Pythagoras

2. Pythagoras

3. Menu Brief History A Pythagorean Puzzle Pythagoras’ Theorem Using Pythagoras’ Theorem Finding the shorter side Further examples

4. Pythagoras (~560-480 B.C.) Pythagoras was a Greek philosopher and religious leader. He was responsible for many important developments inmaths, astronomy, andmusic.

5. The Secret Brotherhood His students formed a secret society called the Pythagoreans. As well as studying maths, they were a political and religious organisation. Members could be identified by a five pointed star they wore on their clothes.

6. The Secret Brotherhood They had to follow some unusual rules. They were not allowed to wear wool, drink wine or pick up anything they had dropped! Eating beans was also strictly forbidden!

7. A Pythagorean Puzzle A right angled triangle

8. A Pythagorean Puzzle Draw a square on each side.

9. A Pythagorean Puzzle Measure the length of each side c b a

10. A Pythagorean Puzzle Work out the area of each square. C² c b b² a a²

11. A Pythagorean Puzzle c² b² a²

12. A Pythagorean Puzzle

13. 1 A Pythagorean Puzzle 

14. 1 A Pythagorean Puzzle 2 

15. 1 A Pythagorean Puzzle  2

16. 2 1 A Pythagorean Puzzle  3

17. 2 1 A Pythagorean Puzzle  3

18. 2 3 1 A Pythagorean Puzzle  4

19. 2 3 1 A Pythagorean Puzzle 4

20. 4 2 3 1 A Pythagorean Puzzle 5

21. 5 4 2 3 1 A Pythagorean Puzzle What does this tell you about the areas of the three squares? The red square and the yellow square together cover the green square exactly. The square on the longest side is equal in area to the sum of the squares on the other two sides.

22. 5 4 2 3 1 A Pythagorean Puzzle Put the pieces back where they came from.

23. 5 4 2 3 1 A Pythagorean Puzzle Put the pieces back where they came from.

24. 5 4 2 3 1 A Pythagorean Puzzle Put the pieces back where they came from.

25. 5 4 2 3 1 A Pythagorean Puzzle Put the pieces back where they came from.

26. 5 4 2 3 1 A Pythagorean Puzzle Put the pieces back where they came from.

27. 5 4 2 3 1 A Pythagorean Puzzle Put the pieces back where they came from.

28. A Pythagorean Puzzle c² b² c²=a²+b² This is called Pythagoras’ Theorem. a²

29. Pythagoras’ Theorem This is the name of Pythagoras’ most famous discovery. It only works with right-angled triangles. The longest side, which is always opposite the right-angle, has a special name: hypotenuse

30. Pythagoras’ Theorem c a b c²=a²+b²

31. c²=a²+b² Pythagoras’ Theorem c c b a y a a b b a c c

32. Using Pythagoras’ Theorem 1m 8m What is the length of the slope?

33. Using Pythagoras’ Theorem c a= 1m b= 8m c²=a²+ b² ? c²=1²+ 8² c²=1 + 64 c²=65

34. We need to use the square root button on the calculator. √ √ Using Pythagoras’ Theorem c²=65 How do we find c? It looks like this = , Enter65 Press So c= √65 = 8.1 m (1 d.p.)

35. 9cm 12cm Example 1 c c²=a²+ b² b c²=12²+ 9² a c²=144 + 81 c²= 225 c = √225= 15cm

36. 4m 6m s Example 2 a b c²=a²+ b² s²=4²+ 6² c s²=16 + 36 s²= 52 s = √52 =7.2m (1 d.p.)

37. 7m h 5m Finding the shorter side c c²=a²+ b² 7²=a²+ 5² a 49=a² + 25 ? b

38. + 25 Finding the shorter side 49 = a² + 25 We need to get a² on its own. Remember, change side, change sign! 49 - 25= a² a²= 24 a = √24 = 4.9 m (1 d.p.)

39. 13m 6m 169 = a² + 36 w Change side, change sign! Example 1 c c²= a²+ b² 13²= a²+ 6² b 169 = w² + 36 a 169 – 36 = a² a²= 133 a = √133 = 11.5m (1 d.p.)

40. Q P 9cm 81 11cm Change side, change sign! R Example 2 c²= a²+ b² b 11²= 9²+ b² a 121 = 81 + b² c 121 – 81 = b² b²= 40 b = √40 = 6.3cm (1 d.p.)

41. r 5m 14m ½ of 14 Example 1 c²=a²+ b² c²=5²+ 7² c²=25 + 49 c c²= 74 r 5m b c = √74 =8.6m (1 d.p.) ? 7m a

42. 38cm p 23cm + 529 Change side, change sign! Example 2 c²= a²+ b² 38²= a²+ 23² 1444 = a²+ 529 c 1444 – 529 = y² 38cm a²= 915 a a = √915 So a =2 x √915 = 60.5cm (1d.p.) 23cm b

43. HOMEWORK