Geometry (Circles)

Tim Brzezinski, Deirdre Longo, Jun 6, 2018

[b]This GeoGebra Book contains lots of discovery-based learning activities, investigations, and meaningful remediation worksheets that were designed to help enhance students' learning of geometry concepts both inside and outside of the mathematics classroom. [color=#1551b5]This GeoGebra Book also contains contributions from from Dr. Ted Coe ([url]https://www.geogebra.org/tedcoe[/url])[/color] [/b]

Table of Contents

  1. HSG.C.A.1
    1. Congruent Circles: Definition
    2. Animation 89
    3. Similar Circles?
  2. HSG.C.A.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. Constructing Tangents: Ex. 12
    7. Tangents to Circles: Investigation
    8. Where to Sit? (I)
    9. Where to Sit (II)?
    10. Inscribed Angle Theorem (V1)
    11. Animation 12
    12. Inscribed Angle Theorem: Take 2!
    13. Inscribed Angle Theorems: Take 3!
    14. Proof Exercise: Inscribed Angle Theorem (Case 1)
    15. Proof Exercise: Inscribed Angle Theorem (Case 2)
    16. Proof Exercise: Inscribed Angle Theorem (Case 3)
    17. Inscribed Angle Theorems: Take 4!
    18. Inscribed Angle Theorem: Corollary 1
    19. Inscribed Angle Theorem Dance: Take 2!
    20. Animation 20 (Inscribed Angle Dance!)
    21. Find that Angle! (GoGeometry Action 144)
    22. Thales' Theorem (VA)
    23. Animation 116
    24. Thales' Theorem (VB)
    25. Animation 54
    26. Angle Formed by Chord & Tangent (I)
    27. Animation 50
    28. Angle Formed by Chord & Tangent (II)
    29. Animation 78
    30. Congruent Chords: Quick Investigation
    31. Congruent Chords (I)
    32. Animation 76
    33. Diameters & Chords I (VA)
    34. Diameters & Chords I (VB)
    35. Animation 24
    36. Congruent Chords Action (A)
    37. Animation 96
    38. Equidistant Chord Action
    39. Animation 101
    40. Angle Formed by 2 Chords (I)
    41. Animation 32
    42. Angle Formed by 2 Chords (II)
    43. Animation 33
    44. Angles from Secants and Tangents (V1)
    45. Animation 61
    46. Angle From 2 Secants (V2)
    47. Animation 45
    48. Crossing Chords: Proof Hint
    49. Animation 79
    50. Crossing Chords Property & Proof Start
    51. Crossing Chords Property & Proof Start (II)
    52. Secants: Proof Hint
    53. Animation 88
    54. Animation 92
  3. HSG.C.A.3
    1. Circumcircle: Construction Exercise (VA)
    2. Circumcircle: Construction Exercise (VB)
    3. Circumcircle: Construction Exercise (VC)
    4. Making Perpendicular Bisectors: Ex. 9
    5. Incircle: Construction Exercise (VA)
    6. Incircle: Construction Exercise (VB)
    7. Constructing Angle Bisectors: Ex. 11
    8. Cyclic Quadrilaterals (IAT: Corollary 3)
    9. Animation 43
    10. Cyclic Quadrilateral: Quick Warm Up (GoGeometry Action 149!)
    11. Where in NYC? (VA)
    12. Where in NYC? (VB)
    13. Where in NYC? (VC)
  4. HSG.C.A.4
    1. Tangent to Circle: Construction 1
    2. Tangent to Circle: Construction 2
    3. Geometric Mean Illustration
    4. Animation 211
  5. HSG.C.B.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)
  6. Links to Other CCSS High School: Geometry Standards
    1. Geometry Resources
  7. Additional - Links to CCSS High School: Functions Resources
    1. Functions Resources

1.

HSG.C.A.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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1.3.

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2.

HSG.C.A.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. Constructing Tangents: Ex. 12
  • 7. Tangents to Circles: Investigation
  • 8. Where to Sit? (I)
  • 9. Where to Sit (II)?
  • 10. Inscribed Angle Theorem (V1)
  • 11. Animation 12
  • 12. Inscribed Angle Theorem: Take 2!
  • 13. Inscribed Angle Theorems: Take 3!
  • 14. Proof Exercise: Inscribed Angle Theorem (Case 1)
  • 15. Proof Exercise: Inscribed Angle Theorem (Case 2)
  • 16. Proof Exercise: Inscribed Angle Theorem (Case 3)
  • 17. Inscribed Angle Theorems: Take 4!
  • 18. Inscribed Angle Theorem: Corollary 1
  • 19. Inscribed Angle Theorem Dance: Take 2!
  • 20. Animation 20 (Inscribed Angle Dance!)
  • 21. Find that Angle! (GoGeometry Action 144)
  • 22. Thales' Theorem (VA)
  • 23. Animation 116
  • 24. Thales' Theorem (VB)
  • 25. Animation 54
  • 26. Angle Formed by Chord & Tangent (I)
  • 27. Animation 50
  • 28. Angle Formed by Chord & Tangent (II)
  • 29. Animation 78
  • 30. Congruent Chords: Quick Investigation
  • 31. Congruent Chords (I)
  • 32. Animation 76
  • 33. Diameters & Chords I (VA)
  • 34. Diameters & Chords I (VB)
  • 35. Animation 24
  • 36. Congruent Chords Action (A)
  • 37. Animation 96
  • 38. Equidistant Chord Action
  • 39. Animation 101
  • 40. Angle Formed by 2 Chords (I)
  • 41. Animation 32
  • 42. Angle Formed by 2 Chords (II)
  • 43. Animation 33
  • 44. Angles from Secants and Tangents (V1)
  • 45. Animation 61
  • 46. Angle From 2 Secants (V2)
  • 47. Animation 45
  • 48. Crossing Chords: Proof Hint
  • 49. Animation 79
  • 50. Crossing Chords Property & Proof Start
  • 51. Crossing Chords Property & Proof Start (II)
  • 52. Secants: Proof Hint
  • 53. Animation 88
  • 54. 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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2.43.

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2.48.

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2.52.

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2.53.

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2.54.

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3.

HSG.C.A.3

  • 1. Circumcircle: Construction Exercise (VA)
  • 2. Circumcircle: Construction Exercise (VB)
  • 3. Circumcircle: Construction Exercise (VC)
  • 4. Making Perpendicular Bisectors: Ex. 9
  • 5. Incircle: Construction Exercise (VA)
  • 6. Incircle: Construction Exercise (VB)
  • 7. Constructing Angle Bisectors: Ex. 11
  • 8. Cyclic Quadrilaterals (IAT: Corollary 3)
  • 9. Animation 43
  • 10. Cyclic Quadrilateral: Quick Warm Up (GoGeometry Action 149!)
  • 11. Where in NYC? (VA)
  • 12. Where in NYC? (VB)
  • 13. 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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3.11.

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3.12.

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3.13.

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4.

HSG.C.A.4

  • 1. Tangent to Circle: Construction 1
  • 2. Tangent to Circle: Construction 2
  • 3. Geometric Mean Illustration
  • 4. Animation 211

4.1.

Tangent to Circle: Construction 1

[color=#980000][b]Directions: [/b][/color][br][br][color=#000000]Use the limited toolbar in the applet below to construct a line that is tangent to the circle that passes through the [/color][color=#ff00ff][b]PINK POINT.[/b][/color] [color=#000000][br][br]Use the [b]Angle[/b] tool to check the accuracy of your construction afterwards.[/color] [br][br][color=#000000][i]Do not open another tab on your internet browser to look it up either. [br]Think about what you've learned about lines drawn tangent to circles! [/i][/color]
– Tim Brzezinski

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5.

HSG.C.B.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)

5.1.

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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6.

Links to Other CCSS High School: Geometry Standards

  • 1. Geometry Resources

6.1.

Geometry Resources

[list][*][b][url=https://www.geogebra.org/m/z8nvD94T]Congruence (Volume 1)[/url][/b][/*][*][b][url=https://www.geogebra.org/m/munhXmzx]Congruence (Volume 2)[/url][/b][/*][*][b][url=https://www.geogebra.org/m/dPqv8ACE]Similarity, Right Triangles, Trigonometry[/url][/b][/*][*][b][url=https://www.geogebra.org/m/C7dutQHh]Circles[/url][/b][/*][*][b][url=https://www.geogebra.org/m/K2YbdFk8]Coordinate and Analytic Geometry[/url][/b][/*][*][b][url=https://www.geogebra.org/m/xDNjSjEK]Area, Surface Area, Volume, 3D, Cross Section[/url] [/b][/*][*][b][url=https://www.geogebra.org/m/NjmEPs3t]Proof Challenges[/url]  [/b][/*][/list]
What phenomenon is dynamically being illustrated here? (Vertices are moveable.)
What phenomenon is dynamically being illustrated here? (Vertices are moveable.)
– Tim Brzezinski

7.

Additional - Links to CCSS High School: Functions Resources

  • 1. Functions Resources

7.1.

Functions Resources

[list][*][b][url=https://www.geogebra.org/m/k6Dvu9f3]Interpreting Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/uTddJKRC]Building Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/GMvvpwrm]Linear, Quadratic, and Exponential Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/aWuJMDas]Trigonometric Functions[/url][/b][/*][/list]
Half-life function: Quick Exploration. (Large point & slider moveable.)
What does it mean for a function to be odd? (Points moveable.)
– GeoGebra Team, Tim Brzezinski

Information