Many people of the time could
not follow the intricate math
theroems, and unfortunately
thought that Pythagoras was
crazy! Even if some
questioned his sanity,
Pythagoras attracted like-minded
individuals where they
continued to learn his
teachings in secret as to not
Name: Date:! ! ! ! Class:
Lingley 8 Math
be considered evil. This secret
group called themselves The
Pythagoreans. Their
meetings became so secret,
that they developed their
own language, and
even had their own
seal, etched above
the doors where their
meetings were held.
The Pythagoreans thought
that all problems could be
solved by numbers. This must
be how they discovered how
to correctly build their iconic
Greek columns. Before news
of Pytagoras’ Theorem
spread, all buildings were
formed with crooked bottoms,
since there was no tool of
measurement to ensure that
their bases were straight.
!
The Pythagoreans traveled
throughout Greece with their
special measurement tool: the
12 knot rope. With this they
were able to solve many
building mistakes.
2. x x ✓
Lingley 8 Math
Solving the Theorem
If
only these columns
were straight!
Understanding the Pythagorean Theorem
Pythagoras saw that the crooked columns casted a triangular
shadow on the ground. Using his knowledge of geometry, he
saw that the crooked columns casted an acute triangle. He then
discovered that the only triangle that will work with his theorem is
a right angle triangle.
hypotenuse
leg
right angle
isosceles acute right
Once Pythagoras switched to only using the right angle
triangle, he soon found a relationship between the legs on
either side of the right angle, and the hypotenuse.
The Secrets behind the Theorem
1. Squares can be formed around each side of the triangle.
2. The sum of the small and medium square areas’ equals the area
around the hypotenuse.
3. This relationship is only true for right angle triangles.
4. The theorem is used when solving for an unknown length of a
triangle.
5. The theorem will also work for any regular polygon around the sides of
the triangle.
3. Example 1
36 cm2
Lingley 8 Math
Applying the Theorem
Using the Pythagorean Theorem, find the
value of the hypotenuse.
6 cm
8 cm
h
h
8 cm
8 cm
6 cm
h
6 cm
1. Draw squares around all sides of the right
angle triangle, and label them.
2. Find the areas of each of those squares.
+ 64 cm2 = 100 cm2
6 cm
6 cm
8 cm
8 cm
3. We now know that the area of the purple
square is 100 cm2. However this is only
the area! We now need to take the
square root to find the side length!
4. Take the square root of the hypotenuse √100 cm2 = 10 cm
Final Answer!
Using the Pythagorean Theorem, find the missing values of the triangles below.
Your Turn!
4 cm
4 cm
h
5 cm
10 cm
h
6 cm
x
9 cm
Do your work on the next page.
If you find the area is a non-square... approximate!
