1. Chapter 9 Circles Define a circle and a sphere.
Apply the theorems that relate tangents, chords and radii.
Define and apply the properties of central angles and arcs.

2. 9.1 Basic Terms Objectives
Define and apply the terms that describe a circle.

3. The Circle
is a set of points in a plane at a given distance from a given point in that plane B A
Blackout slide to show points

4. The Circle
The given distance from the center to any point on the circle is a radius (plural radii)
How many radii does a circle have?
Are they all the same length? B
radius A
center A
“Circle with center A”

5. What is the difference between a line and a line segment?
Think – Pair - Share
What is the difference between a line and a line segment?

6. Chord
any segment whose endpoints are on the circle. C
chord B A

7. Diameter A chord that contains the center of the circle B A
What is another name for half of the diameter?
diameter C

8. Secant
any line that contains a chord of a circle. C
secant B A

9. Tangent any line that contains exactly one point on the circle.
BC is tangent to A B A
tangent C

10. Point of Tangency B
Point of tangency A

11. Common Tangents are lines tangent to more than one coplanar circle.
Common External Tangents
Common Internal Tangents Y B X B R R A A X X B D

12. Tangent Circles are circles that are tangent to each other.
Internally Tangent Circles
Externally Tangent Circles B B R R A A

13. Sphere
is the set of all points in space equidistant from a given point. B A

14. Sphere
Radii
Diameter
Chord
Secant
Tangent C E B A F D

15. Congruent Circles (or Spheres)
WHAT DO THEY HAVE?
have equal radii. B E A D

16. Concentric Circles (or Spheres)
share the same center. G O
Who can think of a real world example?
Think , throwing of pointy objects…. Q

17. Inscribed/Circumscribed
A polygon is inscribed in a circle and the circle is circumscribed about the polygon if each vertex of the polygon lies on the circle.

18. M Q O N R L P Z
Name…
3 radii
Diameter
Chord
Secant
Tangent
What is the name for ZX?
What should point M be called? X

19. White Boards…. Draw the following… An inscribed triangle
A circle circumscribed about a quadrilateral
2 circles with common external tangents
2 circle that are internally tangent to each other

20. 9.2 Tangents
Objectives
Apply the theorems that relate tangents and radii

21. Experiment Supplies: Pencil, protractor, compass
Draw a circle with center A
Draw Point B on the bottom of your circle
Create line BC tangent to the circle at Point B
Draw the radius to the point of tangency and measure the angle formed by the tangent and the radius (L ABC)
Compare your measurements with those around you…

22. If the radius and tangent meet at the point of tangency they form a right angle (perpendicular).B
What can we conclude based on our experiment?
Point of tangency A
radius
tangent C

23. How do we know the 2 right triangles are congruent?X
Dunce Cap Rule
Tangents to a circle from a common point are congruent.
tangent
tangent Z Y A
How do we know the 2 right triangles are congruent?

24. Inscribed/Circumscribed
When each side of a polygon is tangent to a circle, the polygon is said to be circumscribed about the circle and the circle is inscribed in the polygon.
Each side of the poly, is what to the circle?

25. GIVEN BC is tangent to A Radius = 6 BC = 8 Find AC
What allows you to come up with the correct answer?
LC = 45
Radius = 4
Find AC and BC A B C
BC is tangent to A

26. Whiteboards
Line XY is tangent to D at X. The radius of the circle is 7cm and the length of segment XY is 4cm. FIND THE LENGTH OF SEGMENT DY.

27. Whiteboards Create a diagram of the following…
A triangle circumscribed about a circle
A pentagon inscribed inside a circle

28. 9.3 Arcs and Central Angles
Objectives
Define and apply the properties of arcs and central angles.

29. Experiment

30. Think – pair - share
How is the angle measurement that you just created related to the measurement of a circle?

31. Central Angle Arc Sides – 2 radii Vertex – center of circle L C
It’s like cutting out a slice of pizza!!
an arc is part of a circle like a segment is part of a line. A B
This represents the crust of your pizza

32. Central Angle / Arc Measure
the measure of an arc is given by the measure of its central angle. (or vise versa) 80 A C
The central angle tells us how much of the 360 ◦ of crust we are using from our pizza. 80
m L
= 80 B

33. Minor Arc
< 180 C A B

34. Semicircle “_____” a circle.
What indicates that we have half a circle? B A C

35. Major Arc
> 180
We only know how to measure angles up to 180, so how do you find the measure of a major arc? B A C D

36. Practice
Name two minor arcs C R O A S

37. Practice
Name two major arcs C R O A S

38. White Board Practice
Name two semicircles C R O A S

39. Angle addition postulate?
Skip Remember….
Adjacent angles ?
Angle addition postulate?
Smartboard

40. Skip Adjacent Arcs
arcs that have exactly one point in common. D A C B

41. Arc Addition PostulateB C

42. Congruent Central L’s / Arcs Theorem
In the same circle or in congruent circles….
congruent arcs = congruent central angles D E A C B

43. Dissecting a Circle Diagram
FREE VISUAL EVIDENCE !!!
Central angles = minor arcs
All the arcs = 360
Diameters = straight lines = 180
Vertical angles / adjacent angles H C R O A S

44. White Board Practice Find the following measurements… H C 30 100 50
210 R O RH
AOR
HCA 50 A S

45. Pg. 340
#8 - 13

46. Section 9-4 Arcs and Chords
Objectives
Define the relationships between arcs and chords.

47. REVIEW
WHAT IS A CHORD?
WHAT IS AN ARC?

48. Relationship between a chord and an arc
The minor arc between the endpoints of a chord is called the arc of the chord, and the chord between the endpoints of an arc is the chord of the arc. D
Chord BD “cuts off” 2 arcs..
BD and BFD A F B

49. Congruent Chords/Arcs Theorem
In the same circle or in congruent circles…
congruent arcs have congruent chords
congruent chords have congruent arcs. D
If arc BD is congruent to arc FC then… C B A F.

50. Skip - Midpoint/ Bisector of an Arc
Just as we have learned about the bisectors and midpoints of angles and line segments, an arc can be bisected into two congruent arcs D
If D is the midpoint of arc BDC, then……. C A B

51. Circle Handout Experiment
Label the center A
DRAW A CHORD AND LABEL IT DC
FIND THE MIDPOINT OF THE CHORD AND LABEL IT X
DRAW A RADIUS THAT PASSES THROUGH THE MIDPOINT AND INTERSECTS THE ARC OF THE CIRCLE AT Y
USE A PROTRACTOR AND MEASURE LAXC

52. Think – Pair - Share
What facts can you conclude about the arcs, chords, or any other segments?
What is congruent to what?
What about perpendicular?

53. Chord Bisector Theorem
Any line perpendicular to a chord bisects the chord and its arc. D Y X C A B
EC: What other 2 segments do I know are congruent that are not explicitly shown?

54. Remember
When you measure distance from a point to a line, you have to make a perpendicular line. A

55. Putting Pythagorean to Work..
Can we use the given information to make a conclusion about the chords shown? (i.e. length of the chords)
Hint: Just because something is not shown, doesn’t mean it doesn’t exist! (other radii)
AY = 3 AX = 3 E D Y X B C A 4

56. Chord Distance Theorem
In the same circle or in congruent circles:
Chords equally distant from the center are congruent
Congruent chords are equally distant from the center. E D Y B X C A

57. Whiteboards THE HEAD BAND PROBLEM:
Find the length of a chord 3cm from the center of a circle with a radius of 7cm.

58. WHITEBOARDS Solve for x and y x = 12 y = 12C
x = 12
y = 12 5 A 13
IF arc DB is 55 degrees, then arc CB is?

59. WHITEBOARDS
Solve for x and y
x = 8
y = 16 8 y x

60. WARM - UP What does the term inscribed mean to you in your own words?
Describe the placement of the vertices of an inscribed triangle
What do we call the 2 sides and vertex (in circle terms) of a central angle that you learned in 9.3?
What is the measure of the angle equal to?

61. 9.5 Inscribed Angles Objectives
Solve problems and prove statements about inscribed angles.
Solve problems and prove statements about angles formed by chords, secants and tangents.

62. What we’ve learned…
Inscribed means that something is inside of something else – we have looked at inscribed polys and circles
We know that an angle by definition has a vertex and 2 sides that meet at the vertex
In a central angle…
The vertex is the center and the sides are radii

63. Inscribed Angle Sides – two chords Vertex – point on the circle A
“Intercepted arc” C
What do you think the sides are in ‘circle terms’? Where is the vertex? B

64. Experiment Measure the inscribed angle created with a protractor
Using the endpoints of the intercepted arc, draw 2 radii to create a central angle and then measure.
Compare the measurement of the inscribed angle with that of the central angle measure.
Discuss with your partner

65. Theorem
inscribed angle = half the measure of the intercepted arc. A
___ 2 C B

66. Corollary
If two inscribed angles intercept the same arc, then they are congruent. A D C B
Don’t write down, just recgonize

67. WHITEBOARDS r = 50 s = 50 x = 160 y = 100 z = 100
Find the values of r, s, x, y , and z
Take inventory of the diagram before trying to solve!
Concentrate on parts of the whole y◦ r◦
r = 50
s = 50
x = 160
y = 100
z = 100 x◦ O 80 s◦ z◦

68. Corollary
If the intercepted arc is a semicircle, the inscribed angle must = 90. A
What is the measure of an inscribed angle whose intercepted arc has the endpoints of the diameter? B O C

69. Corollary
Quad inscribed = opposite angles supplementary. A B O D C

70. Treat this angle the same as you would an inscribed angle!
Theorem
___ 2 A B O C D F
Treat this angle the same as you would an inscribed angle!

71. Whiteboards
Page 353
#5, 6, 7

72. X = 30 Y = 60 Z = 150 WHITEBOARDS Find the values of x, y , and z 60◦

73. 9.6 Other Angles Objectives
Solve problems and prove statements involving angles formed by chords, secants and tangents.

74. 1 2 4 3 WARM UP: For this diagram….
Write down the different equations that represent the angle relationships shown.
There’s more than one! 1 2 4 3

75. _________ 2 _________ 2 The angle formed by two intersecting chords =
Partners: How do you think we can determine the measure of L1?
The angle formed by two intersecting chords =
half the sum of the intercepted arcs. B
_________ 2 C 1 A 2
_________ 2 E D

76. Angles formed from a point outside the circle…
2 secants
2 tangents
In each circle, 2 arcs are being intercepted by the angle.
1tangent
1 secant
The larger arc is always further away from the vertex.

77. _________ 2 Vertex outside the circle =
half the difference of the intercepted arcs.
_________ 2 B E
large
small 1 C A F D

78. WHITEBOARDS ONE PARTNER OPEN BOOK TO PG. 358 ANSWER #1 35 ANSWER # 640
ANSWER # 4 80
ANSWER # 7
X=50

79. AB is tangent to circle O
AB is tangent to circle O. AF is a diameter m AG = 100 m CE = 30 m EF = 25 B D C 6 30 E 8 25 O 3 5 A F 7 2 1 4
100 G

80. WARM – UP
READ PG. 361 – 363
Identify what elements are involved in each of the 3 theorems in this section
Example: “Theorem 9-11 refers to the relationship of 2 ___________ intersecting”
What is the idea behind this section…. Angles, segments, circles?

81. ANGLES QUIZ x C D 8 B 4 9 E H 2 7 K 3 A 6 5 1 F G I
Identify a numbered angle that represents each of the bullet points.
Write an equation representing the measure of the angle
i.e. m L12 =
X is center of circle C D 8 B 4 9 E H 2 7 K 3 A x 6 5 1
Central angle
Inscribed angle
Angle formed inside
Angle formed outside
Name a 90 angle…. F G I