Pythagoras and the

Pythagorean TheoremGrade 8-9 Lesson

By Lindsay Kallish

Biography of Pythagoras

•Pythagoras was a Greek mathematician and a philosopher, but was best known for his Pythagorean Theorem.

•He was born around 572 B.C. on the island of Samos.

• For about 22 years, Pythagoras spent time traveling though Egypt and Babylonia to educate himself.

•At about 530 B.C., he settled in a Greek town in southern Italy called Crotona.

•Pythagoras formed a brotherhood that was an exclusive society devoted to moral, political and social life. This society was known as Pythagoreans.

Biography of Pythagoras• The Pythagorean School excelled in many

subjects, such as music, medicine and mathematics.

• In the society, members were known as mathematikoi.

• History tells us that this theorem has been introduced through drawings, texts, legends, and stories from Babylon, Egypt, and China, dating back to 1800-1500 B.C.

• Unfortunately, no one is sure who the true founder of the Pythagorean Theorem is. But it does seem certain through many history books that some time in the sixth century B.C., Pythagoras derives a proof for the Pythagorean Theorem.

Venn Diagram Homework Assignment• http://www.arcytech.org/java/pythagoras/history.ht

ml• http://www-groups.dcs.st-and.ac.uk/~history/Biogra

phies/Pythagoras.html

1st Website

2nd Website

Can Any Three Numbers Make A Triangle????Side lengths (cm) Area of square (cm²) Sum of the

areas of smaller squares (cm²)

Sum of the areas of smaller squares (cm²)

Angle name of the triangle

Column 1 Column 2 Column 3 Column 4 Column 5

a b c a² b² c² a² + b² a² +b² ? c² (>,<,=)

Acute, right, obtuse

3 5 7 9 25 49 34 < obtuse

3 4 5 9 16 25 25 = right

Write a sentence to describe the relationship between the sum of the areas of smaller and middle size squares compared to the area of the largest squares for:

1) An acute triangle 2) A right triangle 3) An obtuse triangle

The Pythagorean Theorem• The sum of the squares of

each leg of a right angled triangle equals to the square of the hypotenuse

a² + b² = c²

Many Proofs of the Pythagorean TheoremEuclidean Proof• First of all, ΔABF = ΔAEC by SAS. This is

because, AE = AB, AF = AC, andBAF = BAC + CAF = CAB + BAE = CAE.

• ΔABF has base AF and the altitude from B equal to AC. Its area therefore equals half that of square on the side AC.

• On the other hand, ΔAEC has AE and the altitude from C equal to AM, where M is the point of intersection of AB with the line CL parallel to AE.

• Thus the area of ΔAEC equals half that of the rectangle AELM. Which says that the area AC² of the square on side AC equals the area of the rectangle AELM.

• Similarly, the are BC² of the square on side BC equals that of rectangle BMLD. Finally, the two rectangles AELM and BMLD make up the square on the hypotenuse AB.

• QED• http://www.sunsite.ubc.ca/LivingMathematics/

V001N01/UBCExamples/Pythagoras/pythagoras.html

Many Proofs of the Pythagorean TheoremIndian Proof• Area of the original square is A = c²

• Looking at the first figure, the area of the large triangles is 4 (1/2)ab

• The area of the inner square is (b-a) ²

• Therefore the area of the original square is A=4(1/2)ab + (b-a) ²

• This equation can be worked out as 2ab + b² - 2ab + a² = b² + a²

• Since the square has the same area no matter how you find it, we conclude that

A = c² = a² + b²

Many Proofs of the Pythagorean Theorem• Throughout many texts, there are about

400 possible proofs of the Pythagorean Theorem known today.

• It is not a wonder that there is an abundance of proofs due to the fact that there are numerous claims of different authors to this significant geometric formula.

• Specifically looking at the Pythagorean Theorem, this unique mathematical discovery proves that there is a limitless amount of possibilities of algebraic and geometric associations with the single theorem.

• http://www.cut-the-knot.org/pythagoras/index.shtml

Connection to Technology

• Geometer’s SketchPad– Students can see the

Pythagorean Theorem work using special triangles with 45-45-90 degree angles and 30-60-90 degree angles

Pythagoras Board GameRules:• To begin, roll 2 dice. The person with the highest sum

goes first.

• To move on the board, roll both dice. Substitute the numbers on the dice into the Pythagorean Theorem for the lengths of the legs to find the value of the length of the hypotenuse.

• Using the Pythagorean Theorem a²+b²=c², a player moves around the board a distance that is the integral part of c.

• For example, if a 1 and a 2 were rolled, 1²+2²=c²; 1+4=c²; 5=c²; Since c = √5 or approximately 2.236, the play moves two spaces. Always round the value down.

• When the player lands on a ‘?’ space, a question card is drawn. If the player answers the question correctly, he or she can roll one die and advance the resulting number of places.

• Each player must go around the board twice to complete the game. A play must answer a ‘?’ card correctly to complete the game and become a Pythagorean

Pythagoras Board GameWhat are the lengths of the legs of a 30-60-90 degree triangle with a hypotenuse of length 10?Answer: 5 and 5√3

If you hiked 3 km west and the 4 km north, how far are you from your starting point?Answer: 5 km

The square of the ______ of a right triangle equals the sum of the squared of the lengths of the two legs.Answer: hypotenuse

Find the missing member of the Pythagorean triple (7, __,, 25).Answer: 24

What is the length of the legs in a 45-45-90 degree right triangle with hypotenuse of length √2?Answer: 1

Using a²+b²=c, find b if c = 10 and a = 6Answer: b=8²

True or false? Pythagoras lives circa A.D. 500Answer: false (500 B.C.)

Have the person to your left pick two numbers for the legs of a right triangle. Compute the hypotenuse

Can an isosceles triangle be a right triangle?Answer: yes

Pythagoras was of what nationality?Answer: Greek

Is (7, 8, 11) a Pythagorean triple?Answer: no

How do you spell Pythagoras?

The Pythagorean Theorem is applicable for what type of triangle?Answer: a right triangle

What is the name of the school that Pythagoras founded?Answer: The Pythagorean School

True or false? Pythagoras considered number to be the basis of creation?Answer: true

True or false? Pythagoras formulated the only proof of the Pythagorean Theorem?Answer: false (there are about 400 possible proofs)

References:DeLacy, E. A. (1963). Euclid and geometry

(2nd ed.). USA: Franklin Watts, Inc. Ericksen, D., Stasiuk, J., & Frank, M. (1995).

Bringing pythagoras to life. The Mathematics Teacher, 88(9), 744.

Gow, J. (1968). A short story of greek mathematics. New York: Chelsea Publishing Company.

Katz, V. (1993). A history of mathematics (2nd ed.). USA: Addison Wesley Longman, Inc.

Swetz, F. J., & Kao, T. I. (1977). Was pythagoras chinese? an examination of right triangle theory in ancient china. USA: The Pennsylvania State University.

Veljan, D. (2000). The 2500-year-old pythagorean theorem. Mathematics Magazine, 73(4), 259.