Circle

1
Circle
Form 4 Chapter Four
Software, Geometers Sketchpad, is required.
2
Basic Terms
Centre
Circumference
Radius
Diameter
3
Basic Terms
For any two points A and B,
Major Arc
Can the two arcs be equal ?
the circle is divided into two parts -------
Minor Arc
4
Basic Terms
When AB is the diameter,
A
two arcs are equal.
Semi-circle
B
5
Basic Terms
For any two points A and B,
the region enclosed by radii OA, OB
Major Sector
and the arc AB is called a
O
B
Sector
Minor Sector
A
6
Basic Terms
For any two points A and B,
the line joining A, B is called a
CHORD
7
Basic Terms
A chord also divides the circle into two regions
---
Major Segment
CHORD
Minor Segment
8
Basic Terms
D
We can have different chords.
P
Minimum 0
C
Maximum length of diameter
B
Minimum length ? Maximum length ?
Q
A
9
Review
Circumference
Centre
Radius
Diameter
O
Chord
10
Review
Arc
Sector
Chord
Segment
O
11
Chord
P
Distance of a point P from a line
perpendicular distance PN
N
12
Chord
If
(1) AB is a chord
(2) ON ? AB
Then
O
AN BN
N
Reason line from centre ? chord bisects chord
13
Chord (example)
If AB 12,
OF 8,
A
OF ? AB,
O
find radius.
F
B
14
AF 6
(line from centre ? chord bisects chord)
radius2 AF2 OF2
A
(Pythagoras thm)
O
?radius ?(62 82)
F
10
B
15
Chord
If
(1) AB is a chord
(2) M is mid-point
Then
O
OM ? AB
M
Reason line joining centre to mid-pt of chord ?
chord
16
Chord (example)
B
If AN NB 4,
N
radius 5,
O
A
find distance of centre from AB
17
If AN NB 4,
?ANO 90
(line joining centre to mid-pt of chord ? chord)
radius 5,
B
N
Distance
O
A
ON
?(radius2 - AN2)
?(52 - 42) 3
18
Chord
O
Any perpendicular bisector of chord passes
through the centre.
19
Chord
C
O
D
Any perpendicular bisector of chord passes
through the centre.
20
Chord (example)
B
Let A, B and C be 3 points on a circle.
How to find the centre and draw the circle ?
A
C
21
Chord (example)
B
Joint AB and BC,
their perpendicular bisectors intersect at the
centre, O.
O
A
Take OA as radius
to draw the circle.
C
22
Chord
AB,CD are chords
B
OM,ON are perpendiculars
D
M
If AB ? CD , then
A
O
N
OM ? ON
C
23
Chord
If AB CD, then ...
B
M
OM ON
A
O
C
N
Reason equal chords, equidistant from centre
D
24
Chord
If OM 3 , ON 2
B
then
D
M
AB CD ?
A
O
N
C
25
Chord
If, OM ON then ...
B
M
AB CD
A
O
C
N
Reason chords equidistant from centre are equal
D
26
line from centre ? chord bisects chord
line joining centre to mid-pt of chord ? chord
equal chords, equidistant from centre
chords equidistant from centre are equal
27
END