5.1 Tangents to a Circle

5.2 Tangents to a Circle from an External 　 　　　 Point

5.3 Angle in the Alternate Segment

Contents5 Basic Properties of Circles

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5.4 Euclidean Geometry

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5.1 Tangents to a circle

A straight line is called a tangent to a circle if and only if it touches the circle at one and only one point.

Definition 5.1

Fig. 5.6

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).

Theorem 5.1

..

OTABOTABTOAB

ly,Symbolical radius the to larperpendicu is then ,

at centre withcircle the to tangent a is If

Fig. 5.9(Reference: tangent perp. to radius)

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Theorem 5.2:

circle. the to tangent a is then , to larperpendicu is If .at circle the touches that line straight a is

and centre withcircle the of radius a is

ABOTABTAB

OOT

Fig. 5.10

5.1 Tangents to a circle

circle. the totangent a is then, if words,other In ABOTAB

(Reference: converse of tangent perp. to radius)

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Theorem 5.3:

then , point external an from centre withcircle the to drawn

tangents two the are and if 5.50, Fig. In

TO

TBTA

Fig. 5.50

;

TBTA is, that equal, are tangents two the of length the(a)

; TOBTOA is, that centre, the at angles equal subtend tangents two the (b)

.OTBOTA is, that lines, tangents two theby included angle the of bisector angle

the is circle the of centre the to point external the joining line the (c)

5.2 Tangents to a Circle from an External Point

(Reference: tangent properties)

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5.3 Angle in the Alternate Segment

Fig 5.86

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.

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.

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Theorem 5.4:

A tangent-chord angle of a circle is equal to an angle in the alternate segment.

Symbolically, a = b p = q

) :(Reference segmentaltin .

5.3 Angle in the Alternate Segment

Fig. 5.89 Fig. 5.90

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Theorem 5.5:

Fig. 5.92. at circle the to tangent a is then , if words,other In

ATAyx

5.3 Angle in the Alternate Segment

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.

). segmentaltinofconverse :(Reference

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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.

5.4 Euclidean Geometry

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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.’

.,,, DACDBCABABCD :boundaries four withfigure a is 5.128, Fig. In

Fig. 5.128

5.4 Euclidean Geometry

(a) Definition

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Content Referring to Fig. 5.129,1. The one line is the circumference of the circle.2. The particular point is the centre of the circle.3. The equal length is the radius of the circle.

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

5.4 Euclidean Geometry

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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.’

.

segment of area the to equals segment of area the that such and segments

two in circle the cuts circle. the of diameter a is 5.130, Fig. In ABAB

Fig. 5.130

5.4 Euclidean Geometry

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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

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

5.4 Euclidean Geometry

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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.’

triangle. isosceles an is 5.132, Fig. In ABC

Fig. 5.132

5.4 Euclidean Geometry

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(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.’

Postulate 2:

‘A finite straight line can be produced continuously in a straight line.’

Fig. 5.133

Fig. 5.134

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

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

5.4 Euclidean Geometry

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Postulate 3:

‘A circle may be described with any centre and distance.’

.GHFCDBGHECDA 5.136, Fig. In

Fig 5.135

Fig. 5.136

Postulate 4:

‘All right angles are equal to one another.’

5.4 Euclidean Geometry

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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.’

.180

FCDFACFEBCBE

and of direction the in extended arethey whenmeet will and then , if 5.137, Fig. In

Fig. 5.137

5.4 Euclidean Geometry

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(C) AxiomAn 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.’

Axiom 4:‘Things which coincide with one another equal one another.’

Axiom 5:‘The whole is greater than the part.’

5.4 Euclidean Geometry