Upload
nubia
View
68
Download
1
Tags:
Embed Size (px)
DESCRIPTION
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. 7.4Basic Properties of a Cyclic Quadrilateral. Chapter Summary. - PowerPoint PPT Presentation
Citation preview
7
7.1 Chords of a Circle
7.2 Angles of a Circle
Chapter Summary
Case Study
Basic Properties of Circles (1)
7.3 Relationship among the Chords, Arcs and Angles of a Circle
7.4 Basic Properties of a Cyclic Quadrilateral
P. 2
In order to find the centre of the circular plate:
Step 1: Draw an arbitrary triangle inscribed in the circular plate.
Case Study
You need to find the centre of the circular plate first.
I found a fragment of a circular plate. How can I know its original size?
Step 2: Find the circumcentre of the triangle, i.e., the centre of the circular plate, by drawing 3 perpendicular bisectors.
P. 3
A. Basic Terms of a CircleA. Basic Terms of a Circle7.1 Chords of a Circle7.1 Chords 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. 4
A. Basic Terms of a CircleA. Basic Terms of a Circle7.1 Chords of a Circle7.1 Chords of a Circle
Arc: portion of the circumference
Angle at the centre: angle subtended by an arc or a chord at the centre
minor arc (e.g. AYB)shorter than half of the circumference
(
major arc (e.g. AXB)longer than half of the circumference
(
P. 5
B. Chords a CircleB. Chords a Circle7.1 Chords of a Circle7.1 Chords of a Circle
If we draw a chord AB on a circle and fold the paper as shown below:
Then the crease passes through the centre of the circle;
bisects the chord AB. is perpendicular to the chord AB;
A, B coincide
P. 6
B. Chords a CircleB. Chords a Circle7.1 Chords of a Circle7.1 Chords of a Circle
Properties about a perpendicular line from the centre to a chord:1. Perpendicular Line from Centre to a Chord
Theorem 7.1If a perpendicular line is drawn from a centre ofa circle to a chord, then it bisects the chord. In other words, if OP AB,
then AP BP. (Reference: line from centre chordbisects chord)
This theorem can be proved by considering AOP and BOP.
P. 7
B. Chords a CircleB. Chords a Circle7.1 Chords of a Circle7.1 Chords of a Circle
This theorem can be proved by considering OAP and OBP.
Theorem 7.2If a line is joined from the centre of a circleto the mid-point of a chord, then it isperpendicular to the chord. In other words, if AP BP,
then OP AB. (Reference: line from centre to mid-pt. of chord chord)
The converse of Theorem 7.1 is also true.
P. 8
B. Chords a CircleB. Chords a Circle7.1 Chords of a Circle7.1 Chords of a Circle
From Theorem 7.1 and Theorem 7.2, we obtain an important property of chords:
The perpendicular bisector of any chord ofa circle passes through the centre.
P. 9
Example 7.1T
Solution:
In the figure, O is the centre of the circle. AP PB 5 cm and OP 12 cm. Find PQ.
B. Chords a CircleB. Chords a Circle7.1 Chords of a Circle7.1 Chords of a Circle
∵ AP PB ∴ OP AB (line from centre to mid-pt. of chord chord)
In OAP,OA2 OP2 AP2 (Pyth. Theorem)
OA cm22 512 13 cm OQ OA (radii)
13 cm∴ PQ (13 – 12) cm
1 cm
P. 10
Example 7.2T
Solution:
In the figure, O is the centre of the circle. AOB is a straight line and OM BC. Show that ABC OBM.
B. Chords a CircleB. Chords a Circle7.1 Chords of a Circle7.1 Chords of a Circle
∵ OM BC ∴ BM MC (line from centre chord bisects chord)∴ BC : BM 2 : 1
∵ OB OA (radii)∴ AB : OB 2 : 1OBM ABC (common )
∴ ABC OBM (ratio of 2 sides, inc. )
P. 11
Example 7.3T
Solution:
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.
B. Chords a CircleB. Chords a Circle7.1 Chords of a Circle7.1 Chords of a Circle
In OCP,OC2 OP2 PC2 (Pyth. Theorem)
(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.
r2 (r – 4)2 82 8r 80
r 10 ∴ The radius of the circle is 10 cm.
P. 12
B. Chords a CircleB. Chords a Circle7.1 Chords of a Circle7.1 Chords of a Circle
Properties about a perpendicular line from the centre to a chord:2. Distance between Chords and Centre
This theorem can be proved by considering OAP and OCQ.
Theorem 7.3If the lengths of two chords are equal, thenthey are equidistant from the centre. In other words, if AB CD,
then OP OQ. (Reference: equal chords, equidistantfrom centre)
P. 13
B. Chords a CircleB. Chords a Circle7.1 Chords of a Circle7.1 Chords of a Circle
This theorem can be proved by considering OAP and OCQ.
The converse of Theorem 7.3 is also true.
Theorem 7.4If two chords are equidistant from the centreof a circle, then their lengths are equal. In other words, if OP OQ,
then AB CD. (Reference: chords equidistant from centre are equal)
P. 14
Example 7.4T
Solution:
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.)
B. Chords a CircleB. Chords a Circle7.1 Chords of a Circle7.1 Chords of a Circle
∵ AB CD ∴ OM ON (equal chords, equidistant from centre)
ON cm262
∵ All of the interior angles of the quadrilateral OMPN are right angles and OM ON.
∴ OMPN is a square.In ONP,
OP2 ON2 NP2 (Pyth. Theorem) 2ON2
cm
23
P. 15
A. The Angle at the CircumferenceA. The Angle at the Circumference7.2 Angles of a Circle7.2 Angles of a Circle
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 2. (Reference: at the centre twice at ⊙ce)
P. 16
A. The Angle at the CircumferenceA. The Angle at the Circumference7.2 Angles of a Circle7.2 Angles of a Circle
This theorem can be proved by constructing a diameter PQ.
Since OA OP (radii), AOP is isosceles.∴ OAP OPA a.Hence the exterior angle of AOQ 2a.
In the left semicircle:
Similarly, in the right semicircle, BOQ 2b.
∵ 2a 2b and a b ∴ 2
P. 17
Example 7.5T
Solution:
In the figure, AB and CD are two parallel chords of the circle with centre O. BOD 70 and MDO 10. Find ODC.
∵ BOD 2 BAD ( at the centre twice at ⊙ce) ∴ BAD 35
A. The Angle at the CircumferenceA. The Angle at the Circumference7.2 Angles of a Circle7.2 Angles of a Circle
ODC 10 BAD (alt. s, AB // CD) ODC 10 35 ODC 25
P. 18
B. The Angle in a SemicircleB. The Angle in a Semicircle7.2 Angles of a Circle7.2 Angles of a Circle
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 7.6The 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. 19
Example 7.6T
Solution:
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.
(a) Since AP is a diameter, ACP 90. ( in semicircle)
7.2 Angles of a Circle7.2 Angles of a Circle
PBC 20
B. The Angle in a SemicircleB. The Angle in a Semicircle
∵ AC BC∴ PAC PBC (base s, isos. )
In ACB,
PAC PBC ACB 180 ( sum of ) 2PBC (90 50) 180
(b) APC PBC PCB (ext. of ) 70
P. 20
C. Angle in the Same SegmentC. Angle in the Same Segment7.2 Angles of a Circle7.2 Angles of a Circle
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. 21
C. Angle in the Same SegmentC. Angle in the Same Segment7.2 Angles of a Circle7.2 Angles of a Circle
Theorem 7.7The 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)
The angles in the same segment of a circle are equal.
This theorem can be proved by considering the angle at the centre.
P. 22
Example 7.7T
Solution:
In the figure, AC and BD are two chords that intersect at P. (a) Show that ABP DCP. (b) If AP 8, BP 12 and PC 6, find PD.
(a) In ABP and DCP,
7.2 Angles of a Circle7.2 Angles of a Circle
A D (s in the same segment)
(b) ∵ ABP DCP
PD 4
C. Angle in the Same SegmentC. Angle in the Same Segment
B C (s in the same segment)APB DPC (vert. opp. s)
∴ ABP DCP (AAA)
PCPB
PDPA
∴ (corr. sides, s)
6128
PD
P. 23
7.3 Relationship among the Chords,7.3 Relationship among the Chords, Arcs and Angles of a Circle Arcs and Angles of a Circle1. Equal Chords and Equal Angles at the Centre
Theorem 7.8In 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 subtendequal 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. 24
7.3 Relationship among the Chords,7.3 Relationship among the Chords, Arcs and Angles of a Circle Arcs and Angles of a Circle2. Equal Angles at the Centre and Equal Arcs
Theorem 7.9In 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 subtendequal angles at the centre, that is,
if AB CD,then p q.
(Reference: equal arcs, equal s)
( (
( (
P. 25
7.3 Relationship among the Chords,7.3 Relationship among the Chords, Arcs and Angles of a Circle Arcs and Angles of a Circle3. Equal Chords and Equal Arcs
Theorem 7.10In a circle, equal chords cut arcs with equallengths, that is,
if AB CD,
then AB CD. (Reference: equal chords, equal arcs)
Conversely, equal arcs in a circle subtendequal chords, that is,
if AB CD,then AB CD.
(Reference: equal arcs, equal chords)
( (
( (
P. 26
7.3 Relationship among the Chords,7.3 Relationship among the Chords, Arcs and Angles of a Circle Arcs and Angles of a CircleThe above theorems are summarized in the following diagram:
Equal Chords
Equal Arcs Equal AnglesTheorem 7.10
Theorem 7.8Theore
m 7.9
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. 27
Example 7.8T
Solution:(a) ∵ AB BC CD DE EF FA ∴ AOB BOC COD DOE EOF FOA
(equal chords, equal s)
(b) ∵ AB BC CD DE EF FA
7.3 Relationship among the Chords,7.3 Relationship among the Chords, Arcs and Angles of a Circle Arcs and Angles of a Circle
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.
(
∴ AOB 360 6
∴ AB BC CD DE EF FA (equal chords, equal arcs)
( ( ( ( ( (
∴ AB (30 6) cm
(
5 cm
60
P. 28
7.3 Relationship among the Chords,7.3 Relationship among the Chords, Arcs and Angles of a Circle Arcs and Angles of a Circle4. Arcs Proportional to Angles at the Centre
Theorem 7.11In a circle, arcs are proportional to the anglesat the centre, that is,
AB : PQ : .(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. 29
Example 7.9T
Solution:
7.3 Relationship among the Chords,7.3 Relationship among the Chords, Arcs and Angles of a Circle Arcs and Angles of a Circle
In the figure, O is the centre of the circle. APB 15 cm, PB 6 cm and POB 80. Find AOP.
( (
AP 9 cm
(
AOP : POB AP : PB (arcs prop. to s at centre)
( (
AOP : 80 9 cm : 6 cm AOP 120
P. 30
Example 7.10T
Solution:
7.3 Relationship among the Chords,7.3 Relationship among the Chords, Arcs and Angles of a Circle Arcs and Angles of a Circle
AOB : BOC AB : BC (arcs prop. to s at centre)
( (
40 : BOC 2 : 3 BOC 60
In the figure, O is the centre of the circle. 3AB 2BC and
AOB 40. Find ABC : AEDC.
(
(
( ∵ 3AB 2BC
( (
∴ AB : BC 2 : 3
( (
∴ AOC 100 and Reflex AOC 260
5 : 13∴ ABC : AEDC 100 : 260 (arcs prop. to s at centre)
(
P. 31
7.3 Relationship among the Chords,7.3 Relationship among the Chords, Arcs and Angles of a Circle Arcs and Angles of a Circle5. Arcs Proportional to Angles at the Circumference
Notes:In a circle, chords are not proportional to the angles subtend at the circumference.
Theorem 7.12In a circle, arcs are proportional to the anglessubtended at the circumference, that is,
AB : PQ a : b.(Reference: arcs prop. to s at ⊙ce)
( (
This theorem can be proved by constructing the corresponding angles at the centre for each arcs.
P. 32
Example 7.11T
Solution:
7.3 Relationship among the Chords,7.3 Relationship among the Chords, Arcs and Angles of a Circle Arcs and Angles of a Circle
In AOE, OEC 80 20 (ext. of )
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.
(
(
(a) ∵ AOB 2 ACB ( at the centre twice at ⊙ce) ∴ ACB
40
100 In BCE,OEC ACB CBD (ext. of ) 100 40 CBDCBD 60
12 cm : CD 40 : 60(b) AB : CD ACB : CBD (arcs prop. to s at ⊙ce)
( (
∴ CD 18 cm
P. 33
7.4 Basic Properties of a Cyclic7.4 Basic Properties of a Cyclic Quadrilateral QuadrilateralCyclic quadrilateral:quadrilateral with all vertices lying on a circle
A. Opposite Angles of a Cyclic QuadrilateralA. Opposite Angles of a Cyclic Quadrilateral
Two pairs of opposite angles: BAD and DCB ABC and CDA
Theorem 7.13The opposite angles in a cyclic quadrilateralare 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. 34
Example 7.12T
Solution:
In the figure, ABCD is a cyclic quadrilateral. AD is a diameter of the circle and DAC 35. Find ABC.
7.4 Basic Properties of a Cyclic7.4 Basic Properties of a Cyclic Quadrilateral QuadrilateralA. Opposite Angles of a Cyclic QuadrilateralA. Opposite Angles of a Cyclic Quadrilateral
∵ AD is a diameter.∴ ACD 90 ( in semicircle)
In ACD,35 ACD ADC 180 ( sum of )
∴ ABC ADC 180 (opp. s, cyclic quad.)
35 90 ADC 180 ADC 55
ABC 125
P. 35
7.4 Basic Properties of a Cyclic7.4 Basic Properties of a Cyclic Quadrilateral QuadrilateralFrom Theorem 7.13, we obtain the following relationship between the exterior angle and the interior opposite angle of a cyclic quadrilateral:
B. Exterior Angles of a Cyclic QuadrilateralB. Exterior Angles of a Cyclic Quadrilateral
Theorem 7.14The exterior angle of a cyclic quadrilateralis equal to the interior opposite angle,that is, .(Reference: ext. , cyclic quad.)
P. 36
Example 7.13T
Solution:
In the figure, two circles meet at C and D. ADE and BCF are straight lines. If BAD 105, find DEF.
7.4 Basic Properties of a Cyclic7.4 Basic Properties of a Cyclic Quadrilateral Quadrilateral
FCD BAD (ext. , cyclic quad.)
B. Exterior Angles of a Cyclic QuadrilateralB. Exterior Angles of a Cyclic Quadrilateral
105 ∴ DEF FCD 180 (opp. s, cyclic quad.)
DEF 75
P. 37
7.4 Basic Properties of a Cyclic7.4 Basic Properties of a Cyclic Quadrilateral QuadrilateralPoints are said to be concyclic if they lie on the same circle.
C. Tests for Concyclic PointsC. Tests for Concyclic Points
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. 38
7.4 Basic Properties of a Cyclic7.4 Basic Properties of a Cyclic Quadrilateral QuadrilateralC. Tests for Concyclic PointsC. 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. 39
Example 7.14T
Solution:
In the figure, APB and RDQC are straight lines. If AD // PQ, show that P, Q, C and B are concyclic.
7.4 Basic Properties of a Cyclic7.4 Basic Properties of a Cyclic Quadrilateral Quadrilateral
ADR ABC (ext. , cyclic quad.)
∴ ABC PQR
C. Tests for Concyclic PointsC. Tests for Concyclic Points
ADR PQR (corr. s, AD // PQ)
∴ P, Q, C and B are concyclic. (ext. int. opp. )
P. 40
Example 7.15T
Solution:
Consider the cyclic quadrilateral PQCD.(a) Find y.(b) Write down another four concyclic points.
7.4 Basic Properties of a Cyclic7.4 Basic Properties of a Cyclic Quadrilateral QuadrilateralC. Tests for Concyclic PointsC. Tests for Concyclic Points
(a) y 110 180 (opp. s, cyclic quad.)
(b) ∵ ABQ QPD 70 ∴ A, B, Q and P are concyclic. (ext. int. opp. )
y 70
P. 41
7.1 Chords of a Circle
Chapter Summary
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. 42
7.2 Angles of a Circle
Chapter Summary
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, thenAPB 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. 43
Chapter Summary7.3 Relationship among the Chords, Arcs and Angles of a Circle
1. Equal angles at the centre stand on equal chords.2. Equal angles at the centre stand on equal arcs.3. 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. 44
7.4 Basic Properties of a Cyclic Quadrilateral
Chapter Summary
If ABCD is a cyclic quadrilateral, then(a) a b 180 and(b) a c.