1. Ronald Hui Tak Sun Secondary School
HKDSE Mathematics
Ronald Hui
Tak Sun Secondary School

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

3. Ronald HUI
14 September 2015

4. Ronald HUI
14 September 2015

5. Ronald HUI
14 September 2015

6. Relationships among Arcs, Chords and Angles

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

8. Arcs Proportional to Angles at the Centre
In the figure, OAB is a sector of the circle.
If we duplicate sector OAB four times like this: D
40
40 R O A B
40
40 Q P C
40

9. Arcs Proportional to Angles at the Centre
In the figure, OAB is a sector of the circle.
If we duplicate sector OAB four times like this: D
160
∠COD = 160
Obviously, we have: O A B
40
AB : CD = 1 : 4
and
∠AOB : ∠COD = 1 : 4
i.e. AB : CD = ∠AOB : ∠COD C

10. Theorem 1.16
In a circle, arcs are proportional to their corresponding angles at the centre. O x y D C B A y AB
: CD
= x
: y x
Abbreviation:
arcs prop. to s at centre

11. Yes, we can extend Theorem 1.16 as follows.
Arcs Proportional to Angles at the
Circumference
Are arcs proportional to their corresponding angles at the circumference?
Yes, we can extend Theorem 1.16 as follows.

12. By using ‘ at centre twice  at ce’, we can show that
Arcs Proportional to Angles at the
Circumference
By using ‘ at centre twice  at ce’, we can show that
AB : CD = m : n. n D C B A m P Q O

13. Arcs Proportional to Angles at the Circumference
Let ∠AOB = x and ∠COD = y. n D C B A m P Q
x = 2m  at centre
twice  at ce
y = 2n  at centre
twice  at ce O
AB : CD = x : y arcs prop. to
s at centre x y
= 2m : 2n
= m : n

14. Theorem 1.17
In a circle, arcs are proportional to their corresponding angles at the circumference. n D C B A m P Q AB
: CD
= m
: n n m
Abbreviation:
arcs prop. to s at ce

15. arcs prop. to s at centre
Example:
Find x in the figure. A B C D
80
11 cm
8 cm x O
arcs prop. to s at centre ∴

16. Follow-up question Find x in the figure. ∠CAD = 40°
arcs prop. to s at ce
Find x in the figure. D
10
8 cm A
40° x
ext.  of △
∠CAD = 40°
2 cm K
In △AKD, C B
AKC and BKD are straight lines.

17. Ronald HUI
14 September 2015

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