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. Circle C C
Diameter = _ radius

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

5. A secant is X Y C A B YX AB

6. A tangent is C Y X XY AB A B

7. Common Tangent Lines
internal tangents

8. Common Tangent Lines
external tangents

10. 2 points of intersection
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. no point of intersection
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 Ð
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. 13. Find CA. C 7 D B 15
What is DA? A

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

22. 15. Is AB a tangent?
How do we test if 3 segments create a right triangle? C 10 7 26 B 6 24 15 A

23. 16. Is AB a tangent? C 8 7 17 B 6 12 15 A

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

25. One endpoint is the point of tangency.
A tangent segment A B C
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. 18. Find x. B
7x - 2 A C
3x + 8

28. 19. Find x. B
x2 + 25 A 50 C

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

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

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

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

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

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

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

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

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

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

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

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.
2x x

48. 1. Find x.
x = 60°
120° x°

49. 2. Find x.
x = 140° x°
70°

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

51. 3. Find x and y. x°
45° y°

52. Inscribed
Pentagon

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

54. Theorem 10.10
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.C A y° 42 D x° B

56. Theorem 10.11 If a quadrilateral is inscribed in a circle, then its
opposite angles are supplementary. 2 1 4 3
m 1 + m 3 = 180º
m 2 + m 4 = 180º

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

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

60. Angle Relationships in Circles
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. If a tangent and a chord intersect at a point on a circle, then...
Theorem 10.12
If a tangent and a chord intersect at a point on a circle, then...
GSP

62. … the measure of each angle formed is half
Theorem 10.12
… the measure of each angle formed is half
the measure of its
intercepted arc.

63. B C 2 1 A

64. 1. Find m 1 and m 2. B C
100° 2 1 A

65. 2. Find and mACB and mAB
95° A B C

66. 5x° A B C
(9x + 20)˚
3. Find x

67. If 2 chords intersect inside a circle, then…
Theorem 10.13
If 2 chords intersect inside a circle, then… A C 1 B D

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

69. 4. Find x.
100° A B C D x°
120°

70. 5. Find x.
130° A B C D x°
160°

71. 6. Find x. A C x°
80° y°
90° B D

72. 7. Find x. x° A B C D
100°
120°

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

74. If a tangent and a secant,
Theorem 10.14
If a tangent and a secant,
two tangents, or two secants intersect outside a circle, then… A C D 1

75. If a tangent and a secant,
Theorem 10.14
If a tangent and a secant,
two tangents, or two secants intersect outside a circle, then… A B C 1

76. If a tangent and a secant,
Theorem 10.14
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. A B C D x°
20°
80°

79. 10. Find x.
24°
90° A B C D x°

80. 11. Find x. x°
200°

81. 12. Find x. A C D
135° x°

82. 13. Find x.
100° 2 3
60°
100° 1
100°

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 10.15
If 2 chords intersect inside a circle, then the product of their “segments” are equal.

86. a c d b
a · b = c · d

87. 1. Find x. 6 x 8 4

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

89. 3. Find x. 2x 18 x 4

90. Theorem 10.16 If 2 secant segments share the same endpoint outside a circle, then… GSP

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

92. c a b d
a · c = b · d

93. 5 x 4 6
3. Find x.

94. 9 10 x 20
4. Find x.

95. Theorem 10.17 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 b d
a · a = b · d
a2 = b · d

98. 5 4 x
5. Find x.

99. 15 x 10
6. Find x.

100. ♫ ♪
Quadratic Formula?

101. 15 x 10
6. Find x.

102. x 20 31
7. Find x.

103. 3 4 8 x
8. Find x.

104. 10 x 8
9. Find x.

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.

112. 4.Write an equation of the circle.

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 = 100
Center: (1, -3)
radius = 10
6. x2 + (y - 7)2 = 8
Center: (0, 7)
radius ≈ 2.83
7. (x + 1)2 + y2 = ¼
radius = ½
Center: (-1, 0)

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)
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 = 100
11. (11, 13)
12. (6, -5)
13. (19, - 4)

123. Circumference and Area of Circles
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. 1. Find the length of AB A B
50°
7 cm

127. 2. Find the radius A
10 cm
85° B

128. 3. Find the
circumference.

129. Area

130. A region bound by two radii & their intercepted arc.
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.
50° B
7 cm

133. A B
100°
4. Find the radius.

134. 3. Find the
area.

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