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HKDSE Mathematics Ronald Hui Tak Sun Secondary School
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8 September 2015Ronald HUI Missing Homework SHW1-01 SHW1-01 Summer Holiday Homework Summer Holiday Homework
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8 September 2015Ronald HUI Chapter 1 Properties of Circles Properties of Circles What is a circle? What is a circle?
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Book 5A Chapter 1 Basic Terms of a Circle
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What is a Circle? A circle is a plane closed curve where all the points on it are at the same distance from a fixed point. O A B C D E centre The fixed point is called the centre of the circle, which is denoted by ‘O’.
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What is a Circle? A line segment joining the centre to any point on the circle is called a radius e.g. OA. Note that OA = OB = OC = OD = OE. O A B C D E centre radius
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An arc is the part of a circle which lies between any two points on the circle e.g. curve AB. What is an Arc of a Circle? A B M The length of is shorter than half of the circumference. AB AMB
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An arc is the part of a circle which lies between any two points on the circle e.g. curve AB. What is an Arc of a Circle? A B AB Minor arc AB M AMB So, is called a minor arc. AB
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What is an Arc of a Circle? A B M Minor arc AB AMB The length of is longer than half of the circumference. AMB An arc is the part of a circle which lies between any two points on the circle e.g. curve AB.
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What is an Arc of a Circle? A B M Minor arc AB AMB Major arc AMB So, is called a major arc. AMB An arc is the part of a circle which lies between any two points on the circle e.g. curve AB.
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What is a Chord of a Circle? A chord is a line segment that joins any two points on a circle e.g. AB. O B A a chord P Q a diameter In particular, a chord passing through the centre O is called a diameter.
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What is a Chord of a Circle? A chord is a line segment that joins any two points on a circle e.g. AB. O B A a chord P Q a diameter In particular, a chord passing through the centre O is called a diameter. Diameter is the longest chord of a circle.
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What is a Sector of a Circle? A sector is a region bounded by an arc and two radii of a circle e.g. OAPB. A B P O Q sectors
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What is a Sector of a Circle? A sector is a region bounded by an arc and two radii of a circle e.g. OAPB. A B P O Q sectors The area of sector OAPB is less than half of the circle.
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What is a Sector of a Circle? A sector is a region bounded by an arc and two radii of a circle e.g. OAPB. A B P O Q sectors Minor sector OAPB So, sector OAPB is called a minor sector.
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What is a Sector of a Circle? A sector is a region bounded by an arc and two radii of a circle e.g. OAPB. A B P O Q Minor sector OAPB The area of sector OAQB is greater than half of the circle.
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What is a Sector of a Circle? A B P O Q Minor sector OAPB So, sector OAQB is called a major sector. Major sector OAQB A sector is a region bounded by an arc and two radii of a circle e.g. OAPB.
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A segment is a region bounded by a chord and an arc e.g. APB. What is a Segment of a Circle? A B Q O P segments
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What is a Segment of a Circle? A B Q O P segments The area of segment APB is less than half of the circle. A segment is a region bounded by a chord and an arc e.g. APB.
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What is a Segment of a Circle? A B Q O P segments So, segment APB is called a minor segment. Minor segment APB A segment is a region bounded by a chord and an arc e.g. APB.
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What is a Segment of a Circle? A B Q O P The area of segment AQB is greater than half of the circle. Minor segment APB A segment is a region bounded by a chord and an arc e.g. APB.
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What is a Segment of a Circle? A B Q O P So, segment AQB is called a major segment. Minor segment APB Major segment AQB A segment is a region bounded by a chord and an arc e.g. APB.
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Special Types of Circles (a) Concentric circles They have the same centre but different radii.
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Special Types of Circles (b) Equal circles (or congruent circles) They have equal radii.
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Special Types of Circles (c) Circumcircles (or circumscribed circles) A circumcircle of a polygon passes through all the vertices of the polygon. A B C P Q R S △ ABC and quadrilateral PQRS are inscribed in circles, they are called inscribed polygons.
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Special Types of Circles (d) Inscribed circles An inscribed circle of a polygon is enclosed by the polygon such that each side of the polygon touches the circle at only one point. C A B △ ABC and quadrilateral PQRS are circumscribed about circles, they are called circumscribed polygons. P Q R S
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Book 5A Chapter 1 Chords of a Circle
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O AB N Consider △ ONA and △ ONB. ON = ON common side OA ONA △ ONB RHSRHS ∴ △ ONA ∴ AN = BNcorr. sides, △ s = OB radii = ONB = 90 given In the figure, N is a point on the chord AB such that ON AB. i.e. N is the mid-point of AB. Is N the mid-point of AB? Perpendiculars to Chords
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O N AB Theorem 1.1 Abbreviation: line from centre chord bisects chord If ON AB, then AN = BN.
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AB O N Conversely, if AN = BN, ? is ON perpendicular to AB? Yes, you can prove that ON AB by using the properties of congruent triangles.
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Abbreviation: line joining centre to mid-pt. of chord chord Theorem 1.2 (Converse of Theorem 1.1) AB O N If AN = BN, then ON AB.
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O AB N Consider △ ONA and △ ONB. ON = ON common side OA △ ONB SSS ∴ △ ONA ∴ ∠ ANO = ∠ BNO corr. ∠ s, △ s = OB radii Proof of Theorem 1.2 AN= BN given ∵ ∠ ANO + ∠ BNO = 180 adj. ∠ s on st. line 2 ∠ BNO = 180 ∠ BNO = 90 ∴ ON AB
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x = ONA NOB = 90 40 = 50 Find x in the figure. A B O N 40 x ext. of △ Let us use Theorem 1.2 to solve the following problem. ∴ ONA = 90 line joining centre to mid-pt. of chord chord ∵ AN = BN
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A B Follow-up question O M In the figure, M is a point on the chord AB. Find the radius of the circle. 4 cm 6 cm ∵ OM AB Join OB. ∴ The radius of the circle is 5 cm. ∴ AM = BM In △ OMB, line from centre chord bisects chord Pyth. theorem
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AB O N (i)If ON AB, then AN = BN. (ii)If AN = BN, then ON AB. Theorem 1.1 Theorem 1.2 From the results of (i) and (ii), ON is the perpendicular bisector of chord AB. In summary, we have:
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AB Theorem 1.3 O N The perpendicular bisector of a chord passes through the centre of the circle. Abbreviation: bisector of chord passes through centre
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O P Q A B Let M be the foot of perpendicular from O on AB. Then, OM is the distance between the centre O and the chord AB. M Distance between O and AB Reflect AB and OM about diameter PQ to form CD and ON respectively. Distances between Chords and Centre
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ON is the distance between the centre O and the chord CD. By symmetry, we have: and OM = ON AB = CD Equal chords O P Q A B C D N M Distance between O and AB Distance between O and CD This suggests that equal chords of a circle are equidistant from the centre.
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Theorem 1.4 Abbreviation: equal chords, equidistant from centre O N A B M C D and ON CD, If AB = ON. OM AB then OM = CD,
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Conversely, given that OM AB, ON CD and OM = ON, is it true that AB = CD? Yes, the converse of Theorem 1.4 is also true. O C D A B M N P Q
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O C D A B M N P Q Conversely, given that OM AB, ON CD and OM = ON, is it true that AB = CD? You can try to show that MB = ND first by considering △ OMB and △ OND.
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Consider △ OMB and △ OND. OM = ON given OB = OD radii OMB = OND = 90 given ∴ △ OMB △ OND RHS ∴ MB = ND corr. sides, △ s ∵ OM ⊥ AB and ON ⊥ CD given ∴ AM = MB and CN = ND line from centre chord bisects chord ∴ AB = 2MB = 2ND = CD O N D M B A C
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Theorem 1.5 (Converse of Theorem 1.4) Abbreviation: chords equidistant from centre are equal O N C D A B M and ON CD, = CD. If OM then AB = ON, OM AB
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Follow-up question O C D A B M Find OM in the figure. N 5 cm 2 cm CN = DN = 5 cm ∴ BM = AM = 5 cm ∴ AB = CD ∴ OM = ON = 2 cm ∵ OM ⊥ AB and ON ⊥ CD line from centre ⊥ chord bisects chord equal chords, equidistant from centre
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4 September 2015Ronald HUI
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8 September 2015Ronald HUI Chapter 1 SHW1-A1 SHW1-A1 Due date? Due date?
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