1. Starter Draw a right angled triangle, where the two shorter sides are 7cm and 13cm, and measure the hypotenuse 7cm 13cm ?

2. Right-angled triangles

3. Using just the sides:Using sides and angles: Right-angled triangles

4. Using just the sides: Pythagoras’ theorem Using sides and angles: Right-angled triangles

5. Using just the sides: Pythagoras’ theorem Using sides and angles: Trigonometry Right-angled triangles

6. Using just the sides: Pythagoras’ theorem The theorem Using sides and angles: Trigonometry Right-angled triangles

7. Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Using sides and angles: Trigonometry Right-angled triangles

8. Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Using sides and angles: Trigonometry Right-angled triangles

9. Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Using sides and angles: Trigonometry Right-angled triangles

10. Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Pythagoras and the circle Using sides and angles: Trigonometry Right-angled triangles

11. Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Pythagoras and the circle Using sides and angles: Trigonometry Sine, cosine and tangent Right-angled triangles

12. Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Pythagoras and the circle Using sides and angles: Trigonometry Sine, cosine and tangent Finding an angle Right-angled triangles

13. Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Pythagoras and the circle Using sides and angles: Trigonometry Sine, cosine and tangent Finding an angle Finding a side Right-angled triangles

14. Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Pythagoras and the circle Using sides and angles: Trigonometry Sine, cosine and tangent Finding an angle Finding a side Looking at the graphs Right-angled triangles

15. Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Pythagoras and the circle Using sides and angles: Trigonometry Sine, cosine and tangent Finding an angle Finding a side Looking at the graphs Trigonometry beyond 90° Right-angled triangles

16. Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Pythagoras and the circle Using sides and angles: Trigonometry Sine, cosine and tangent Finding an angle Finding a side Looking at the graphs Trigonometry beyond 90° Right-angled triangles

17. Pythagoras’ Theorem 5 13 12

18. 5 13 12 Pythagoras’ Theorem

19. 5 13 12 Pythagoras’ Theorem

20. 5 13 12 Pythagoras’ Theorem

21. 5 13 12 5 13 Pythagoras’ Theorem

22. 5 13 12 5 13 25 Pythagoras’ Theorem

23. 5 13 12 5 13 25 144 Pythagoras’ Theorem

24. 5 13 12 5 13 25 144 169 Pythagoras’ Theorem

25. 5 13 12 5 13 25 144 169 25 + 144 = 169 Pythagoras’ Theorem

26. a c b

27. a c b a2a2 b2b2 c2c2

28. a c b a2a2 b2b2 c2c2 a 2 + b 2 = c 2 Pythagoras’ Theorem

29. Finding the hypotenuse 4cm 3cm a 24cm 7cm b 13cm c

30. a 2 = 3 2 + 4 2 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse

31. a 2 = 3 2 + 4 2 a 2 = 9 + 16 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse

32. a 2 = 3 2 + 4 2 a 2 = 9 + 16 a 2 = 25 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse

33. a 2 = 3 2 + 4 2 a 2 = 9 + 16 a 2 = 25 a = 5 cm 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse

34. a 2 = 3 2 + 4 2 a 2 = 9 + 16 a 2 = 25 a = 5 cm b 2 = 24 2 + 7 2 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse

35. a 2 = 3 2 + 4 2 a 2 = 9 + 16 a 2 = 25 a = 5 cm b 2 = 24 2 + 7 2 b 2 = 576 + 49 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse

36. a 2 = 3 2 + 4 2 a 2 = 9 + 16 a 2 = 25 a = 5 cm b 2 = 24 2 + 7 2 b 2 = 576 + 49 b 2 = 625 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse

37. a 2 = 3 2 + 4 2 a 2 = 9 + 16 a 2 = 25 a = 5 cm b 2 = 24 2 + 7 2 b 2 = 576 + 49 b 2 = 625 b = 25 cm 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse

38. a 2 = 3 2 + 4 2 a 2 = 9 + 16 a 2 = 25 a = 5 cm b 2 = 24 2 + 7 2 b 2 = 576 + 49 b 2 = 625 b = 25 cm c 2 = 7 2 + 13 2 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse

39. a 2 = 3 2 + 4 2 a 2 = 9 + 16 a 2 = 25 a = 5 cm b 2 = 24 2 + 7 2 b 2 = 576 + 49 b 2 = 625 b = 25 cm c 2 = 7 2 + 13 2 c 2 = 49 + 169 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse

40. a 2 = 3 2 + 4 2 a 2 = 9 + 16 a 2 = 25 a = 5 cm b 2 = 24 2 + 7 2 b 2 = 576 + 49 b 2 = 625 b = 25 cm c 2 = 7 2 + 13 2 c 2 = 49 + 169 c 2 = 218 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse

41. a 2 = 3 2 + 4 2 a 2 = 9 + 16 a 2 = 25 a = 5 cm b 2 = 24 2 + 7 2 b 2 = 576 + 49 b 2 = 625 b = 25 cm c 2 = 7 2 + 13 2 c 2 = 49 + 169 c 2 = 218 c = 14.8 cm (to 1 d.p.) 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse

42. a 2 = 3 2 + 4 2 a 2 = 9 + 16 a 2 = 25 a = 5 cm b 2 = 24 2 + 7 2 b 2 = 576 + 49 b 2 = 625 b = 25 cm c 2 = 7 2 + 13 2 c 2 = 49 + 169 c 2 = 218 c = 14.8 cm (to 1 d.p.) Exercise 15B Page 278 Questions 1-4 Finding the hypotenuse

43. Finding a shorter side a 3cm 5cm 9cm b 11cm

44. 5 2 = a 2 + 3 2 a 3cm 5cm 9cm b 11cm Finding a shorter side

45. 5 2 = a 2 + 3 2 25 = a 2 + 9 a 3cm 5cm 9cm b 11cm Finding a shorter side

46. 5 2 = a 2 + 3 2 25 = a 2 + 9 a 2 = 16 a 3cm 5cm 9cm b 11cm Finding a shorter side

47. 5 2 = a 2 + 3 2 25 = a 2 + 9 a 2 = 16 a = 4 cm a 3cm 5cm 9cm b 11cm Finding a shorter side

48. 5 2 = a 2 + 3 2 25 = a 2 + 9 a 2 = 16 a = 4 cm 11 2 = b 2 + 9 2 a 3cm 5cm 9cm b 11cm Finding a shorter side

49. 5 2 = a 2 + 3 2 25 = a 2 + 9 a 2 = 16 a = 4 cm 11 2 = b 2 + 9 2 121 = b 2 + 81 a 3cm 5cm 9cm b 11cm Finding a shorter side

50. 5 2 = a 2 + 3 2 25 = a 2 + 9 a 2 = 16 a = 4 cm 11 2 = b 2 + 9 2 121 = b 2 + 81 b 2 = 40 a 3cm 5cm 9cm b 11cm Finding a shorter side

51. 5 2 = a 2 + 3 2 25 = a 2 + 9 a 2 = 16 a = 4 cm 11 2 = b 2 + 9 2 121 = b 2 + 81 b 2 = 40 b = 6.3 cm (to 1 d.p.) a 3cm 5cm 9cm b 11cm Finding a shorter side

52. 5 2 = a 2 + 3 2 25 = a 2 + 9 a 2 = 16 a = 4 cm 11 2 = b 2 + 9 2 121 = b 2 + 81 b 2 = 40 b = 6.3 cm (to 1 d.p.) Exercise 15C Page 280 Questions 1-6 Finding a shorter side

53. Pythagoras’ theorem applied twice a 6cm 5cm b 14cm

54. a 2 = 5 2 + 6 2 a 2 = 25 + 36 a 2 = 61 a = 7.8 cm a 6cm 5cm b 14cm Pythagoras’ theorem applied twice

55. a 2 = 5 2 + 6 2 a 2 = 25 + 36 a 2 = 61 a = 7.8 cm 14 2 = b 2 + 7.8 2 196 = b 2 + 61 b 2 = 135 b = 11.6 cm (to 1 d.p.) a 6cm 5cm b 14cm Pythagoras’ theorem applied twice

56. a 2 = 5 2 + 6 2 a 2 = 25 + 36 a 2 = 61 a = 7.8 cm 14 2 = b 2 + 7.8 2 196 = b 2 + 61 b 2 = 135 b = 11.6 cm (to 1 d.p.) a 6cm 5cm b 14cm Pythagoras’ theorem applied twice

57. a 2 = 5 2 + 6 2 a 2 = 25 + 36 a 2 = 61 a = 7.8 cm 14 2 = b 2 + 7.8 2 196 = b 2 + 61 b 2 = 135 b = 11.6 cm (to 1 d.p.) Exercise 15D Page 281 Question 1 Pythagoras’ theorem applied twice

58. The distance between two points What is the distance between (2, 3) and (11, -2) ?

59. The distance between two points

60. What is the distance between (2, 3) and (11, -2) ? The distance between two points

61. What is the distance between (2, 3) and (11, -2) ? The first number: How far apart are 2 & 11? The distance between two points

62. What is the distance between (2, 3) and (11, -2) ? The first number: How far apart are 2 & 11? 9 9 The distance between two points

63. What is the distance between (2, 3) and (11, -2) ? The first number: How far apart are 2 & 11? 9 The second number: How far apart are 3 & -2? 9 The distance between two points

64. What is the distance between (2, 3) and (11, -2) ? The first number: How far apart are 2 & 11? 9 The second number: How far apart are 3 & -2? 5 9 5 The distance between two points

65. x 2 = 9 2 + 5 2 x 2 = 81 + 25 x 2 = 106 x = 10.3 units What is the distance between (2, 3) and (11, -2) ? The first number: How far apart are 2 & 11? 9 The second number: How far apart are 3 & -2? 5 9 5 The distance between two points

66. x 2 = 9 2 + 5 2 x 2 = 81 + 25 x 2 = 106 x = 10.3 units What is the distance between (2, 3) and (11, -2) ? The first number: How far apart are 2 & 11? 9 The second number: How far apart are 3 & -2? 5 Exercise 15E Page 282 Questions 1-3 The distance between two points

67. Pythagoras’ theorem in 3- D

68. 4 5 8

69. 4 5 8 H E F G D C B A

70. 4 5 8 H E F G D C B A

71. 4 5 8 H E F G D C B A 5 4 E FG H EG 2 = 5 2 + 4 2 EG 2 = 25 + 16 EG 2 = 41 EG = 6.4 cm

72. Pythagoras’ theorem in 3- D 4 5 8 H E F G D C B A EG 2 = 5 2 + 4 2 EG 2 = 25 + 16 EG 2 = 41 EG = 6.4 cm A E G 8 6.4 AG 2 = 8 2 + 6.4 2 AG 2 = 64 + 41 AG 2 = 105 AG = 10.2 cm

73. Pythagoras’ theorem in 3- D 4 5 8 H E F G D C B A EG 2 = 5 2 + 4 2 EG 2 = 25 + 16 EG 2 = 41 EG = 6.4 cm A E G 8 6.4 AG 2 = 8 2 + 6.4 2 AG 2 = 64 + 41 AG 2 = 105 AG = 10.2 cm Note: 5 2 + 4 2 + 8 2 = 25 + 16 + 64 = 105

74. The equation of a circle

76. x 2 + y 2 = r 2