1. HKDSE Mathematics Ronald Hui Tak Sun Secondary School

2. 14 September 2015Ronald HUI Missing Homework SHW1-A1 SHW1-A1 10 Sep (Last week!) 10 Sep (Last week!) 10, 24 10, 24 SHW1-B1 SHW1-B1 14 Sep (Today!) 14 Sep (Today!) Summer Holiday Homework Summer Holiday Homework 25 Sep (Fri) 25 Sep (Fri)

3. 14 September 2015Ronald HUI

4. 14 September 2015Ronald HUI

5. Book 5A Chapter 1 Relationships among Arcs, Chords and Angles

6. Equal Arcs, Equal Chords and Equal Angles We have learnt that arc length. r O θ is the angle at the centre subtended by the arc. arc length

7. Equal Arcs, Equal Chords and Equal Angles The length of an arc depends on its angle (θ) at the centre and the radius (r) of the circle. r O θ is the angle at the centre subtended by the arc. arc length

8. Consider two arcs AB and CD. r O r A D B C If ∠ AOB = ∠ COD = θ, then, AB = CD.

9. Consider two arcs AB and CD. r O r A D B C then, ∠ AOB = ∠ COD. If AB = CD =, Can you explain the above two results? If ∠ AOB = ∠ COD = θ, then, AB = CD.

10. O A B C D x y If x = y, then AB = CD. Theorem 1.10 Abbreviation: equal  s, equal arcs From the results above, we have:

11. O A B C D x y If x = y, then AB = CD. Theorem 1.10 Abbreviation: equal  s, equal arcs Theorem 1.11 (Converse of Theorem 1.10) Abbreviation: equal arcs, equal  s From the results above, we have: If AB = CD, then x = y.

12. Are there any theorems relating to chords and angles at the centre subtended by the chords? Yes. Let’s consider this figure. O A D B C

13. O A D B C Consider △ OAB and △ OCD, with ∠ AOB = ∠ COD. Then, we have: OA = OC radii ∠ AOB = ∠ COD given OB = OD radii ∴ △ OAB △ OCD SAS Hence, AB = CD. corr. sides, △

14. O A D B C x y If x = y, then Theorem 1.12 Abbreviation: equal  s, equal chords AB = CD. In fact, the converse of Theorem 1.12 is also true.

15. O A D B C x y If x = y, then Theorem 1.12 Abbreviation: equal  s, equal chords AB = CD. x = y. Theorem 1.13 (Converse of Theorem 1.12) Abbreviation: equal chords, equal  s If AB = CD, then

16. If, is it true that AB = CD? CDAB = A B C D Yes, by Theorem 1.11 and Theorem 1.12, we have: O x y x = y Theorem 1.11 AB = CD Theorem 1.12 CDAB =

17. A B C D Theorem 1.15 (Converse of Theorem 1.14) AB = CD. Abbreviation: equal chords, equal arcs If AB = CD, then AB = CD. Theorem 1.14 Abbreviation: equal arcs, equal chords If, then AB =CD

18. Equal angles at the centre Equal arcs Equal chords equal  s, equal chords equal chords, equal  s equal  s, equal arcs equal arcs, equal  s equal arcs, equal chords equal chords, equal arcs We can summarize the relationships among angles at the centre, arcs and chords of a circle as follows:

19. Find x and y in the figure. C A B O x cm y 70  D 6 cm equal arcs, equal chords equal arcs, equal  s Example: CDAB = ∵ = 70  y and 6 = x ∴

20. Follow-up question In the figure, AOD is a straight line. AB = CD and  AOB = 32 . Find  BOC. ∵ CD = AB (given) O B A C D 32  = 32  ∴  COD =  AOB (equal chords, equal  s) (adj.  s on st. line) 32 

21. 14 September 2015Ronald HUI

22. 14 September 2015Ronald HUI Chapter 1 SQ1: 2/10 (Fri) SQ1: 2/10 (Fri) Revision Ex: 30/9 (Wed) Revision Ex: 30/9 (Wed) Time to work harder please!!! Time to work harder please!!!