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PYTHAGORAS

- and the Pythagorean Theorem

a² b² c²

Objective/Intended Audience

- Objectives
- (SOL 8.11) The student will verify the

Pythagorean Theorem by measuring then applying

the theorem to find the missing length of a side

of a right triangle when the lengths of the other

two sides are given. - The student will increase his knowledge about the

life and contributions of Pythagoras. - Intended Audience Eighth Grade Math Students

Lesson Objectives

- Who was Pythagoras?
- What was important about
- his life?
- Why was he famous?
- What is the Pythagorean Theorem?
- How is his Theorem applied?

Pythagorass Roots and Routes

Homonyms are cool!

- Pythagoras lived from about 560 to 500 B.C.
- He was born on the Aegean island of Samos, but

traveled to Egypt, Babylon, India, and Syria to

further his studies. - Eventually, he settled in the Greek colony of

Croton in Southern Italy. (see map for details)

Birthplace of Pythagoras

Croton

- Croton was a Greek colony located on the southern

tip of Italy. - In this colony, Pythagoras established the

Pythagorean School.

Syria

- Pythagoras traveled widely as a child with his

father who was a merchant. - It was in Syria that he was tutored by the

Chaldean.

Egypt

- It was here that Pythagoras studied the pyramids

to learn more about Geometry. - His studies helped him to develop his theorem.

Babylon

- It is believed that Pythagorass travel to

Babylon advanced his knowledge in astronomy

because of their expertise at the time. - The Pythagoreans were the first to use the term

cosmos to mean an orderly universe that could

be expressed in numbers. Certain numbers were

given mystical or holy values.

India

- Because of his time spent in India, Pythagoras

adopted the Hindu belief that souls transmigrate

after death.

Important Contributions

- Pythagoras found mathematical relationships as

the essence of all things. - He believed that the earth was spherical and that

the sun, moon, and planets have movements of

their own. This contributed to the discovery

that the earth revolved around the sun. - Hes credited with developing the Pythagorean

Theorem which is a formula that is used to

determine the lengths of the sides of right

triangles. - His discoveries laid the foundation for all

subsequent developments in Geometry.

The Pythagorean Order

- In Croton, Pythagoras set up a school which was

based on a religious order, or brotherhood, that

was made up of his followers. - It was known as the Pythagorean Order, and they

were expected to not eat meat or drink alcohol.

They also had to wear simple clothing and go

barefoot.

The Pythagorean Order

- Students at the Pythagorean School had to live by

strict rules since they all strove to create pure

minds and bodies. - They all thought that the mind could be purified

through studying Geometry, mathematics, music,

and astronomy.

The Pythagorean Order

- Women were allowed to join the Order, but they

were kept separated from the rest of the Order.

Principles of Pythagoras

- At its deepest level, reality is mathematical in

nature. - Philosophy can be used for spiritual

purification. - The soul can rise to union with the divine.
- Certain symbols have a mystical significance.
- All brothers and sisters of the Order should

observe strict loyalty and secrecy.

Pythagorean TheoremWhat does the formula mean?

- a² b² c² means that in a right the sum

of the squares of the lengths of two legs equals

the square of the length of the hypotenuse.

c

a

For a more in-depth look at how this formula was

derived, check out this site http//forum.swarth

more.edu/sarah/hamilton/ham.rttriangles.html

b

There are Many Types of TrianglesCan you name

them?Click on the names for the answers.

- Scalene
- Isosceles
- Equilateral
- Acute
- Obtuse
- Equiangular Triangle
- Right Triangle

What is a Right Triangle?

- The Pythagorean Theorem applies only to right

triangles. - A right triangle is a triangle that has a 90

degree right angle. It has two legs and a

hypotenuse. - The hypotenuse is the side opposite the right

angle and is always the longest. - The variables a b are used for the legs and c

is the variable for the hypotenuse.

a

c

b

Lets See How it Works!Find the Hypotenuse

Question If a 3 and b 4, then what does

c equal?

- a² b² c²
- (3)2 (4)2 c²
- (3x3) (4x4) c²
- 9 16 c²
- 25 c2
- ?25 ?c²
- 5 c

a

b

How to Find a Leg

If a 5 and c 13, then what does b equal?

- a² b² c²
- 52 b² 13²
- (5x5) b² (13x13)
- 25 b² 169
- 25 b² - 25 169 - 25
- b² 144
- ?b² ?144
- b 12

Now You Can Try!

- . Find the missing hypotenuse c
- a 6
- b 8
- 2. Find the missing leg a
- c 26
- b 24

c

6

8

26

a

24

Answers!

- a² b² c²
- (6)2 (8)2 c²
- (6x6) (8x8) c²
- 36 64 c²
- 100 c²
- ? ?100 ?c²
- 10 c

Answers!

- a² b² c²
- a² (24)2 (26)2
- a² (24x24) (26x26)
- a² 576 676
- a² 576 - 576 676 - 576
- a² 100
- ? a² ? 100
- a 10

More Practice

- Middle School Math Course 3 by Scott Foresman -

Addison Wesley - (Textbook) Exercises and Applications 1-16,

pages 377 and 378. - (Practice Workbook) Practice 7-9, page 73.
- Buckle Down on Virginia Mathematics, SOL 8, 2nd

Edition by Buckle Down Publishing Co. - Lesson 7, page 67-70

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