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Lesson 10.1 Parts of a Circle Today, we are going to… > identify segments and lines related to circles > use properties of tangents to a circle.

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Presentation on theme: "Lesson 10.1 Parts of a Circle Today, we are going to… > identify segments and lines related to circles > use properties of tangents to a circle."— Presentation transcript:

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2 Lesson 10.1 Parts of a Circle Today, we are going to… > identify segments and lines related to circles > use properties of tangents to a circle

3 C Circle C Diameter = _ radius

4 C A chord is YX AB A B X Y N BN

5 C A secant is A B X Y YX AB

6 C A tangent is AB A B Y X XY

7 internal tangents Common Tangent Lines

8 external tangents

9

10 Two circles can intersect in 2, 1, or 0 points. Draw 2 circles that have 2 points of intersection

11 internally tangent circles Draw two circles that have 1 point of intersection

12 externally tangent circles Draw two circles that have 1 point of intersection

13 concentric circles Draw two circles that have no point of intersection

14 9. What are the center and radius of circle A? Center: Radius =

15 10. What are the center and radius of circle B? Center: Radius =

16 11. Identify the intersection of the two circles.

17 12. Identify all common tangents of the two circles.

18 m  ABC = A B C

19 Theorem 10.1 & 10.2 A line is tangent to a circle if and only if it is _____________ to the radius from the point of tangency. A B C

20 7 13. Find CA. 15 D C B A What is DA?

21 7 14. Find x. 15 x 6 C B A xx 16 8 What is CA?

22 C B A How do we test if 3 segments create a right triangle? 15. Is AB a tangent?

23 C B A Is AB a tangent?

24 17. Find the slope of line t. A C A (3,0) and C (5, -1) Slope of AC? Slope of line t? t

25 C A tangent segment A B One endpoint is the point of tangency.

26 Theorem 10.3 If 2 segments from the same point outside a circle are tangent to the circle, then they are congruent.

27 7x - 2 3x Find x. A C B

28 x Find x. A C B

29

30 Lesson 10.2 Arcs and Chords Today, we are going to… > use properties of arcs and chords of circles

31 C An angle whose vertex is the center of a circle is a central angle. A B

32 C Minor Arc - Major Arc A B D Minor Arc AB Major Arc ADB

33 C A B D 60˚ m AB = Measures of Arcs

34 C Semicircle m AED = m ABD = m AD A B D E

35 Find the measures of the arcs. 1. m BD 2. m DE 3. m FC 4. m BFD D E F B C 100˚ 52˚ 68˚ 53˚ ?

36 AD and EB are diameters. F A B D E C 5. Find x, y, and z. 30˚ z˚z˚ x˚x˚ y˚y˚ x = y = z =

37 Theorem 10.4 Two arcs are congruent if and only if their chords are congruent.

38 (2x + 48)° (3x + 11)° B A D C 6. Find m AB

39 Theorem 10.5 & 10.6 A chord is a diameter if and only if it is a perpendicular bisector of a chord and bisects its arc.

40 7. Is AB a diameter? A B

41 8. Is AB a diameter? A B 8 8

42 9. Is AB a diameter? A B

43 Theorem 10.7 Two chords are congruent if and only if they are equidistant from the center.

44 AB = Find CG. DE = 12 7 D G B A C F E 6 x ?

45 Lesson 10.3 Inscribed Angles Today, we are ALSO going to… > use properties of inscribed angles to solve problems

46 An inscribed angle is an angle whose vertex is on the circle and whose sides contain chords of the circle.

47 Theorem 10.8 If an angle is inscribed, then its measure is half the measure of its intercepted arc. xx 2x 

48 1. Find x. x°x° 120° x = 60°

49 2. Find x. x°x° 70° x = 140°

50 Theorem 10.9 If 2 inscribed angles intercept the same arc, then the angles are congruent.

51 3. Find x and y. y°y° 45° x°x°

52 Inscribed Pentagon

53 x°x° A D C B 4. DC is a diameter. Find x.

54 Theorem If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.

55 5. Find the values of x and y. x°x° y°y° A 42  D C B

56 Theorem If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary m  1 + m  3 = 180º m  2 + m  4 = 180º

57 6. Find the values of x and y. x°x° 110° 80° y°y°

58 7. Find the values of x and y. x°x° 120° 100° y°y°

59

60 Lesson 10.4 Angle Relationships in Circles Today, we are going to… > use angles formed by tangents and chords to solve problems > use angles formed by intersecting lines to solve problems

61 Theorem If a tangent and a chord intersect at a point on a circle, then... GSP

62 Theorem … the measure of each angle formed is half the measure of its intercepted arc.

63 1 A B C 2

64 1 A B C 2 1. Find m  1 and m  °

65 2. Find and mACB and mAB 95° A B C

66 3. Find x 5x° A B C (9x + 20)˚

67 Theorem If 2 chords intersect inside a circle, then… A B C D 1

68 B C A D 1 …the measure of the angle is half the sum of the intercepted arcs.

69 A B C D x°x° 4. Find x. 100° 120°

70 A B C D x°x° 5. Find x. 130° 160°

71 A B C D x°x° 6. Find x. 80° 90° y°y°

72 A B C D x°x° 7. Find x. 100° 120°

73 A B C D x°x° 8. Find x. 52° 74° Do you notice a pattern?

74 Theorem If a tangent and a secant, two tangents, or two secants intersect outside a circle, then… A C D 1

75 Theorem If a tangent and a secant, two tangents, or two secants intersect outside a circle, then… A B C 1

76 Theorem If a tangent and a secant, two tangents, or two secants intersect outside a circle, then… A B C D 1

77 A B C D 1 …the measure of the angle is half the difference of the intercepted arcs.

78 9. Find x. 20° 80° A B C D x°x°

79 10. Find x. 24° 90° A B C D x°x°

80 11. Find x. 200° x°x°

81 A C D 12. Find x. 135° x°x°

82 13. Find x. 100° °

83

84 Lesson 10.5 Segment Lengths in Circles Today, we are going to… > find the lengths of segments of chords, tangents, and secants

85 Theorem If 2 chords intersect inside a circle, then the product of their “segments” are equal.

86 a · b = c · da · b = c · d a b c d

87 1. Find x x

88 2. Find x. 3x 18 2x 3

89 3. Find x. 2x 18 x 4

90 Theorem If 2 secant segments share the same endpoint outside a circle, then… GSP GSP

91 …one secant segment times its external part equals the other secant segment times its external part.

92 a · c = b · d b a c d

93 3. Find x. 5 x 4 6

94 4. Find x x 20

95 Theorem Theorem If a secant segment and a tangent segment share an endpoint outside a circle, then…

96 …the length of the tangent segment squared equals the length of the secant segment times its external part.

97 a · a = b · d d b a a2 = b · da2 = b · d

98 5 4 x 5. Find x.

99 15 x Find x.

100 Quadratic Formula? ♫♪♫♪♫♪♫♪♫♪♫♪♫♪♫♪♫♪♫♪♫♪♫♪

101 15 x Find x.

102 x Find x.

103 8. Find x x

104 10 x 8 9. Find x.

105

106 Lesson 10.6 Equations of Circles Today, we are going to… > write the equation of a circle

107 Standard Equation for a Circle with Center: (0,0) Radius = r

108 1. Write an equation of the circle.

109 2. Write an equation of the circle.

110 Standard Equation for a Circle with Center: (h,k) Radius = r

111 3.Write an equation of the circle. C = r =

112 4.Write an equation of the circle. C = r =

113 Graph (x – 3) 2 + (y + 2) 2 = 9 Center? Radius =

114 Identify the center and radius of the circle with the given equation. 5. (x – 1) 2 + (y + 3) 2 = x 2 + (y - 7) 2 = 8 7. (x + 1) 2 + y 2 = ¼ Center: (1, -3)radius = 10 Center: (0, 7)radius ≈ 2.83 Center: (-1, 0) radius = ½

115 Write the standard equation of the circle with a center of (5, -1) if a point on the circle is (1,2).

116 8. Write the standard equation of the circle with a center of (-3, 4) if a point on the circle is (2,-5).

117 Is (-2,-10) on the circle (x + 5) 2 + (y + 6) 2 = 25?

118 9. Is (0, - 6) on the circle (x + 5) 2 + (y – 5) 2 = 169?

119 10. Is (2, 5) on the circle (x – 7) 2 + (y + 5) 2 = 121?

120 > < =

121 Would the point be inside the circle, outside the circle, or on the circle? (x – 13) 2 + (y - 4) 2 = (11, 13) 12. (6, -5) 13. (19, - 4)

122

123 Lessons 11.4 & 11.5 Circumference and Area of Circles Today, we are going to… > find the length around part of a circle and find the area of part of a circle

124 Circumference

125 Arc Length = A B

126 A B 50° 7 cm 1. Find the length of AB

127 A B 85° 10 cm 2. Find the radius

128 3. Find the circumference.

129 Area

130 Sector of a circle A region bound by two radii & their intercepted arc. A slice of pizza!

131 Area of a Sector =

132 3. Find the area of the sector. A B 50° 7 cm

133 4. Find the radius. A B 100°

134 3. Find the area.

135 Workbook P. 211 (1 – 10) P. 215 (1 – 6)

136


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