CCSS High School: Geometry (Challenges)

Tim Brzezinski, Mª Esperanza Gesteira, May 3, 2017

[color=#1551b5][b]This GeoGebra book contains links to a plethora of problems teachers can use to provide challenging enrichment activities for students that excel within the regular curriculum. [/b] Many of the problems in this book come from those posted on Twitter by Antonio Gutierrez ([url]https://twitter.com/gogeometry[/url]) and Alexander Bogomolny ([url]https://twitter.com/CutTheKnotMath[/url]). [b]Each challenge (and corresponding animation) is mapped to one (or more) of the CCSS High School: Geometry standards.[/b] [/color]

Table of Contents

  1. HSG.CO.C.9
    1. Cut-The-Knot-Action (5)!
    2. Animation 214
    3. Cut-the-Knot-Action (3)!
    4. Animation 188
  2. HSG.CO.C.10
    1. Medians & Equal Areas!
    2. Animation 74
    3. Animation 90
    4. Euler Line (Informal Investigation)
    5. Euler Line Generator
    6. Morley Action!
    7. Animation 113
    8. Napoleon Motion!
    9. Animation 14
    10. Hexagonal Napoleon Theorem?
    11. Animation 172
    12. Vivani Action!
    13. Equilateral Action (1)!
    14. Animation 91
    15. Equilateral Action (2)!
    16. Animation 94
    17. Equilateral Action (3)!
    18. Animation 107
    19. Equilateral Action (4)!
    20. Animation 108
    21. Equilateral Triangle and Rectangle (Area Problem)
    22. Animation 154
    23. Equal Triangular Areas in a Regular Pentagon
    24. Animation 194
    25. NCTM Calendar Problem (11-2-2016)
    26. Animation 139
    27. 9 Point Circle Action
    28. Animation 34
    29. 9-Point Circle Action (Part 2)
    30. Animation 106
    31. 9-Point Circle Action (Part 3A)
    32. 9-Point Circle (Informal Investigation)
    33. Animation 71
    34. 9-Point Circle Action (Part 3B)
    35. 9-Point Circle Action (Part 4)
    36. Animation 53
    37. 9-Point Circle Action (Part 5)
    38. Animation 115
    39. Isosceles Triangle: Surprising Theorem!
    40. Animation 121
    41. Cut-The-Knot Action !!!
    42. More Triangle Action!!!
    43. Animation 163
    44. Triangle Median & Circles Action!!!
    45. Animation 164
    46. Desargues Action!!!
    47. Animation 209
    48. GoGeometry Action 10!
    49. Animation 219
    50. GoGeometry Action 14!
    51. Animation 223
    52. GoGeometry Action 17!
    53. Animation 228
    54. GoGeometry Action 18!
    55. Animation 229
    56. Cut-The-Knot-Action 8!
    57. Animation 230
    58. GoGeometry Action 20!
    59. Animation 233
    60. Cut-The-Knot Action 9!
    61. Animation 234
    62. Carnot Action! = GoGeometry Action 27!
    63. Animation 241
    64. GoGeometry Action 29!
    65. Animation 244
    66. GoGeometry Action 30!
    67. Animation 245
    68. Incircle + Excircle = GoGeometry Action 31!
    69. Animation 246
    70. GoGeometry Action 32!
    71. Animation 247
    72. Cut-The-Knot Action 10!
    73. Animation 248
    74. GoGeometry Action 34!
    75. Animation 250
    76. Cut-The-Knot-Action 12!
    77. Animation 251
    78. GoGeometry Action 35!
    79. Animation 252
    80. GoGeometry Action 37!
    81. Animation 254
    82. GoGeometry Action 38!
    83. Animation 255
  3. HSG.CO.C.11
    1. Finsler-Hadwiger Action!!!
    2. Animation 110
    3. Thebault Action!!!
    4. Animation 83
    5. Van Aubel Action!!!
    6. Animation 109
    7. Square Problem
    8. 3 Squares Problem
    9. Animation 120
    10. Harmonic Mean In an Isosceles Trapezoid
    11. Square Action! (Proof Problem 1)
    12. Cut-The-Knot Action (4)!
    13. Animation 213
    14. Rhombus Action + Sequel = GoGeometry Action 7!
    15. Animation 215
    16. GoGeometry Action 11!
    17. Animation 220
    18. GoGeometry Action 12!
    19. Animation 221
    20. GoGeometry Action 18!
    21. Animation 231
    22. GoGeometry Action 21!
    23. Animation 235
    24. GoGeometry Action 22!
    25. Animation 236
    26. GoGeometry Action 23!
    27. Animation 237
    28. GoGeometry Action 24!
    29. Animation 238
    30. GoGeometry Action 36!
    31. Animation 253
  4. HSG.CO.D.12
    1. Bizzare Trisection?
    2. Animation 198
  5. HSG.C.A.2
    1. Butterfly Theorem Action!
    2. Animation 132
    3. Not Your Everyday Chord & Tangent Theorem
    4. Animation 133
    5. Cut-The-Knot Action (2) !!!
    6. Animation 152
    7. Int+Ext Tangent Action!
    8. Animation 155
    9. Square, Tangent, and Two Congruent & Tangent Circles
    10. Animation 153
    11. GoGeometry Action 1!
    12. Animation 205
    13. GoGeometry Action 2!
    14. Animation 206
    15. GoGeometry Action 3!
    16. Animation 207
    17. GoGeometry Action 4!
    18. Animation 208
    19. GoGeometry Action 5!
    20. Animation 210
    21. GoGeometry Action 6!
    22. Animation 212
    23. GoGeometry Action 8!
    24. Animation 216
    25. Cut-The-Knot Action (6)!
    26. Animation 217
    27. GoGeometry Action 9!
    28. Animation 218
    29. GoGeometry Action 13!
    30. Animation 222
    31. GoGeometry Action 15!
    32. Animation 224
    33. GoGeometry Action 16!
    34. Animation 226
    35. Cut-The-Knot Action 7!
    36. Animation 227
    37. GoGeometry Action 19!
    38. Animation 232
    39. Thales Action + Sequel = GoGeometry Action 25!
    40. Animation 239
    41. GoGeometry Action 26!
    42. Animation 240
    43. Carnot Action! = GoGeometry Action 27!
    44. Animation 241
    45. Incircle + Excircle = GoGeometry Action 31!
    46. Animation 246
    47. Cut-The-Knot Action 10!
    48. Animation 248
    49. Cut-The-Knot-Action 11!
    50. Animation 249
  6. Links to Other CCSS High School: Geometry Standards
    1. Geometry Resources

1.

HSG.CO.C.9

  • 1. Cut-The-Knot-Action (5)!
  • 2. Animation 214
  • 3. Cut-the-Knot-Action (3)!
  • 4. Animation 188

1.1.

Cut-The-Knot-Action (5)!

Creation of this applet was inspired by a [url=https://twitter.com/CutTheKnotMath/status/839945580088602629]tweet[/url] from [url=https://twitter.com/CutTheKnotMath]Alexander Bogomolny[/url]. [br][br]In the applet below, a [b][color=#6aa84f]regular pentagon[/color][/b] and a [color=#bf9000][b]regular decagon[/b][/color] share a common side. [br][br][color=#ff00ff][b]What is the measure of the pink angle? [/b][/color] [br][br][color=#0000ff][b]How can you formally prove what this applet informally illustrates? [/b][/color]
Quick (Silent) Demo
– Tim Brzezinski

1.2.

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1.3.

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1.4.

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2.

HSG.CO.C.10

  • 1. Medians & Equal Areas!
  • 2. Animation 74
  • 3. Animation 90
  • 4. Euler Line (Informal Investigation)
  • 5. Euler Line Generator
  • 6. Morley Action!
  • 7. Animation 113
  • 8. Napoleon Motion!
  • 9. Animation 14
  • 10. Hexagonal Napoleon Theorem?
  • 11. Animation 172
  • 12. Vivani Action!
  • 13. Equilateral Action (1)!
  • 14. Animation 91
  • 15. Equilateral Action (2)!
  • 16. Animation 94
  • 17. Equilateral Action (3)!
  • 18. Animation 107
  • 19. Equilateral Action (4)!
  • 20. Animation 108
  • 21. Equilateral Triangle and Rectangle (Area Problem)
  • 22. Animation 154
  • 23. Equal Triangular Areas in a Regular Pentagon
  • 24. Animation 194
  • 25. NCTM Calendar Problem (11-2-2016)
  • 26. Animation 139
  • 27. 9 Point Circle Action
  • 28. Animation 34
  • 29. 9-Point Circle Action (Part 2)
  • 30. Animation 106
  • 31. 9-Point Circle Action (Part 3A)
  • 32. 9-Point Circle (Informal Investigation)
  • 33. Animation 71
  • 34. 9-Point Circle Action (Part 3B)
  • 35. 9-Point Circle Action (Part 4)
  • 36. Animation 53
  • 37. 9-Point Circle Action (Part 5)
  • 38. Animation 115
  • 39. Isosceles Triangle: Surprising Theorem!
  • 40. Animation 121
  • 41. Cut-The-Knot Action !!!
  • 42. More Triangle Action!!!
  • 43. Animation 163
  • 44. Triangle Median & Circles Action!!!
  • 45. Animation 164
  • 46. Desargues Action!!!
  • 47. Animation 209
  • 48. GoGeometry Action 10!
  • 49. Animation 219
  • 50. GoGeometry Action 14!
  • 51. Animation 223
  • 52. GoGeometry Action 17!
  • 53. Animation 228
  • 54. GoGeometry Action 18!
  • 55. Animation 229
  • 56. Cut-The-Knot-Action 8!
  • 57. Animation 230
  • 58. GoGeometry Action 20!
  • 59. Animation 233
  • 60. Cut-The-Knot Action 9!
  • 61. Animation 234
  • 62. Carnot Action! = GoGeometry Action 27!
  • 63. Animation 241
  • 64. GoGeometry Action 29!
  • 65. Animation 244
  • 66. GoGeometry Action 30!
  • 67. Animation 245
  • 68. Incircle + Excircle = GoGeometry Action 31!
  • 69. Animation 246
  • 70. GoGeometry Action 32!
  • 71. Animation 247
  • 72. Cut-The-Knot Action 10!
  • 73. Animation 248
  • 74. GoGeometry Action 34!
  • 75. Animation 250
  • 76. Cut-The-Knot-Action 12!
  • 77. Animation 251
  • 78. GoGeometry Action 35!
  • 79. Animation 252
  • 80. GoGeometry Action 37!
  • 81. Animation 254
  • 82. GoGeometry Action 38!
  • 83. Animation 255

2.1.

Medians & Equal Areas!

In the app below, the LARGE VERTICES are moveable. The smaller points shown on the triangle's sides are midpoints. Interact with the app below and observe what happens. [br][br]Be sure to move the triangle's large vertices around as you explore!
How would you describe the first three segments drawn? What are they? How do you know this?
What does this animation suggest about all six non-overlapping triangles? Explain how this animation helps suggest your assertion.
– Tim Brzezinski

2.2.

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2.5.

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2.6.

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2.7.

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2.9.

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2.10.

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2.11.

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2.12.

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2.13.

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2.14.

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2.15.

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2.17.

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2.24.

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2.26.

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2.27.

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2.29.

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2.31.

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2.32.

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2.33.

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2.35.

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2.36.

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2.37.

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2.38.

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2.39.

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2.41.

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2.42.

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2.48.

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2.49.

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2.51.

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2.54.

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2.55.

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2.57.

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2.60.

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2.62.

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2.63.

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2.66.

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2.67.

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2.68.

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2.69.

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2.70.

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2.73.

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2.80.

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2.81.

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2.82.

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2.83.

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3.

HSG.CO.C.11

  • 1. Finsler-Hadwiger Action!!!
  • 2. Animation 110
  • 3. Thebault Action!!!
  • 4. Animation 83
  • 5. Van Aubel Action!!!
  • 6. Animation 109
  • 7. Square Problem
  • 8. 3 Squares Problem
  • 9. Animation 120
  • 10. Harmonic Mean In an Isosceles Trapezoid
  • 11. Square Action! (Proof Problem 1)
  • 12. Cut-The-Knot Action (4)!
  • 13. Animation 213
  • 14. Rhombus Action + Sequel = GoGeometry Action 7!
  • 15. Animation 215
  • 16. GoGeometry Action 11!
  • 17. Animation 220
  • 18. GoGeometry Action 12!
  • 19. Animation 221
  • 20. GoGeometry Action 18!
  • 21. Animation 231
  • 22. GoGeometry Action 21!
  • 23. Animation 235
  • 24. GoGeometry Action 22!
  • 25. Animation 236
  • 26. GoGeometry Action 23!
  • 27. Animation 237
  • 28. GoGeometry Action 24!
  • 29. Animation 238
  • 30. GoGeometry Action 36!
  • 31. Animation 253

3.1.

Finsler-Hadwiger Action!!!

[color=#000000]In the applet below, simply slide the slider very slowly and enjoy the phenomena you witness. [br][br]After doing so, feel free to adjust the [b][color=#38761D]green [/color][/b]and [b][color=#BF9000]yellow[/color][/b] sliders to change the sizes of the [b][color=#38761D]green [/color][/b]and [b][color=#BF9000]yellow[/color][/b] squares, respectively. You can also change the locations of any of the white points. [br][br]Interact with this applet for a few minutes. Then answer the questions that follow. [/color][br][br]
Write the phenomena you've witnessed several times as a conditional ("if-then") statement.
Can you use coordinate geometry to formally prove what this applet informally illustrates? [br](For starters, why not let the common vertex be (0,0) and go from there?)
– Tim Brzezinski

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3.27.

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3.28.

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3.29.

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3.31.

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4.

HSG.CO.D.12

  • 1. Bizzare Trisection?
  • 2. Animation 198

4.1.

Bizzare Trisection?

[color=#000000]The following applet illustrates an alternate means to [b][color=#0000ff]trisect a segment [/color][/b]without using parallel lines. [br][br][b][color=#0000ff]KEY QUESTION(S): [/color][/b][br]Why does this method work? [br]What previously learned theorem(s) justify your conclusions?[br]Explain. [/color]
Quick (Silent) Demo
– Tim Brzezinski

4.2.

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5.

HSG.C.A.2

  • 1. Butterfly Theorem Action!
  • 2. Animation 132
  • 3. Not Your Everyday Chord & Tangent Theorem
  • 4. Animation 133
  • 5. Cut-The-Knot Action (2) !!!
  • 6. Animation 152
  • 7. Int+Ext Tangent Action!
  • 8. Animation 155
  • 9. Square, Tangent, and Two Congruent & Tangent Circles
  • 10. Animation 153
  • 11. GoGeometry Action 1!
  • 12. Animation 205
  • 13. GoGeometry Action 2!
  • 14. Animation 206
  • 15. GoGeometry Action 3!
  • 16. Animation 207
  • 17. GoGeometry Action 4!
  • 18. Animation 208
  • 19. GoGeometry Action 5!
  • 20. Animation 210
  • 21. GoGeometry Action 6!
  • 22. Animation 212
  • 23. GoGeometry Action 8!
  • 24. Animation 216
  • 25. Cut-The-Knot Action (6)!
  • 26. Animation 217
  • 27. GoGeometry Action 9!
  • 28. Animation 218
  • 29. GoGeometry Action 13!
  • 30. Animation 222
  • 31. GoGeometry Action 15!
  • 32. Animation 224
  • 33. GoGeometry Action 16!
  • 34. Animation 226
  • 35. Cut-The-Knot Action 7!
  • 36. Animation 227
  • 37. GoGeometry Action 19!
  • 38. Animation 232
  • 39. Thales Action + Sequel = GoGeometry Action 25!
  • 40. Animation 239
  • 41. GoGeometry Action 26!
  • 42. Animation 240
  • 43. Carnot Action! = GoGeometry Action 27!
  • 44. Animation 241
  • 45. Incircle + Excircle = GoGeometry Action 31!
  • 46. Animation 246
  • 47. Cut-The-Knot Action 10!
  • 48. Animation 248
  • 49. Cut-The-Knot-Action 11!
  • 50. Animation 249

5.1.

Butterfly Theorem Action!

[color=#cc0000][b]Note: [/b][/color][br]Creation of this applet was inspired by a [url=https://twitter.com/CutTheKnotMath/status/791398769459924998]tweet [/url]from [url=https://twitter.com/CutTheKnotMath]Alexander Bogomolny[/url] (at [url=http://www.cut-the-knot.org/]Cut-the-Knot[/url].) [br][br]The applet below dynamically illustrates a theorem known as the [b]Butterfly Theorem[/b]. [br][br]Interact with this applet for a few minutes. As you do, feel free to change the location(s) of any 1 (or more) of the [color=#674ea7][b]BIG POINT(S) that are already there[/b][/color][color=#274e13][b] (or will soon appear)[/b][/color]! [br][br]How can you formally prove what this applet informally illustrates?
– Tim Brzezinski

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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

Information