Transcript Book 4 Chapter 7 Basic Properties of Circles (1)

7
Basic Properties of
Circles (1)
Case Study
7.1
Chords of a Circle
7.2
Angles of a Circle
7.3
Relationship among the Chords, Arcs and Angles
of a Circle
7.4
Basic Properties of a Cyclic Quadrilateral
Chapter Summary
Case Study
I found a fragment of a
circular plate. How can I
know its original size?
You need to find the centre
of the circular plate first.
In order to find the centre of the circular plate:
Step 1: Draw an arbitrary triangle inscribed in the
circular plate.
Step 2: Find the circumcentre of the triangle,
i.e., the centre of the circular plate, by
drawing 3 perpendicular bisectors.
P. 2
7.1 Chords of a Circle
A. Basic Terms of a Circle
Circle: closed curve in a plane where every point on the
curve is equidistant from a fixed point.
Centre: fixed point
Circumference: curve or the length of the curve
Chord: line segment with two end points on the circumference
Radius: line segment joining the centre to any point on
the circumference
Diameter: chord passing through the centre
Remarks:
1. The length of a radius is half that of a diameter.
2. A diameter is the longest chord in a circle.
P. 3
7.1 Chords of a Circle
A. Basic Terms of a Circle
Arc: portion of the circumference
(
minor arc (e.g. AYB)
shorter than half of the circumference

major arc (e.g. AXB)
longer than half of the circumference
(

Angle at the centre: angle subtended by an arc or
a chord at the centre
P. 4
7.1 Chords of a Circle
B. Chords a Circle
If we draw a chord AB on a circle and fold the paper as shown
below:
A, B coincide
Then the crease
 passes through the centre of the circle;
 is perpendicular to the chord AB;
 bisects the chord AB.
P. 5
7.1 Chords of a Circle
B. Chords a Circle
Properties about a perpendicular line from the centre to a chord:
1. Perpendicular Line from Centre to a Chord
Theorem 7.1
If a perpendicular line is drawn from a centre of
a circle to a chord, then it bisects the chord.
In other words, if OP  AB,
then AP  BP.
(Reference: line from centre  chord
bisects chord)
This theorem can be proved by considering
DAOP and DBOP.
P. 6
7.1 Chords of a Circle
B. Chords a Circle
The converse of Theorem 7.1 is also true.
Theorem 7.2
If a line is joined from the centre of a circle
to the mid-point of a chord, then it is
perpendicular to the chord.
In other words, if AP  BP,
then OP  AB.
(Reference: line from centre to mid-pt. of
chord  chord)
This theorem can be proved by considering
DOAP and DOBP.
P. 7
7.1 Chords of a Circle
B. Chords a Circle
From Theorem 7.1 and Theorem 7.2, we obtain an
important property of chords:
The perpendicular bisector of any chord of
a circle passes through the centre.
P. 8
7.1 Chords of a Circle
B. Chords a Circle
Example 7.1T
In the figure, O is the centre of the circle. AP  PB  5 cm
and OP  12 cm. Find PQ.
Solution:
∵ AP  PB
∴ OP  AB
(line from centre to mid-pt. of chord  chord)
In DOAP,
OA2  OP2  AP2 (Pyth. Theorem)
OA  122  52 cm
 13 cm
OQ  OA
(radii)
 13 cm
∴ PQ  (13 – 12) cm
 1 cm
P. 9
7.1 Chords of a Circle
B. Chords a Circle
Example 7.2T
In the figure, O is the centre of the circle. AOB is a straight
line and OM  BC. Show that DABC  DOBM.
Solution:
∵ OM  BC
∴ BM  MC
∴ BC : BM  2 : 1
∵ OB  OA
∴ AB : OB  2 : 1
OBM  ABC
∴ DABC  DOBM
(line from centre  chord bisects chord)
(radii)
(common  )
(ratio of 2 sides, inc.  )
P. 10
7.1 Chords of a Circle
B. Chords a Circle
Example 7.3T
In the figure, O is the centre and AB is a diameter of the circle.
AB  CD, PB  4 cm and CD  16 cm.
(a) Find the length of PC.
(b) Find the radius of the circle.
Solution:
(a) ∵ OB  CD
∴ PC  PD
(line from centre  chord bisects chord)
 8 cm
(b) Let r cm be the radius of the circle.
Then OC  r cm and OP  (r – 4) cm.
In DOCP,
OC2  OP2  PC2
(Pyth. Theorem)
r2  (r – 4)2 
8r  80
82
r  10
∴ The radius of the circle is 10 cm.
P. 11
7.1 Chords of a Circle
B. Chords a Circle
Properties about a perpendicular line from the centre to a chord:
2. Distance between Chords and Centre
Theorem 7.3
If the lengths of two chords are equal, then
they are equidistant from the centre.
In other words, if AB  CD,
then OP  OQ.
(Reference: equal chords, equidistant
from centre)
This theorem can be proved by considering
DOAP and DOCQ.
P. 12
7.1 Chords of a Circle
B. Chords a Circle
The converse of Theorem 7.3 is also true.
Theorem 7.4
If two chords are equidistant from the centre
of a circle, then their lengths are equal.
In other words, if OP  OQ,
then AB  CD.
(Reference: chords equidistant from centre
are equal)
This theorem can be proved by considering
DOAP and DOCQ.
P. 13
7.1 Chords of a Circle
B. Chords a Circle
Example 7.4T
In the figure, O is the centre of the circle. AB  CD,
AB  CD, OM  AB and ON  CD. If OP  6 cm,
find ON. (Give the answer in surd form.)
Solution:
∵ AB  CD
∴ OM  ON
(equal chords, equidistant from centre)
∵ All of the interior angles of the quadrilateral OMPN are
right angles and OM  ON.
∴ OMPN is a square.
In DONP,
OP2  ON2  NP2
 2ON2
ON  62  2 cm
 3 2 cm
(Pyth. Theorem)
P. 14
7.2 Angles of a Circle
A. The Angle at the Circumference
Angle at the circumference:
angle subtended by an arc (or a chord) at the circumference
Angle at the centre:
angle subtended by an arc (or a chord) at the centre
Relationship between these angles:
Theorem 7.5
In each of the above figures, the angle at the centre subtended
by an arc is twice the angle at the circumference subtended by
the same arc. This means that q  2f.
(Reference:  at the centre twice  at ⊙ce)
P. 15
7.2 Angles of a Circle
A. The Angle at the Circumference
This theorem can be proved by constructing a diameter PQ.
In the left semicircle:
Since OA  OP (radii), DAOP is isosceles.
∴ OAP  OPA  a.
Hence the exterior angle of AOQ  2a.
Similarly, in the right semicircle, BOQ  2b.
∵ q  2a  2b and f  a  b
∴ q  2f
P. 16
7.2 Angles of a Circle
A. The Angle at the Circumference
Example 7.5T
In the figure, AB and CD are two parallel chords of the circle with
centre O. BOD  70 and MDO  10. Find ODC.
Solution:
∵ BOD  2  BAD
∴ BAD  35
ODC  10  BAD
ODC  10  35
ODC  25
( at the centre twice  at ⊙ce)
(alt. s, AB // CD)
P. 17
7.2 Angles of a Circle
B. The Angle in a Semicircle
In the figure, if AB is a diameter of the circle with centre O,
then the arc APB is a semicircle and APB is called the
angle in a semicircle.
Since the angle at the centre AOB  180,
the angle at the circumference APB  90.
( at the centre twice  at ⊙ce)
Theorem 7.6
The angle in a semicircle is 90.
That is, if AB is a diameter,
then APB  90.
(Reference:  in semicircle)
Conversely, if APB  90,
then AB is a diameter.
(Reference: converse of  in semicircle)
P. 18
7.2 Angles of a Circle
B. The Angle in a Semicircle
Example 7.6T
In the figure, AP is a diameter of the circle with
centre O and AC  BC. If PCB  50, find
(a) PBC and
(b) APC.
Solution:
(a) Since AP is a diameter, ACP  90. ( in semicircle)
In DACB,
∵ AC  BC
∴ PAC  PBC
(base s, isos. D)
PAC  PBC  ACB  180
( sum of D)
2PBC  (90  50)  180
PBC  20
(b) APC  PBC  PCB
 70
(ext.  of D)
P. 19
7.2 Angles of a Circle
C. Angle in the Same Segment
Segment:
region enclosed by a chord and the corresponding
arc subtended by the chord
 Major segment APB
area greater than half of the circle
 Minor segment AQB
area less than half of the circle
Angles in the same segment:
angles subtended on the same side of a chord at the
circumference
Notes:
We can construct infinity many angles in the
same segment.
P. 20
7.2 Angles of a Circle
C. Angle in the Same Segment
The angles in the same segment of a circle are equal.
Theorem 7.7
The angles in the same segment of a circle
are equal, that is, if AB is a chord,
then APB  AQB.
(Reference: s in the same segment)
This theorem can be proved by considering the angle
at the centre.
P. 21
7.2 Angles of a Circle
C. Angle in the Same Segment
Example 7.7T
In the figure, AC and BD are two chords that intersect at P.
(a) Show that DABP  DDCP.
(b) If AP  8, BP  12 and PC  6, find PD.
Solution:
(a) In DABP and DDCP,
A  D
(s in the same segment)
B  C
(s in the same segment)
APB  DPC
(vert. opp. s)
∴ DABP  DDCP (AAA)
(b) ∵ DABP  DDCP
PA PB

∴
(corr. sides, Ds)
PD PC
8 12

PD 6
PD  4
P. 22
7.3 Relationship among the Chords,
Arcs and Angles of a Circle
1.
Equal Chords and Equal Angles at the Centre
Theorem 7.8
In a circle, if the angles at the centre are equal,
then they stand on equal chords, that is,
if x  y,
then AB  CD.
(Reference: equal s, equal chords)
Conversely, equal chords in a circle subtend
equal angles at the centre, that is,
if AB  CD,
then x  y.
(Reference: equal chords, equal s)
This theorem can be proved using congruent triangles.
P. 23
7.3 Relationship among the Chords,
Arcs and Angles of a Circle
2.
Equal Angles at the Centre and Equal Arcs
Theorem 7.9
In a circle, if the angles at the centre are equal,
then they stand on equal arcs, that is,
if p  q,
(
(
then AB  CD.
(Reference: equal s, equal arcs)
Conversely, equal arcs in a circle subtend
equal angles at the centre, that is,
(
(
if AB  CD,
then p  q.
(Reference: equal arcs, equal s)
P. 24
7.3 Relationship among the Chords,
Arcs and Angles of a Circle
3.
Equal Chords and Equal Arcs
Theorem 7.10
In a circle, equal chords cut arcs with equal
lengths, that is,
if AB  CD,
(
(
then AB  CD.
(Reference: equal chords, equal arcs)
Conversely, equal arcs in a circle subtend
equal chords, that is,
(
(
if AB  CD,
then AB  CD.
(Reference: equal arcs, equal chords)
P. 25
7.3 Relationship among the Chords,
Arcs and Angles of a Circle
The above theorems are summarized in the following diagram:
Equal Chords
Equal Arcs
Theorem 7.10
Equal Angles
Example: In the figure, the chords AB, BC and CA are of the same length.
∴ Each of the angles at the centre are equal,
i.e., AOB  BOC  COA  120.
∴ Each of the arcs are equal,
(
(
(
i.e., AB  BC  CA  9 cm.
P. 26
7.3 Relationship among the Chords,
Arcs and Angles of a Circle
Example 7.8T
(
In the figure, O is the centre of the circle with circumference 30 cm.
A regular hexagon ABCDEF is inscribed in the circle.
(a) Find AOB.
(b) Find the length of AB.
Solution:
(a) ∵ AB  BC  CD  DE  EF  FA
∴ AOB  BOC  COD  DOE  EOF  FOA
(equal chords, equal s)
∴ AOB  360  6
 60
(
(
(
(
(
(
(b) ∵ AB  BC  CD  DE  EF  FA
∴ AB  BC  CD  DE  EF  FA
∴ AB  (30  6) cm
 5 cm
(equal chords, equal arcs)
(
P. 27
7.3 Relationship among the Chords,
Arcs and Angles of a Circle
4.
Arcs Proportional to Angles at the Centre
Theorem 7.11
In a circle, arcs are proportional to the angles
at the centre, that is,
(
(
AB : PQ  q : f.
(Reference: arcs prop. to s at centre)
Notes:
1. In a circle, chords are not proportional to the angles subtend at the
centre.
2. In a circle, chords are not proportional to the arcs.
P. 28
7.3 Relationship among the Chords,
Arcs and Angles of a Circle
(
(
Example 7.9T
In the figure, O is the centre of the circle. APB  15 cm, PB  6 cm
and POB  80. Find AOP.
Solution:
(
(
(
AP  9 cm
AOP : POB  AP : PB
AOP : 80  9 cm : 6 cm
AOP  120
(arcs prop. to s at centre)
P. 29
7.3 Relationship among the Chords,
Arcs and Angles of a Circle
(
(
Example 7.10T
(


In the figure, O is the centre of the circle. 3AB  2BC and
AOB  40. Find ABC : AEDC.
Solution:
(
(
(
(
∵ 3AB  2BC
∴ AB : BC  2 : 3
(
(
(


AOB : BOC  AB : BC
(arcs prop. to s at centre)
40 : BOC  2 : 3
BOC  60
∴
AOC  100 and Reflex AOC  260
∴ ABC : AEDC  100 : 260
 5 : 13
(arcs prop. to s at centre)
P. 30
7.3 Relationship among the Chords,
Arcs and Angles of a Circle
5.
Arcs Proportional to Angles at the Circumference
Theorem 7.12
In a circle, arcs are proportional to the angles
subtended at the circumference, that is,
(
(
AB : PQ  a : b.
(Reference: arcs prop. to s at ⊙ce)
Notes:
In a circle, chords are not proportional to the angles subtend at the
circumference.
This theorem can be proved by constructing the corresponding
angles at the centre for each arcs.
P. 31
7.3 Relationship among the Chords,
Arcs and Angles of a Circle
Example 7.11T
(
In the figure, O is the centre of the circle. AOB  80,
OAC  20 and AB  12 cm.
(a) Find CBD.
(b) Find the length of CD.
(
Solution:
(a) ∵ AOB  2  ACB
∴ ACB  40
In DAOE, OEC  80  20
 100
In DBCE,
OEC  ACB  CBD
100  40  CBD
CBD  60
(
(
(b)
AB : CD  ACB : CBD
12 cm : CD  40 : 60
∴
CD  18 cm
( at the centre twice  at ⊙ce)
(ext.  of D)
(ext.  of D)
(arcs prop. to s at ⊙ce)
P. 32
7.4 Basic Properties of a Cyclic
Quadrilateral
A. Opposite Angles of a Cyclic Quadrilateral
Cyclic quadrilateral:
quadrilateral with all vertices lying on a circle
Two pairs of opposite angles:
 BAD and DCB
 ABC and CDA
Theorem 7.13
The opposite angles in a cyclic quadrilateral
are supplementary.
Symbolically, BAD  DCB  180 and
ABC  CDA  180.
(Reference: opp. s, cyclic quad.)
This theorem can be proved by constructing the corresponding angles
at the centre.
P. 33
7.4 Basic Properties of a Cyclic
Quadrilateral
A. Opposite Angles of a Cyclic Quadrilateral
Example 7.12T
In the figure, ABCD is a cyclic quadrilateral. AD is a
diameter of the circle and DAC  35. Find ABC.
Solution:
∵ AD is a diameter.
∴ ACD  90
In DACD,
35  ACD  ADC  180
35  90  ADC  180
ADC  55
∴ ABC  ADC  180
ABC  125
( in semicircle)
( sum of D)
(opp. s, cyclic quad.)
P. 34
7.4 Basic Properties of a Cyclic
Quadrilateral
B. Exterior Angles of a Cyclic Quadrilateral
From Theorem 7.13, we obtain the following relationship
between the exterior angle and the interior opposite angle of
a cyclic quadrilateral:
Theorem 7.14
The exterior angle of a cyclic quadrilateral
is equal to the interior opposite angle,
that is, f  q.
(Reference: ext. , cyclic quad.)
P. 35
7.4 Basic Properties of a Cyclic
Quadrilateral
B. Exterior Angles of a Cyclic Quadrilateral
Example 7.13T
In the figure, two circles meet at C and D. ADE and BCF
are straight lines. If BAD  105, find DEF.
Solution:
FCD  BAD
(ext. , cyclic quad.)
 105
∴ DEF  FCD  180
(opp. s, cyclic quad.)
DEF  75
P. 36
7.4 Basic Properties of a Cyclic
Quadrilateral
C. Tests for Concyclic Points
Points are said to be concyclic if they lie on the same circle.
To test whether a given set of 4 points are concyclic
(or a given quadrilateral is cyclic):
Theorem 7.15 (Converse of Theorem 7.7)
In the figure, if p  q,
then A, B, C and D are concyclic.
(Reference: converse of s in the same segment)
P. 37
7.4 Basic Properties of a Cyclic
Quadrilateral
C. Tests for Concyclic Points
Theorem 7.16 (Converse of Theorem 7.13)
In the figure, if a  c  180 (or b  d  180),
then A, B, C and D are concyclic.
(Reference: opp. s supp.)
Theorem 7.17 (Converse of Theorem 7.14)
In the figure, if p  q,
then A, B, C and D are concyclic.
(Reference: ext.   int. opp.  )
P. 38
7.4 Basic Properties of a Cyclic
Quadrilateral
C. Tests for Concyclic Points
Example 7.14T
In the figure, APB and RDQC are straight lines. If AD // PQ,
show that P, Q, C and B are concyclic.
Solution:
ADR  ABC
ADR  PQR
∴ ABC  PQR
∴ P, Q, C and B are concyclic.
(ext. , cyclic quad.)
(corr. s, AD // PQ)
(ext.   int. opp.  )
P. 39
7.4 Basic Properties of a Cyclic
Quadrilateral
C. Tests for Concyclic Points
Example 7.15T
Consider the cyclic quadrilateral PQCD.
(a) Find y.
(b) Write down another four concyclic points.
Solution:
(a) y  110  180
y  70
(opp. s, cyclic quad.)
(b) ∵ ABQ  QPD  70
∴ A, B, Q and P are concyclic.
(ext.   int. opp.  )
P. 40
Chapter Summary
7.1 Chords of a Circle
1.
If a perpendicular line is drawn from the centre
of the circle to a chord, then it bisects the chord,
and vice versa.
2.
If the lengths of two chords are equal, then they
are equidistant from the centre of the circle, and
vice versa.
P. 41
Chapter Summary
7.2 Angles of a Circle
1.
The angle at the centre is twice the angle at the
circumference subtended by the same arc, that is,
x  2y.
2.
If AB is a diameter, then
APB  90.
Conversely, if the angle at the circumference
APB  90, then AB is a diameter.
3.
The angles in the same segment are equal, that
is, x  y.
P. 42
Chapter Summary
7.3 Relationship among the Chords, Arcs and
Angles of a Circle
1.
2.
3.
Equal angles at the centre stand on equal chords.
Equal angles at the centre stand on equal arcs.
Equal arcs subtend equal chords.
4.
Arcs are proportional to the angles at the centre.
(
(
AB : PQ  x : y
5.
The arcs are proportional to the angles subtended
at the circumference, that is,
(
(
AB : BC  x : y.
P. 43
Chapter Summary
7.4 Basic Properties of a Cyclic Quadrilateral
If ABCD is a cyclic quadrilateral, then
(a) a  b  180 and
(b) a  c.
P. 44