Chapter 6. Sir Migo Mendoza

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1 Circles Chapter 6 Sir Migo Mendoza

2 Central Angles Lesson 6.1 Sir Migo Mendoza

3 Central Angles

4 Definition 5.1 Arc An arc is a part of a circle.

5 Types of Arc Minor Arc Major Arc Semicircle

6 Definition 5.2 Central Angle A central angle of a circle is an angle whose vertex is the center of the circle.

7 Definition 5.2 Central Angle

8 Definition 5.3 Minor Arc Minor arc consists of points M and I and all points of O that are in the interior of central angle MOI.

9 Definition 5.3 Minor Arc

10 Definition 5.4 Major Arc Major arc consists of points M and I and all points of O that are in the exterior of central angle MOI.

11 Definition 5.4 Major Arc

12 Definition 5.5 Semicircle Semicircle consists of endpoints D and M of diameter DM and all points O that lie on one side of DM.

13 Definition 5.5 Semicircle

14 Note: The degree measure of an arc is defined in terms of its central angle.

15 Definition 5.6 Degree Measure of Minor The degree measure of minor is equal to the degree measure of central angle MOI.

16 Definition 5.6 Degree Measure of Minor

17 Definition 5.7 Degree Measure of Major The degree measure of major is equal to 360 minus the degree measure of central angle MOI.

18 Definition 5.7 Degree Measure of Major

19 Definition 5.8 Degree Measure of Semicircle The degree measure of semicircle DIM is equal to 180.

20 Definition 5.8 Degree Measure of Semicircle

21 Definition 5.9 Congruent Circles Congruent Circles are circles with the same radius.

22 Definition 5.10 Congruent Arc Congruent Arcs are arcs with the same measures.

23 Postulate 5.1 The Central Angle- Intercepted Arc Postulate (CA-IA Postulate) The measure of a central angle of a circle is equal to the measure of its intercepted arc.

24 Example

25 Postulate 5.2 The Arc Addition Postulate

26 Postulate 5.3 A diameter divides a circle into two semicircles.

27 Definition 5.11 Arc Length The measure of the central angle can also be used to determine the arc length. The arc length (or length of an arc) is different from the degree measure of an arc. That is, if a circle is made up of string, the length of the arc is the linear distance of the piece of string representing the arc. The length of the arc is a part of the circumference and proportional to the measure of the central angle when compared to the entire circle.

28 Example:

29 Let s Practice:

30 Let s Practice:

31 Let s Practice:

32 Inscribed Angle Lesson 6.2 Sir Migo Mendoza

33 Definition 5.12 Inscribed Angle An inscribed angle in a circle is an angle whose vertex is on the circle and whose sides contain chords of the circle.

34 Example: Considering the definition, which among these three is/are inscribed angle?

35 Inscribed Angle?

36 Theorem 5.1 The Inscribed Angle Theorem The measure of an inscribed angle is one-half the measure of its intercepted arc.

37 Proof:

38 Example:

39 Theorem 5.2 The Semicircle Theorem An angle inscribed in a semicircle is a right angle.

40 Theorem 5.3 Inscribed Angles in the Same Arc Theorem Two or more angles inscribed in the same arc are congruent.

41 Example:

42 Prove This:

43 Direction: Use the given figures to find the value of x.

44 Tangents Lesson 6.3 Sir Migo Mendoza

45 Definition 5.13 Tangent to a Circle A line in the plane of the circle that intersects the circle at exactly one point is called tangent line.

46 Definition 5.14 Point of Tangency It is the point of intersection between a tangent line and a circle.

47 Note: A circle separates a plane into three parts: 1. the interior; 2. the exterior; and 3. the circle itself.

48 Theorem 5.4 The Tangent-Line Theorem If a line is tangent to a circle, then it is perpendicular to the radius at its outer endpoint.

49 Theorem 5.4 The Tangent-Line Theorem

50 Theorem 5.5 The Converse of the Tangent-Line Theorem In a plane, if a line is perpendicular to a radius of a circle at the endpoint on the circle, then the line is a tangent to the circle.

51 Note: Theorem 5.5 can be used to identify tangents to a circle.

52 Example:

53 Theorem 5.6 The Tangent- Segment Theorem If two tangent segments are drawn to a circle from an external point, then: 1. the two tangent segments are congruent, and 2. the angles between the tangent segments and the line joining the external point to the center of the circle are congruent.

54 Proof:

55 Definition 5.14 Common Tangent A line or a segment that is tangent to two circles in the same plane is called a common tangent of the two circles.

56 Types of Common Tangents 1. Common External Tangent 2. Common Interior Tangents

57 Common External Tangent Common external tangents do not intersect the segment whose endpoints are the centers of the circles.

58 Common External Tangent

59 Common Interior Tangents Common interior tangents intersect the segments whose endpoints are the centers of the circles.

60 Common Interior Tangent

61 Theorem 5.7 The Tangent Circles Theorem If two circles are tangent internally or externally, then their line of centers pass through the point of contact.

62 Internally Tangent Circles

63 Externally Tangent Circles

64 Proof:

65 Definition 5.15 Tangent Circles These are two circles whose intersection is exactly one point.

66 Definition 5.16 Line of Centers It is the segment joining the centers of two circles.

67 Definition 5.17 Common Tangent It is a line which is tangent to two circles.

68 Definition 5.18 Common Internal Tangent It is a common tangent which intersects the line of centers.

69 Definition 5.19 Common External Tangent It is a common tangent which does not intersect the line of centers.

70 Definition 5.20 Internally Tangent Circles These are tangent circles whose common tangent does not intersect the line of centers.

71 Definition 5.21 Externally Tangent Circles These are tangent circles whose common tangent intersects the line of centers.

72 Prove This:

73 Let s Practice:

74 Chords and Arcs Lesson 6.4 Sir Migo Mendoza

75 Theorem 5.8 The perpendicular from the center of the circle to any chord bisects the chord.

76 Proof:

77 Theorem 5.9 The line joining the center of the circle to the midpoint of any chord which is not a diameter is perpendicular to the chord.

78 Proof:

79 Theorem 5.10 The perpendicular bisector of a chord of a circle passes through the center of the circle.

80 Proof:

81 Theorem 5.11 The perpendicular bisector of a chord of a circle bisects the central angle subtended by the chord.

82 Proof:

83 Theorem 5.12 The bisector of a central angle subtended by the chord is the perpendicular bisector of the chord.

84 Proof:

85 Theorem 5.13 In the same circle or in congruent circles, chords are congruent if and only if their distances from the center(s) of the circle(s) are equal. Note: Congruent circles are those whose radii are congruent.

86 Proof:

87 Theorem 5.14 In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

88 Proof:

89 Direction: Find x in each figure.

90 Angles Formed by Secants, Tangents, and Chords Lesson 6.5 Sir Migo Mendoza

91 Introduction: There are four ways for the two intersecting lines to intersect a circle. These are:

92 (1) Inscribed Angle

93 (2) an angle formed by a tangent and a secant;

94 (3) an angle formed by two secants intersecting in the interior of the circle; and

95 (4) an angle formed by two secants intersecting in the exterior of the circle.

96 Theorem 5.15 The Intersecting Secants-Exterior Theorem The measure of an angle formed by two secants that intersect in the exterior of a circle is one-half the difference of its intercepted arcs.

97 Proof:

98 Example: The angle between two secants intersecting in the exterior of the circle is 55. If one of the intercepted arcs measures 150, what is the degree measure of the other arc?

99 Theorem 5.16 The measure of an angle formed by a tangent and a secant drawn at the point of contact is one-half the measure of its intercepted arc.

100 Proof:

101 Example:

102 Theorem 5.17 The Intersecting Secants-Interior Theorem The measure of an angle formed by two secants intersecting in the interior of the circle is equal to one-half the sum of the measures of its intercepted arcs.

103 Example:

104 Example:

105 Let s Practice:

106 Let s Practice:

107 The Power Theorems Lesson 6.6 Sir Migo Mendoza

108 Theorem 5.18 The Intersecting Segments of Chords Power Theorem If two chords intersect in the interior of the circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord

109 Example:

110 Theorem 5.19 The Segments of Secants Power Theorem If two secants intersect in the exterior of the circle, the product of the length of one secant segment and the length of its external part is equal to the product of the length of the other secant segment and the length of its external part.

111 Example:

112 Theorem 5.20 The Tangent Secant Segments Power Theorem If a tangent segment and a secant intersect in the exterior of a circle, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external part.

113 Example:

114 Let s Practice: Find the value of x.

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