Download presentation
Presentation is loading. Please wait.
Published byValentine Sullivan Modified over 9 years ago
1
HKDSE Mathematics Ronald Hui Tak Sun Secondary School
2
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!)
3
8 September 2015Ronald HUI Chapter 1 Properties of Circles Properties of Circles What is a circle? What is a circle?
4
4 September 2015Ronald HUI
5
Book 5A Chapter 1 Angles in a Circle
6
Angle at the Centre O A B x The vertex of angle x lies at the centre. P
7
O A B x We call x the angle at the centre subtended by AB. P Angle at the Centre
8
O A B x Similarly, the vertex of angle y lies at the centre. y P Angle at the Centre
9
O A B x y P We call y the angle at the centre subtended by APB.
10
In each of the following, is x the angle at the centre? Yes B C O x B O x A
11
Angle at the Circumference A B The vertex of angle c lies on the circumference. D C c
12
We call c the angle at the circumference subtended by AB. Angle at the Circumference A B c D C
13
The vertex of angle d lies on the circumference. Angle at the Circumference A B c D C d
14
A B c D C d We call d the angle at the circumference subtended by. ACB
15
In each of the following, is y the angle at the circumference? A B O y No Yes A B y C
16
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
17
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
18
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 °
19
P APB = AOB = 90 at centre twice at ce = 180 then AOB = 180 . 180 O AB if AB is a diameter, In particular,
20
if AB is a diameter, then AOB = 180 . In particular, APB is called an angle in a semi-circle. P 180 O AB
21
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
22
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
23
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.
24
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.
25
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:
26
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?
27
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
28
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
29
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
30
8 September 2015Ronald HUI Chapter 1 SHW1-B1 SHW1-B1 Due date? Due date?
Similar presentations
© 2024 SlidePlayer.com. Inc.
All rights reserved.