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1. Circle Properties

2. Circle Properties • In this unit you will learn about: • The names of the parts of a circle • Inscribed regular polygons • The geometry of tangents and chords

3. Circle Definitions radius • The radius is the distance from the centre to the edge. X X X • A circle has the special property that all the points on the outside are the same distance from the centre. X X X X X X X X

4. Circle Definitions • A line joining two points on the circle is called a chord. • If the line also passes through the centre, it is called the diameter. chord X diameter

5. Circle Definitions • The distance round the circle is the circumference. • A part of the circumference is an arc. • A ‘pie slice’ of the circle is called a sector sector arc

6. Chords and Tangents Imagine a line moving towards a circle. At first, the line just touches the circle. This is called a tangent. Then, the line crosses the circle and now has two points where it intersects. It is now a chord. X

7. Chords and Tangents The chord splits the circle into two segments – a major segment and a minor segment. Minor segment Major segment X

8. Useful formulae • Area of circle = p r2 • Circumference =  d • Area of sector = q x p r2 360 • Length of arc = q x p d 360

9. Circle Properties

10. Circle Property Proofs and Definitions Isosceles triangles + perp. bisectors 1 Proof Definition Angle at center and at circumference 2 Proof Definition Angle in a semi circle 3 Proof Definition Angles in the same segment 4 Proof Definition Opposite angles in a cyclic quadrilateral 5 Proof Definition Right angle between a tangent and radius 6 Proof Definition Tangential lines the same length 7 Proof Definition Alternate segment theorem 8 Proof Definition

11. Measure the angles What do you notice?

12. Triangles in circles

13. Circle Property 1 Triangles formed using two radii will form an isosceles triangle. The perpendicular bisector of a chord passes through the the centre of the circle. Remember to spot isosceles triangles and perpendicular bisectors in circle diagrams.

14. x Circle Property Example 2 x = 200 Property 1 1400 x

15. x Circle Property Example 3 x = 540 Property 1 x 630

16. Circle Property 2 The angle subtended by an arc at the centre of a circle, is twice the angle subtended at the circumference. x x 2x The angle at the centre is half the angle on the circumference.

17. Circle Property 2 a a 2b x 2a b b Using Property 1 we can label the angles at the centre. Full Version of Property

18. x Circle Property Example 4 x = 420 Property 2 x 840

19. x Circle Property Example 5 x = 700 Property 2 350 x y = 550 y Property 1

20. x Circle Property Example 7 x = 2600 Property 2 x 1300

21. x Circle Property Example 9 x = 860 Property 2 430 x

22. Circle Property 3 Any angle subtended on the circumference of a semi- circle will be a right angle angle. x 1800 The angle in a semi-circle is a right angle.

23. Circle Property 3 If PQ is a diameter then the angle at the centre is 1800. Using Property 1 ... The angle on the circumference is half of 1800 = 900. a P x 1800 Q The angle in a semi-circle is a right angle.

24. x Circle Property Example 14 x = 420 Property 3 x 150 480 y y = 750 Property 3

25. Circle Property 4 a a b a and b are both “subtended” by the same chord Bitesize

26. Circle Property 4 a a b x x 2a 2a If a is half the angle at the centre then so is b. So b = a. Full Version of Property

27. x Circle Property Example 16 x = 350 x Property 4 350 y y = 700 Property 2

28. x Circle Property Example 17 x = 680 Property 4 y y = 1360 680 x Property 2

29. x Circle Property Example 18 x = 430 x Property 4 430 860 y = 470 y Property 4 + 1

30. x Circle Property Example 19 x = 120 y Property 4 340 x 120 y = 340 Property 4

31. x Circle Property Example 20 x = 700 Property 4 y 640 700 y = 640 x Property 4

32. Circle Property 5 Opposite angles in a cyclic quadrilateral add up to 1800. (A cyclic quadrilateral is a 4 sided shape with all four points on the circumference of a circle.) a x b Opposite angles in a cyclic quadrilateral add to 1800.

33. x Circle Property Example 25 x = 1600 Property 2 x 800 y y = 1000 Property 5

34. x Circle Property Example 26 x = 730 Property 2 x 1460 y y = 1070 Property 5

35. x Circle Property Example 27 x = 830 Property 2 1660 x y y = 970 Property 5

36. x Circle Property Example 28 x = 950 Property 2 1900 x y y = 850 Property 5

37. x Circle Property Example 29 x = 500 x Property 5 y y = 1000 1300 Property 2

38. x Circle Property Example 30 x = 700 x Property 5 y y = 2200 1100 Property 2

39. Exercise 31.1 Page 333

40. Circle Property 6 Q The tangent to a circle is perpendicular to the radius drawn at the point of contact. OPQ = 900 P x O A tangent to a circle is at right angles to its radius.

41. Circle Property Example 33 x = 400 x Property 6 500 x y = 250 y Property 2

42. Circle Property 7 Q Two tangents drawn to a circle from the same point are equal in length. QP = QR QP = QR P x O R The tangents drawn from a point to a circle are equal in length.

43. x Circle Property Example 36 x = 1240 y Property 2 x y = 560 620 Property 6 + 7

44. Circle Property 8 The angle between a tangent and a chord drawn at a point of contact is equal to any angle in the alternate segment. y x x z y The angle between a tangent and a chord is equal to the angle in the alternate segment.

45. Circle Property 8 The angle between a tangent and a chord drawn at a point of contact is equal to any angle in the alternate segment. y x x z y The angle between a tangent and a chord is equal to the angle in the alternate segment.

46. Circle Property 8 This property is called the alternate segment theorem and states that the angle a is equal to the angle b b x a The angle between a tangent and a chord is equal to the angle in the alternate segment. Full Version of Property

47. Circle Property Example 41 x = 700 Property 8 x z y = 600 y 500 Property Triangle 700 z = 1200 Property 8

48. Circle Property Example 42 x = 1200 Property 1 y x y = 600 x 300 Property 2 z z = 600 Property 8

49. Circle Property Example 44 x = 500 Property 8 x x y = 1000 y Property 2 z 500 y = 400 Property 1