3. St. Kentigerns Academy

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

5. Pythagoras was a Greek philosopher and religious leader. He was responsible for many important developments in maths, astronomy, and music. Pythagoras (~560-480 B.C.)

6. 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. The Secret Brotherhood

7. 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! The Secret Brotherhood

8. A right angled triangle A Pythagorean Puzzle Ask for the worksheet and try this for yourself!

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

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

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

12. A Pythagorean Puzzle c² b² a²

13. A Pythagorean Puzzle

14. 1 

15. 1 2 

16. 1 2 

17. 1 2 3 

18. 1 2 3 

19. 1 23 4 

20. 1 23 4

21. 1 23 4 5

22. 1 23 4 5 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. A Pythagorean Puzzle

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

24. 1 23 4 5 Put the pieces back where they came from.

25. 1 23 4 5 A Pythagorean Puzzle Put the pieces back where they came from.

26. 1 23 4 5 A Pythagorean Puzzle Put the pieces back where they came from.

27. 1 23 4 5 A Pythagorean Puzzle Put the pieces back where they came from.

28. 1 23 4 5 A Pythagorean Puzzle Put the pieces back where they came from.

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

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

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

32. c a c c b b b a a c y a c²=a²+b²

33. 1m 8m Using Pythagoras’ Theorem What is the length of the slope?

34. 1m 8m c b= a= c²=a²+ b² c²=1²+ 8² c²=1 + 64 c²=65 ? Using Pythagoras’ Theorem

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

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

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

38. Now try Exercise 4 P156 Then Exercise 5 Problems involving Pythagoras Theorem

39. 7m 5m h c a b c²=a²+ b² 7²=a²+ 5² 49=a² + 25 ? Finding the shorter side

40. 49 = a² + 25 We need to get a² on its own. Remember, change side, change sign! Finding the shorter side + 25 49 - 25 = a² a²= 24 a = √ 24 = 4.9 m (1 d.p.)

41. 169 = w² + 36 c w 6m 13m a b c²= a²+ b² 13²= a²+ 6² 169 – 36 = a² a = √ 133 = 11.5m (1 d.p.) a²= 133 Example 1 169 = a² + 36 Change side, change sign!

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

43. Now try Exercise 6* P 161

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

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

46. Now try Exercise 2 Questions 1 to 5