HKDSE Mathematics Ronald Hui Tak Sun Secondary School

8 September 2015Ronald HUI Missing Homework SHW1-01 SHW1-01 Summer Holiday Homework Summer Holiday Homework

8 September 2015Ronald HUI Chapter 1 Properties of Circles Properties of Circles What is a circle? What is a circle?

Book 5A Chapter 1 Basic Terms of a Circle

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’.

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

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

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

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.

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.

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.

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.

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

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.

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.

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.

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.

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

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.

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.

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.

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.

Special Types of Circles (a) Concentric circles They have the same centre but different radii.

Special Types of Circles (b) Equal circles (or congruent circles) They have equal radii.

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.

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

Book 5A Chapter 1 Chords of a Circle

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

O N AB Theorem 1.1 Abbreviation: line from centre  chord bisects chord If ON  AB, then AN = BN.

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.

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.

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

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

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

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:

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

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

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.

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,

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

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.

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

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

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

4 September 2015Ronald HUI

8 September 2015Ronald HUI Chapter 1 SHW1-A1 SHW1-A1 Due date? Due date?