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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.

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Basic Properties of Circles (1)

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  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

  2. 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. Case Study 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.

  3. 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.

  4. 7.1 Chords of a Circle (  minor arc (e.g. AYB) shorter than half of the circumference (  major arc (e.g. AXB) longer than half of the circumference A. Basic Terms of a Circle Arc: portion of the circumference Angle at the centre: angle subtended by an arc or a chord at the centre

  5. 7.1 Chords of a Circle A, B coincide B. Chords 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;  is perpendicular to the chord AB;  bisects the chord AB.

  6. 7.1 Chords of a Circle 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) B. Chords a Circle Properties about a perpendicular line from the centre to a chord: 1. Perpendicular Line from Centre to a Chord This theorem can be proved by considering DAOP and DBOP.

  7. 7.1 Chords of a Circle 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) B. Chords a Circle The converse of Theorem 7.1 is also true. This theorem can be proved by considering DOAP and DOBP.

  8. 7.1 Chords of a Circle The perpendicular bisector of any chord of a circle passes through the centre. B. Chords a Circle From Theorem 7.1 and Theorem 7.2, we obtain an important property of chords:

  9. OA  cm ∴ PQ  (13 – 12) cm  1 cm 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)  13 cm OQ  OA (radii) 13 cm

  10. 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 thatDABCDOBM. Solution: ∵ OMBC ∴BM MC (line from centre  chord bisects chord) ∴BC: BM  2 : 1 ∵ OB  OA (radii) ∴AB: OB  2 : 1 OBM  ABC(common  ) ∴ DABCDOBM(ratio of 2 sides, inc.  )

  11.  8 cm 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) (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  82 8r 80 r  10 ∴ The radius of the circle is 10 cm.

  12. 7.1 Chords of a Circle 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) B. Chords 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 DOAP and DOCQ.

  13. 7.1 Chords of a Circle 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) B. Chords a Circle The converse of Theorem 7.3 is also true. This theorem can be proved by considering DOAP and DOCQ.

  14. ON  cm  cm 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(Pyth. Theorem) 2ON2

  15. 7.2 Angles of a Circle 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 thatq 2f. (Reference:  at the centre twice at ⊙ce) 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:

  16. 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

  17. ODC  25 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 ( at the centre twice at ⊙ce) ∴BAD  35 ODC 10  BAD (alt. s, AB // CD) ODC 10  35

  18. 7.2 Angles of a Circle 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, thenAB is a diameter. (Reference: converse of  in semicircle) 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)

  19. PBC  20 70 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 (b) APC PBC PCB (ext.  of D)

  20. 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.

  21. 7.2 Angles of a Circle 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) C. Angle 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.

  22. ∴ (corr. sides, Ds) PD  4 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

  23. 7.3 Relationship among the Chords, Arcs and Angles of a Circle 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) 1. Equal Chords and Equal Angles at the Centre This theorem can be proved using congruent triangles.

  24. 7.3 Relationship among the Chords, Arcs and Angles of a Circle 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) ( ( ( ( 2. Equal Angles at the Centre and Equal Arcs

  25. 7.3 Relationship among the Chords, Arcs and Angles of a Circle 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) ( ( ( ( 3. Equal Chords and Equal Arcs

  26. 7.3 Relationship among the Chords, Arcs and Angles of a Circle Equal Chords Theorem 7.8 Theorem 7.9 Theorem 7.10 Equal Arcs Equal Angles ∴ Each of the arcs are equal, i.e., ABBCCA 9 cm. ( ( ( The above theorems are summarized in the following diagram: 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.

  27. 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. (  60 ( ( ( ( ( ( ∴ ABBCCDDEEFFA (equal chords, equal arcs) ( ∴ AB (30  6) cm  5 cm 7.3 Relationship among the Chords, Arcs and Angles of a Circle Example 7.8T Solution: (a) ∵ ABBCCDDEEFFA ∴ AOB BOCCODDOEEOFFOA (equal chords, equal s) ∴ AOB  360  6 (b) ∵ABBCCDDEEFFA

  28. 7.3 Relationship among the Chords, Arcs and Angles of a Circle 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) ( ( 4. Arcs Proportional to Angles at the 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.

  29. ( ( 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 120 7.3 Relationship among the Chords, Arcs and Angles of a Circle Example 7.9T Solution: AOP : 80 9 cm : 6 cm

  30. ( ( In the figure, O is the centre of the circle. 3AB 2BC and AOB 40. Find ABC : AEDC. (  ( ( ∵ 3AB 2BC ( ( ( ( AOB:BOC  AB : BC (arcs prop. to s at centre) ∴AB : BC  2 : 3 (  ∴ABC : AEDC  100 : 260 (arcs prop. to s at centre)  5 : 13 7.3 Relationship among the Chords, Arcs and Angles of a Circle Example 7.10T Solution: 40:BOC  2 : 3 BOC 60 ∴ AOC  100 and Reflex AOC  260

  31. 7.3 Relationship among the Chords, Arcs and Angles of a Circle 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) ( ( 5. Arcs Proportional to Angles at the Circumference 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.

  32. 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. ( ( ( ( (b) AB : CD ACB : CBD (arcs prop. to s at ⊙ce) CBD  60 ∴CD 18 cm 7.3 Relationship among the Chords, Arcs and Angles of a Circle Example 7.11T Solution: (a) ∵ AOB  2 ACB ( at the centre twice at ⊙ce) ∴ACB  40 In DAOE, OEC  80  20 (ext.  of D)  100 In DBCE, OEC  ACB  CBD (ext.  of D) 100 40  CBD 12 cm : CD 40 : 60

  33. 7.4 Basic Properties of a Cyclic Quadrilateral 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.) 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 This theorem can be proved by constructing the corresponding angles at the centre.

  34. ABC  125 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 semicircle) In DACD, 35ACDADC  180 ( sum of D) 3590ADC  180 ADC  55 ∴ ABC  ADC  180(opp. s, cyclic quad.)

  35. 7.4 Basic Properties 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.) 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:

  36. DEF  75 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.)

  37. 7.4 Basic Properties of a Cyclic Quadrilateral 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) 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):

  38. 7.4 Basic Properties of a Cyclic Quadrilateral 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. ) C. Tests for Concyclic Points

  39. 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 (ext. , cyclic quad.) ADR  PQR (corr. s, AD // PQ) ∴ ABC  PQR ∴ P, Q, C and B are concyclic. (ext.   int. opp.  )

  40. y 70 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(opp. s, cyclic quad.) (b) ∵ ABQ  QPD  70 ∴ A, B, Q and P are concyclic. (ext.   int. opp.  )

  41. 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.

  42. 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.

  43. 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. ( ( Chapter Summary 7.3 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.

  44. 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.

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