Circles (C) Cluster

Tim Brzezinski, JohnCMorris, Nov 14, 2017

MAFS.912.G-C.1.1 Prove that all circles are similar. MAFS.912.G-C.1.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. MAFS.912.G-C.1.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. MAFS.912.G-C.2.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Table of Contents

  1. MAFS.912.G-C.1.1
    1. Congruent Circles: Definition
    2. Animation 89
    3. Similar Circles?
  2. MAFS.912.G-C.1.2
    1. Circle Terminology
    2. Circle Tangent-Types Illustrator
    3. Properties of Tangents Drawn to Circles (A)
    4. Properties of Tangents Drawn to Circles (B)
    5. Animation 3
    6. Tangents to Circles: Investigation
    7. Where to Sit? (I)
    8. Where to Sit (II)?
    9. Inscribed Angle Theorem (V1)
    10. Animation 12
    11. Inscribed Angle Theorem: Take 2!
    12. Inscribed Angle Theorems: Take 3!
    13. Proof Exercise: Inscribed Angle Theorem (Case 1)
    14. Proof Exercise: Inscribed Angle Theorem (Case 2)
    15. Proof Exercise: Inscribed Angle Theorem (Case 3)
    16. Inscribed Angle Theorems: Take 4!
    17. Inscribed Angle Theorem: Corollary 1
    18. Animation 20 (Inscribed Angle Dance!)
    19. Thales' Theorem (VA)
    20. Animation 116
    21. Thales' Theorem (VB)
    22. Animation 54
    23. Angle Formed by Chord & Tangent (I)
    24. Animation 50
    25. Angle Formed by Chord & Tangent (II)
    26. Animation 78
    27. Congruent Chords: Quick Investigation
    28. Congruent Chords (I)
    29. Animation 76
    30. Diameters & Chords I (VA)
    31. Diameters & Chords I (VB)
    32. Animation 24
    33. Congruent Chords Action (A)
    34. Animation 96
    35. Equidistant Chord Action
    36. Animation 101
    37. Angle Formed by 2 Chords (I)
    38. Animation 32
    39. Angle Formed by 2 Chords (II)
    40. Animation 33
    41. Angles from Secants and Tangents (V1)
    42. Animation 61
    43. Angle From 2 Secants (V2)
    44. Animation 45
    45. Crossing Chords: Proof Hint
    46. Animation 79
    47. Crossing Chords Property & Proof Start
    48. Crossing Chords Property & Proof Start (II)
    49. Secants: Proof Hint
    50. Animation 88
    51. Animation 92
  3. MAFS.912.G-C.1.3
    1. Circumcircle: Construction Exercise (VA)
    2. Circumcircle: Construction Exercise (VB)
    3. Circumcircle: Construction Exercise (VC)
    4. Incircle: Construction Exercise (VA)
    5. Incircle: Construction Exercise (VB)
    6. Cyclic Quadrilaterals (IAT: Corollary 3)
    7. Animation 43
    8. Where in NYC? (VA)
    9. Where in NYC? (VB)
    10. Where in NYC? (VC)
  4. MAFS.912.G-C.2.5
    1. Radius Unwrapper v2.0
    2. 1 Radian: Clear Definition
    3. Movie: Radians to Revs
    4. Arc length: Quick Investigation
    5. Area of a Sector
    6. Arc Length and Area of a Sector (V1)
    7. Arc Length and Area of a Sector (V2)

1.

MAFS.912.G-C.1.1

  • 1. Congruent Circles: Definition
  • 2. Animation 89
  • 3. Similar Circles?

1.1.

Congruent Circles: Definition

[color=#000000]The applet below demonstrates what it means for 2 circles to be [b]congruent circles. [/b] [br]Interact with this applet for a minute or two, then answer the writing prompt that follows. [br][i]Be sure to change the locations of the points around each time before re-sliding the slider.[/i][/color]
Complete the following sentence definition: [br][br]Two circles are said to be congruent circles [color=#1e84cc][b]if and only if...[/b][/color]
– Tim Brzezinski

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2.

MAFS.912.G-C.1.2

  • 1. Circle Terminology
  • 2. Circle Tangent-Types Illustrator
  • 3. Properties of Tangents Drawn to Circles (A)
  • 4. Properties of Tangents Drawn to Circles (B)
  • 5. Animation 3
  • 6. Tangents to Circles: Investigation
  • 7. Where to Sit? (I)
  • 8. Where to Sit (II)?
  • 9. Inscribed Angle Theorem (V1)
  • 10. Animation 12
  • 11. Inscribed Angle Theorem: Take 2!
  • 12. Inscribed Angle Theorems: Take 3!
  • 13. Proof Exercise: Inscribed Angle Theorem (Case 1)
  • 14. Proof Exercise: Inscribed Angle Theorem (Case 2)
  • 15. Proof Exercise: Inscribed Angle Theorem (Case 3)
  • 16. Inscribed Angle Theorems: Take 4!
  • 17. Inscribed Angle Theorem: Corollary 1
  • 18. Animation 20 (Inscribed Angle Dance!)
  • 19. Thales' Theorem (VA)
  • 20. Animation 116
  • 21. Thales' Theorem (VB)
  • 22. Animation 54
  • 23. Angle Formed by Chord & Tangent (I)
  • 24. Animation 50
  • 25. Angle Formed by Chord & Tangent (II)
  • 26. Animation 78
  • 27. Congruent Chords: Quick Investigation
  • 28. Congruent Chords (I)
  • 29. Animation 76
  • 30. Diameters & Chords I (VA)
  • 31. Diameters & Chords I (VB)
  • 32. Animation 24
  • 33. Congruent Chords Action (A)
  • 34. Animation 96
  • 35. Equidistant Chord Action
  • 36. Animation 101
  • 37. Angle Formed by 2 Chords (I)
  • 38. Animation 32
  • 39. Angle Formed by 2 Chords (II)
  • 40. Animation 33
  • 41. Angles from Secants and Tangents (V1)
  • 42. Animation 61
  • 43. Angle From 2 Secants (V2)
  • 44. Animation 45
  • 45. Crossing Chords: Proof Hint
  • 46. Animation 79
  • 47. Crossing Chords Property & Proof Start
  • 48. Crossing Chords Property & Proof Start (II)
  • 49. Secants: Proof Hint
  • 50. Animation 88
  • 51. Animation 92

2.1.

Circle Terminology

[color=#000000]There are many vocabulary terms we use when talking about a circle. [br]The following app was designed to help you clearly see and interact with each term. [br][br]Explore this app for a few minutes. Then answer the questions that follow. [/color]
Note: LARGE POINTS are moveable.
How would you describe or define a [b]CIRCLE[/b] as a locus (set of points that meets specified criteria)?
How would you describe the term [b][color=#38761d]RADIUS[/color] [/b][i]without using the words "half" or "diameter" [/i]in your description?
What does the term [b][color=#9900ff]CHORD[/color] [/b]mean here in the context of a circle?
How would you describe the term [b][color=#ff7700]DIAMETER[/color] [/b][i]without using the words "two", "double", or "diameter" [/i]in your description?
How would you describe/define the term [b][color=#cc0000]SECANT[/color][/b]?
What does it mean for a line to be [b][color=#1e84cc]TANGENT [/color][/b]to a circle?
– Tim Brzezinski

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3.

MAFS.912.G-C.1.3

  • 1. Circumcircle: Construction Exercise (VA)
  • 2. Circumcircle: Construction Exercise (VB)
  • 3. Circumcircle: Construction Exercise (VC)
  • 4. Incircle: Construction Exercise (VA)
  • 5. Incircle: Construction Exercise (VB)
  • 6. Cyclic Quadrilaterals (IAT: Corollary 3)
  • 7. Animation 43
  • 8. Where in NYC? (VA)
  • 9. Where in NYC? (VB)
  • 10. Where in NYC? (VC)

3.1.

Circumcircle: Construction Exercise (VA)

[color=#000000]Use any of the tools in the limited toolbar below to construct this triangle's circumcircle. [br][br]You can use the slider to change the measure of angle [i]A[/i] at any time. [br]Feel free to move the triangle's white vertices around as well. [br][br]Feel free to reference [url=https://www.geogebra.org/m/ueV9RpZf]this worksheet[/url] at any time. [/color]
[color=#980000][b]Recall that the circumcenter is the center of a triangle's circumcircle. [br][/b][/color][br][b][color=#000000]Questions: [/color][/b][br][br][color=#000000]1) Is it ever possible for a triangle's circumcenter to lie OUTSIDE the triangle? I[br] If so, under what circumstance(s) will this occur?[br][br]2) Is it ever possible for a triangle's circumcenter to lie ON THE TRIANGLE ITSELF? [br] If so, under what circumstance(s) will this occur? [br] [br]3) If your answer for (2) was "YES", where on the triangle did the circumcenter lie?[br] Use the tools of GeoGebra to validate your response. [br][br][/color][color=#000000]4) Is it ever possible for a triangle's circumcenter to lie INSIDE the triangle? [br] If so, under what circumstance(s) will this occur?[/color]
– Tim Brzezinski

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4.

MAFS.912.G-C.2.5

  • 1. Radius Unwrapper v2.0
  • 2. 1 Radian: Clear Definition
  • 3. Movie: Radians to Revs
  • 4. Arc length: Quick Investigation
  • 5. Area of a Sector
  • 6. Arc Length and Area of a Sector (V1)
  • 7. Arc Length and Area of a Sector (V2)

4.1.

Radius Unwrapper v2.0

Try to imagine how you might measure the subtended arc in terms of radius lengths. Once you have taken a guess use the slider to see the unfolded length. Turn on the radius ruler to confirm your guess. Geogebra will measure the angle in terms of radians, but it will restrict the answer on the interval of zero to 2 Pi.
– Ted Coe

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