HKDSE Mathematics Ronald Hui Tak Sun Secondary School

8 September 2015Ronald HUI Missing Homework SHW1-01 SHW1-01 Missing One Missing One Summer Holiday Homework Summer Holiday Homework 25 Sep (Fri) 25 Sep (Fri) SHW1-A1 SHW1-A1 10 Sep (Today!) 10 Sep (Today!)

8 September 2015Ronald HUI Chapter 1 Properties of Circles Properties of Circles What is a circle? What is a circle?

4 September 2015Ronald HUI

Book 5A Chapter 1 Angles in a Circle

Angle at the Centre O A B x The vertex of angle x lies at the centre. P

O A B x We call x the angle at the centre subtended by AB. P Angle at the Centre

O A B x Similarly, the vertex of angle y lies at the centre. y P Angle at the Centre

O A B x y P We call y the angle at the centre subtended by APB.

In each of the following, is x the angle at the centre? Yes B C O x B O x A

Angle at the Circumference A B The vertex of angle c lies on the circumference. D C c

We call c the angle at the circumference subtended by AB. Angle at the Circumference A B c D C

The vertex of angle d lies on the circumference. Angle at the Circumference A B c D C d

A B c D C d We call d the angle at the circumference subtended by. ACB

In each of the following, is y the angle at the circumference? A B O y No Yes A B y C

A B P Q O There is a relationship between the angle at the centre and the angle at the circumference. In △ OPA, ∵ OA = OP ∴  OAP = a  AOQ = a + a = 2a radii base  s, isos. △ ext.  of △ a a 2a2a 2b2b b b Similarly,  BOQ = 2b  AOB = 2a + 2b = 2  APB

Theorem 1.6 Abbreviation:  at centre twice  at  ce The angle at the centre of a circle subtended by an arc is twice the angle at the circumference subtended by the same arc. q = 2p i.e. O A B P q p O A B P q p O A B P q p q = 2p

Follow-up question O A B Find x in the figure. 210  x P  at centre twice  at  ce  AOB = 360  – 210  = 150   s at a pt. = 75  150 °

P  APB =  AOB = 90   at centre twice  at  ce =  180  then  AOB = 180 . 180  O AB if AB is a diameter, In particular,

if AB is a diameter, then  AOB = 180 . In particular,  APB is called an angle in a semi-circle. P 180  O AB

Theorem 1.8 (Converse of Theorem 1.7) O If AB is a diameter, A B then  APB = 90  P Abbreviation:  in semi-circle Theorem 1.7 then AB is a diameter. A B If  APB = 90  P Abbreviation: converse of  in semi-circle

Follow-up question O A B Find x in the figure. 30  x C  in semi-circle x = 180    ACB  ABC = 180   90   30   sum of △  ACB = 90  = 60 

Angles in the Same Segment In the figure, ∠ AEB lies in the segment APB. B A E P ∠ AEB is an angle in segment APB.

B A P C D E These angles all lie in the major segment APB, and are called angles in the same segment. For example: Angles in the Same Segment In segment APB, we can draw many angles at circumference subtended by the same arc AB.

B A Q R Similarly, in segment AQB, we can draw many angles at circumference subtended by the same arc. T S P Angles in the Same Segment These angles all lie in the minor segment AQB, and are also called angles in the same segment. For example:

P A B y Q x Yes P A B Q x y P A B Q x y No In each of the following, are x and y angles in the same segment?

By considering  AOB, we can show that x = y. Is there any relationship among angles in the same segment? x O y P Q A B

Abbreviation:  s in the same segment Theorem 1.9 The angles in the same segment of a circle are equal. i.e.x = y x y P Q A B

Follow-up question P A B Find  PQB in the figure. 60  Q 40   s in the same segment  AQP = 40   AQB = 60   PQB =  AQB +  AQP = 60  + 40  = 100 

8 September 2015Ronald HUI Chapter 1 SHW1-B1 SHW1-B1 Due date? Due date?