Chapter 4 Basic Properties Of Circles (1)

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Basic Properties of Circles (1) Conte nts 4.1 Chords And Arcs 4.2 Angles of a Circle 4.3 Basic Properties of a Cyclic Quadrilateral

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Basic Properties of Circles (1)And Arcs 4.1 Chords A. Basic Terms of a Circle Definition 4.1: •

A circle is a closed curve in a plane where every point on the curve is equidistant from a fixed point.

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The fixed point is called the centre.

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The length of the curve is called the circumference of the circle.

Fig 4.15

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Basic Properties of Circles (1)And Arcs 4.1 Chords A. Basic Terms of a Circle Definition 4.2: •

A chord of a circle is a line segment with two end points on the circumference.

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A radius of a circle is a line segment joining the centre to any point on the circumference.

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A diameter of a circle is a chord passing through the centre.

Fig 4.16

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Basic Properties of Circles (1)And Arcs 4.1 Chords A. Basic Terms of a Circle Definition 4.3: An arc of a circle is a portion of the circumference. ︵ The minor arc (denoted by AB) is shorter than half of ︵ the circumference and the major arc (denoted by AXB) is longer than half of the circumference. Definition 4.4:

Fig 4.17

An angle at the centre is an angle subtended by an arc or a chord at the centre. Content

For example, in Fig. 4.18, ︵ ∠AOB is an angle at the centre subtended by APB (or chord AB). Fig 4.18

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Basic Properties of Circles (1)And Arcs 4.1 Chords B. Basic Terms of a Circle Theorem 4.1: 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, Content

Fig 4.21

if AB = CD, then x = y. (Reference: equal chords, equal ∠s)

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Basic Properties of Circles (1)And Arcs 4.1 Chords Theorem 4.2: 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, ︵

Fig 4.23

︵

if AB = CD, Content

then p = q. (Reference: equal arcs, equal ∠s)

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Basic Properties of Circles (1)And Arcs 4.1 Chords Theorem 4.3: In a circle, equal chords cut arcs with equal length, 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 Content

Fig 4.24

then AB = CD. (Reference: equal arcs, equal chords)

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Basic Properties of Circles (1)And Arcs 4.1 Chords Theorem 4.4: In a circle, arcs are proportional to the angels at the centre, that is, ︵ ︵ AB: PQ = θ: ψ. (Reference: arcs prop. to ∠s at centre) Notes: 1. 1. Content

In a circle, chords are not proportional to the angles they subtend at the centre, that is, AB : PQ ≠ θ: ψ.

Fig 4.31

In a circle, chords are not proportional to the arcs, that is, ︵ ︵ AB : PQ ≠ AB : PQ.

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Basic Properties of Circles (1)And Arcs 4.1 Chords C. Chords of a Circle Theorem 4.5: 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 perp. to chord bisects chord)

Fig 4.41

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Basic Properties of Circles (1)And Arcs 4.1 Chords C. Chords of a Circle Theorem 4.6: 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. Fig 4.43

(Reference: line from centre to mid-pt. of chord perp. to chord) Content

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Basic Properties of Circles (1)And Arcs 4.1 Chords Theorem 4.7: 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)

Fig 4.51

Theorem 4.8: If two chords are equidistant from the centre of a circle, then their lengths are equal. Content

In other words, if OP = OQ, then AB = CD. (Reference: chords equidistant from centre are equal)

Fig 4.53

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Basic Properties of Circles (1) 4.2 Angles of a Circle A. The Angle at the Circumference Definition 4.5: The angle at the circumference is the angle subtended by an arc (or a chord) at the circumference. Theorem 4.9: (i)

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(ii)

Fig 4.85(a)

(iii)

Fig 4.85(b)

Fig 4.85(c)

The angle at the centre subtended by an arc is twice the angle at the circumference subtended by the same arc. This means that θ= 2ψ. ce (Reference: ∠at centre twice ∠at ⊙ )

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Basic Properties of Circles (1) 4.2 Angles of a Circle Theorem 4.10: In a circle, if the angles at the circumference are equal, then they stand on equal chords (or arcs), that is, ︵ ︵ if a = b, then AB = BC (or AB = BC). (Reference: equal ∠s, chords / arcs) Conversely, equal chords (or arcs) in circle subtend equal angles at the circumference, that is, ︵ ︵ if AB = BC (AB = BC), then a = b. Content

Fig 4.88

(Reference: equal chords / arcs, equal ∠s)

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Basic Properties of Circles (1) 4.2 Angles of a Circle Theorem 4.11: Arcs are proportional to the angles they subtended at the circumference, that is, ︵ ︵ AB : PQ = a : b. (Reference: arcs prop. to ∠s at ⊙ce) Fig 4.89

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Basic Properties of Circles (1) 4.2 Angles of a Circle B. The Angle in a Semicircle Definition 4.6: As shown in Fig. 4.96, if AB︵is a diameter of the circle with centre O, then APB is a semicircle and ∠APB is called the angle in a semicircle.

Fig 4.96 Content

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Basic Properties of Circles (1) 4.2 Angles of a Circle Theorem 4.12: 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.

Fig 4. 97

(Reference: converse of ∠ in semicircle)

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Basic Properties of Circles (1) 4.2 Angles of a Circle C. Angles in the same Segment Definition 4.7: In Fig. 4.102, the region enclosed by the chord AB ︵ and the APB is called segment APB. ︵ The region enclosed by the chord AB and the AQB is called segment AQB. Suppose the area of the circle is A. (a) Since the area of segment AQB < Content

Fig 4.102

A , segment AQB is called a 2

minor segment . (b) Since the area of segment APB >

A , segment APB is called a 2

major segment .

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Basic Properties of Circles (1) 4.2 Angles of a Circle Definition 4.8: In Fig. 4.103, ∠APB and ∠AQB are called the angles in the same segment.

Fig 4.103

Theorem 4.13:

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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)

Fig 4.105

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Basic Properties of Circles 4.3 Basic(1) Properties of a Cyclic Quadrilateral A. Opposite Angles of a Cyclic Quadrilateral Definition 4.9: 

If all the vertices of a quadrilateral lie on a circle, then this quadrilateral is called a cyclic quadrilateral.

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In Fig. 4.141, ABCD is a cyclic quadrilateral. ∠A and ∠C are a pair of opposite angles of the cyclic quadrilateral; ∠B and ∠D are another pair of opposite angles.

Fig 4.141

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Basic Properties of Circles 4.3 Basic(1) Properties of a Cyclic Quadrilateral Theorem 4.14 The opposite angles in a cyclic quadrilateral are supplementary. Symbolically, ∠A + ∠C = 180° ∠B + ∠D = 180° (Reference: opp. ∠s, cyclic quad.)

and

Fig 4.143

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Basic Properties of Circles 4.3 Basic(1) Properties of a Cyclic Quadrilateral B. Exterior Angles of a Cyclic Quadrilateral Theorem 4.15: The exterior angle of a cyclic quadrilateral is equal to its interior opposite angle, that is, ψ=θ. (Reference: ext. ∠ , cyclic quad.) Fig 4.149

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Basic Properties of Circles 4.3 Basic(1) Properties of a Cyclic Quadrilateral C. Tests for Concyclic Points Definition 4.10: Points are said to be concyclic if they lie on the same circle. For example, in Fig 4.155, A, B, C, D and E are concyclic. Fig 4.155

Theorem 4.16: (Converse of Theorem 4.13)

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In Fig. 4.156, if p = q, then A, B, C and D are concyclic. (Reference: converse of ∠s in same segment) Fig 4.156

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Basic Properties of Circles 4.3 Basic(1) Properties of a Cyclic Quadrilateral Theorem 4.17: (Converse of Theorem 4.14) In Fig. 4.157, if a + c = 180° (or b + d = 180°), then A, B, C and D are concyclic. (Reference: opp. ∠s supp.) Fig 4.157

Theorem 4.18: (Converse of Theorem 4.15)

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In Fig. 4.158, if p = q, then A, B, C and D are concyclic. (Reference: ext. ∠ = int. opp. ∠)

Fig 4.158

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