Chapter 4 Basic Properties of Circles (1)

4.1 Chords And Arcs

4.2 Angles of a Circle

4.3 Basic Properties of a Cyclic Quadrilateral

Contents4 Basic Properties of Circles (1)

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Basic Properties of Circles (1)

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Definition 4.1:

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

2. The fixed point is called the centre.

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

A. Basic Terms of a Circle

4.1 Chords And Arcs

Fig 4.15

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4.1 Chords And Arcs

Definition 4.2:

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

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

3. A diameter of a circle is a chord passing through the centre. Fig 4.16

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Definition 4.3:

Fig 4.17

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.

︵︵

4.1 Chords And Arcs


Definition 4.4:

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

Fig 4.18

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

︵

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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,

if AB = CD,then x = y.

(Reference: equal chords, equal s∠ )

Fig 4.21

4.1 Chords And Arcs

B. Basic Terms of a Circle

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Theorem 4.2:

Fig 4.23

In a circle, if the angles at the centre are equal,then they stand on equal arcs, that is,

(Reference: equal s, equal arcs∠ )

Conversely, equal arcs in a circle subtend equalangles at the centre, that is,

(Reference: equal arcs, equal s∠ )

if p = q,

then AB = CD.︵︵

if AB = CD,

then p = q.

︵︵

4.1 Chords And Arcs

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Theorem 4.3:

Fig 4.24

In a circle, equal chords cut arcs with equal length, that is

(Reference: equal chords, equal arcs)

Conversely, equal arcs in a circle subtend equalchords, that is,

(Reference: equal arcs, equal chords)

if AB = CD,

then AB = CD.︵︵

if AB = CD

then AB = CD.

︵︵

4.1 Chords And Arcs

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Theorem 4.4:

In a circle, arcs are proportional to the angels at the centre, that is,

(Reference: arcs prop. to s at centre∠ )

Notes:

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

Fig 4.31

AB: PQ = θ: ψ.︵︵

2. In a circle, chords are not proportional to the arcs,

that is,

AB : PQ ≠ AB : PQ.︵︵

4.1 Chords And Arcs

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Theorem 4.5:

If a perpendicular line is drawn from a centre of a circle to a chord, then it bisects the chord.

C. Chords of a Circle

4.1 Chords And Arcs

Fig 4.41

In other words, if OP ⊥ AB, then AP = BP.

(Reference: line from centre perp. to chord bisects chord)

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Theorem 4.6:

If a line is joined from the centre of a circle to themid-point of a chord, then it is perpendicular to thechord.

Fig 4.43

In other words, if AP = BP, then OP ⊥ AB.

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

C. Chords of a Circle

4.1 Chords And Arcs

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Theorem 4.7:

Theorem 4.8:

Fig 4.51

Fig 4.53

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)

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)

4.1 Chords And Arcs

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Definition 4.5:

Theorem 4.9:

A. The Angle at the Circumference


Fig 4.85(a)

The angle at the circumference is the angle subtended by an arc (or a chord) at the circumference.

Fig 4.85(b) Fig 4.85(c)

(i) (ii) (iii)

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.

(Reference: ∠at centre twice a∠ t ⊙ ) ce

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Theorem 4.10:

In a circle, if the angles at the circumference are equal,then they stand on equal chords (or arcs), that is,

(Reference: equal s, chords / arcs∠ )

Conversely, equal chords (or arcs) in circle subtend equal angles at the circumference, that is,

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

Fig 4.88

if a = b, then AB = BC (or AB = BC).︵︵

if AB = BC (AB = BC),then a = b.

︵︵


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Theorem 4.11:

Arcs are proportional to the angles they subtended atthe circumference, that is,

Fig 4.89

AB : PQ = a : b.︵︵

(Reference: arcs prop. to s at ∠ ⊙ce)


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Definition 4.6:

B. The Angle in a Semicircle


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

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Theorem 4.12:

Fig 4. 97

The angle in a semicircle is 90°.

(Reference: in semicircle∠ )

(Reference: converse of in semicircle∠ )

.90APBAB

then diameter, a is if is, That

diameter. a is then , if ,Conversely

ABAPB 90

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Fig 4.102

Definition 4.7:

Suppose the area of the circle is A.

C. Angles in the same Segment

.

a called is segment , segment of area the Since (a)

segment minor

AQBA

AQB2

.

a called is segment , segment of area the Since (b)

segment major

APBA

APB2

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.

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Fig 4.105

Fig 4.103

The angles in the same segment of a circle are equal,that is,

In Fig. 4.103, APB∠ and AQB∠ are called the angles in the same segment.

Definition 4.8:

Theorem 4.13:

if AB is a chord,then ∠APB = ∠AQB.

(Reference: ∠s in the same segment)

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Fig 4.141

A. Opposite Angles of a Cyclic Quadrilateral

Definition 4.9:

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

2. 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.

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Fig 4.143

The opposite angles in a cyclic quadrilateral are supplementary.

(Reference: opp. ∠s, cyclic quad.)

Theorem 4.14

Symbolically, and

180180

DBCA


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Fig 4.149

Theorem 4.15:

The exterior angle of a cyclic quadrilateral is equal to its interior opposite angle, that is, ψ=θ.

B. Exterior Angles of a Cyclic Quadrilateral

(Reference: ext. , cyclic quad∠ .)

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Fig 4.155

Points are said to be concyclic if they lie on the same circle.

C. Tests for Concyclic Points

Definition 4.10:

Theorem 4.16: (Converse of Theorem 4.13)

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

For example, in Fig 4.155, A, B, C, D and E are concyclic.

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Fig 4.158

Fig 4.157



In Fig. 4.158, if p = q, then A, B, C and D are concyclic.

In Fig. 4.157, if a + c = 180° (or b + d = 180°),then A, B, C and D are concyclic.

(Reference: opp. s supp∠ .)

(Reference: ext. = int. opp. ∠ ∠)

Chapter 4 Basic Properties of Circles (1)

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equal chords

equal arcs

equal angles

equal s

equal length

chord ab

basic terms

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