Chapter 5 Basic Properties Of Circles (2)

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Basic Properties of Circles (2)

Contents

5.1 Tangents to a Circle 5.2 Tangents to a Circle from an External 　 Point 5.3 Angle in the Alternate Segment 5.4 Euclidean Geometry

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Basic Properties of Circles (2)

5.1 Tangents to a circle Definition 5.1 A straight line is called a tangent to a circle if and only if it touches the circle at one and only one point. For example, in Fig. 5.6, AB is a tangent to the circle. The point T common to both the circle and the straight line is called the point of contact (or the point of tangency).

Fig. 5.6

Theorem 5.1

Content

If AB is a tangent to the circle with centre O at T , then AB is perpendicu lar to the radius OT . Symbolically, AB ⊥ OT . (Reference: tangent perp. to radius)

Fig. 5.9

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5.1 Tangents to a circle Theorem 5.2:

OT is a radius of the circle with centre O and AB is a straight line that touches the circle at T . If AB is perpendicu lar to OT , then AB is a tangent to the circle. In other words, if AB ⊥ OT , then AB is a tangent to the circle. (Reference: converse of tangent perp. to radius)

Fig. 5.10

Content

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5.2 Tangents to a Circle from an External Point Theorem 5.3:

In Fig. 5.50, if TA and TB are the two tangents drawn to the circle with centre O from an external point T , then (a) the length of the two tangents are equal, that is, TA = TB; (b) the two tangents subtend equal angles at the centre, that is, ∠TOA = ∠TOB;

Content

Fig. 5.50

(c) the line joining the external point to the centre of the circle is the angle bisector of the angle included by the two tangents lines, that is, ∠OTA = ∠OTB. (Reference: tangent properties)

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5.3 Angle in the Alternate Segment In Fig. 5.86, AB is a tangent to the circle at T and PT is a chord of the circle. ∠ PTA and ∠ PTB are formed by the chord and the tangent. ∠ PTA and ∠ PTB are called tangent-chord angle.

Fig 5.86 Content

The chord PT divides the circle into two segments I and II as shown in Fig. 5.86. Segment II lies on the opposite side of ∠ PTA is called the alternate segment with respect to ∠ PTA. Similarly, segment I is called the alternate segment with respect to ∠ PTB.

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5.3 Angle in the Alternate Segment Theorem 5.4: A tangent-chord angle of a circle is equal to an angle in the alternate segment.

Fig. 5.89

Symbolically, Content

a=b

Fig. 5.90

p=q

(Reference : ∠ in alt. segment )

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5.3 Angle in the Alternate Segment Theorem 5.5: A straight line is drawn through an end point of a chord of a circle. If the angle between the straight line and the chord is equal to an angle in alternate segment, then the straight line is a tangent to the circle. In other words, if x = y, then TA is a tangent to the circle at A.

Fig. 5.92

(Reference : converse of ∠ in alt. segment ) Content

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5.4 Euclidean Geometry A. Introduction Elements is a series of books written in about 300 BC by a very famous Greek mathematician called Euclid ( 歐幾里得 ) who developed Euclidean Geometry. Elements consists of 13 books. In these books, Euclid gave a single deductive chain of 465 propositions neatly and systematically. Book I of the series is about the fundamentals of geometry which includes the theories of triangles, parallels and area. There are 23 necessary basic definitions, 5 postulates and 5 axioms in this book.

Content

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5.4 Euclidean Geometry (a) Definition A definition is a statement that requires only an understanding of the terms being used.

Definition 1: ‘A figure is that which is contained by any boundary or boundaries.’

Content

Fig. 5.128

In Fig. 5.128, ABCD is a figure with four boundaries : AB, BC , CD, DA.

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5.4 Euclidean Geometry Definition 2: ‘A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.’

Fig. 5.129 Content

Referring to Fig. 5.129, 2. The one line is the circumference of the circle. 3. The particular point is the centre of the circle. 4. The equal length is the radius of the circle.

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5.4 Euclidean Geometry Definition 3: ‘A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.’

Fig. 5.130 Content

In Fig. 5.130, AB is a diameter of the circle. AB cuts the circle in two segments Ι and ΙΙ such that the area of segment Ι equals to the area of segment ΙΙ.

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5.4 Euclidean Geometry Definition 4: ‘A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.

Fig. 5.131 Content

In Fig. 5.131, we can see that the semicircle and the circle have the same centre.

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5.4 Euclidean Geometry Definition 5: ‘Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle has two of its sides alone equal and a scalene triangle that which has its three sides unequal.’

In Fig. 5.132, ∆ABC is an isosceles triangle.

Content Fig. 5.132

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5.4 Euclidean Geometry (b) Postulate A postulate is a statement that is assumed to be true without proof. The postulates are all specific to the subject matter.

Postulate 1: ‘A straight line can be drawn from any point to any point.’ Fig. 5.133

In Fig. 5.133, we can draw a straight line form A to B.

Postulate 2: ‘A finite straight line can be produced continuously in a straight line.’ Content Fig. 5.134

In Fig. 5.134, we can extend the straight line from AB to CD.

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5.4 Euclidean Geometry Postulate 3: ‘A circle may be described with any centre and distance.’

Postulate 4:

Fig 5.135

‘All right angles are equal to one another.’

Content

In Fig. 5.136, ∠CDA = ∠GHE = ∠CDB = ∠GHF .

Fig. 5.136

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5.4 Euclidean Geometry Postulate 5: ‘If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are together less than two right angles.’

Content

Fig. 5.137

In Fig. 5.137, if ∠CBE + ∠FEB < 180°, then AC and DF will meet when they are extended in the direction of C and F .

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5.4 Euclidean Geometry (C) Axiom An axiom (which Euclid called common notation in his book) is an assertion, the truth of which is taken for granted as being obvious. Axiom 1: ‘Things which equal to same thing also equal one another.’ Axiom 2: ‘If equals are added to equals, then the wholes are equal.’ Axiom 3: ‘If equals are subtracted from equals, then the remainders are equal.’

Content

Axiom 4: ‘Things which coincide with one another equal one another.’ Axiom 5: ‘The whole is greater than the part.’

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