2. The Pythagorean Theorem Geometry Notes – 8.1 Mr. Belanger

3. History Pythagoras was born in the island of Samos in ancient Greece. There is no certainty regarding the exact year when he was born, but it is believed that it was around 570 BC That is about 2,570 years ago! Those were times when a person believed in superstitions and had strong beliefs in gods, spirits, and the mysterious. Religious cults were very popular in those times. Pythagoras of Samos

4. History When Pythagoras was about 18 years old he went to the island of Lesvos where he worked and learned from Anaximander, an astronomer and philosopher, and Thales of Miletus, a very wise philosopher and mathematician.

5. History Thales had visited Egypt and recommended that Pythagoras go to Egypt. Pythagoras arrived in Egypt around 547 BC when he was 23 years old. He stayed in Egypt for 21 years learning a variety of things including geometry from Egyptian priests. It was probably in Egypt where he learned the theorem that is now called by his name. By the time he was about 55 years old he returned to his native land and started a school on the island of Samos. However, because of the lack of students he decided to move to Croton in the south of Italy.

6. History In Croton he started a school which concentrated in the teaching and learning of Mathematics, Music, Philosophy, and Astronomy and their relationship with Religion. It is said that as many as 600 of the worthiest people in the city attended the school. The school reached its highest splendor around the year 490 BC. He taught the young to respect their elders and to develop their mind through learning. Calmness and gentleness were principles encouraged at the school. Pythagoreans became known for their close friendships and devotion to each other. More than anyone before him Pythagoras combined the spiritual teachings with the pursuit of knowledge and science.

7. The Dark Side Pythagoras also headed a cult known as “the secret brotherhood” that worshiped numbers and numerical relationships. They attempted to find mathematical explanations for music, the gods, the cosmos, etc. Pythagoras believed that all relations could be reduced to number relations. At some point Pythagoras was exiled from Croton and had to move to Tarentum. After 16 years he had to move again, this time to Metapontus where he lived four years before he died at the age of 99.

8. The Theorem

9. The Proof

10. The Converse If a Triangle has sides of lengths a, b, c and a 2 + b 2 =c 2, then the triangle is right.

11. Example 1 Finding C 8 6 c 6 2 + 8 2 = c 2 36 + 64 = c 2 100 = c 2 10 = c

12. Example 2 Finding a Leg 5 A 13 52 52 + A2 A2 = 13 2 25 + A 2 = 169 -25 A2 A2 = 144 A = 12

13. Steps: Finding a Hypotenuse: Square the numbers Add them Square root Finding a Leg: Square the numbers Subtract them Square root

14. Pythagorean Triples If a set of three numbers can be lengths of sides in a right triangle the set is called a Pythagorean triple. Examples: 6, 8, 10 or 5, 12, 13 Think about multiples of these… 3, 4, 5 9, 12, 15 10, 24, 26 12, 16, 20

15. Right, Obtuse, Acute??? Do the sides 6, 9, 10 make a right, obtuse or acute triangle? Check Pythagorean Theorem: 36 + 81 = 100??? No, 36 + 81 = 117 which is larger than 100, thus it must be an obtuse triangle. If the sum is smaller than c squared then it makes an acute triangle.