Coordinate & Analytic Geometry

Tim Brzezinski, Sep 30, 2016

Table of Contents

  1. Conic Sections: 3D to 2D
    1. Exploring Conic Section Possibilities
    2. Conic Sections: Introduction
    3. Conic Sections
    4. Conic Section Explorations
  2. Equations of Circles
    1. Conic Sections
    2. Circle Equation: Center (0,0)
    3. Circle Equation: Center NOT (0,0)
    4. Animation 200
    5. Circle Equation: Center (h,k)
    6. Quiz: Circle Graphing - EQ in Standard Form
    7. Quiz: Circle Graphing - EQ in General Form
    8. Sphere Investigation
    9. Animation 156
  3. Parabolas
    1. Parabola: Cause and Effect
    2. Locus Problem (1)
    3. Locus Construction 1
    4. Animation 85
    5. Parabola as a Locus
    6. Animation 4
    7. Locus Constructions (via Paper Folding)
    8. Parabola (Locus Construction)
    9. Parabola (Graph & Equation Anatomy)
    10. Parabolas: Quick Anatomy Exploration
    11. Reflectors, Cookers, and Other Phenomena
    12. Parabola (Locus): Quick Prompt
    13. Graphing Parabolas: Focus and Vertex Placement
    14. Quick Locus Exploration
  4. Ellipses and Hyperbolas
    1. Locus Problem (2)
    2. Locus Construction 2
    3. Ellipse (Locus Construction)
    4. Animation 86
    5. Ellipse: Magic Sum = ?
    6. Ellipse: Magic Sum = ? (V2)
    7. Animation 10
    8. Quiz: Graphing Ellipses (EQ in Standard Form)
    9. Locus Problem (3)
    10. Locus Construction 3
    11. Animation 87
    12. Hyperbola: Difference = ?
    13. Animation 30
    14. Locus Constructions (via Paper Folding)
    15. Conic Sections
    16. Hyperbola (Locus Construction)
  5. Coordinate Geometry Proofs
    1. Parallelogram Creation Exercises (I)
    2. Parallelogram Creation Exercise (II)
    3. Parallelogram Creation Exercise (III)
    4. Parallelogram: Coordinate Geometry Setup
    5. Parallelogram Diagonals
    6. Isosceles Triangle Medians
    7. Rectangle Diagonals
    8. Isosceles Trapezoid Diagonals
    9. Isosceles Trapezoid Midpoints
    10. Rhombus Diagonals
    11. Inscribed Angle Intercepts Semicircle
    12. 3 Special Points!
    13. Finsler-Hadwiger Exercise (VA)
    14. Finsler-Hadwiger Exercise (VB)
    15. Slope to Angle Measure Calculator (I)
    16. Slope to Angle Measure Calculator (II)
  6. Slopes of Parallel and Perpendicular Lines
    1. Slope: Intuitive Introduction
    2. Slope: Intuitive Introduction (II)
    3. Quick Slope Types Illustrator
    4. Slope: Intuitive Intro and Exploration
    5. Quiz: Building Lines with a Given Slope
    6. Rise & Run (Revamped)
    7. Rise & Run
    8. Parallel Lines and Slope: Quick Investigation
    9. Coordinate Geometry: Quick Investigation
    10. Parallel Consequence? (VA)
    11. Parallel Consequence? (VB)
    12. Parallel Lines in the Coordinate Plane (VA)
    13. Parallel Lines in the Coordinate Plane (VB)
    14. Special Pair of Lines in the Coordinate Plane
    15. Parallel Consequence: Quick Reminder
    16. Parallel and Perpendicular Lines in the Coordinate Plane
    17. Parallel and Perpendicular Lines in the Coordinate Plane
    18. Perpendicular Lines in the CP: Quick Investigation
    19. Perpendicular Lines Investigation (VA)
    20. Perpendicular Lines Investigation (VB)
    21. Slope Triangle Rotation (VA)
    22. Animation 103
    23. 90 Degree Rotation about ANY POINT: Quick Exploration
    24. Parallel & Perpendicular Consequence
    25. Open Middle: Parallel and Perpendicular Lines
    26. Slopes and Angles (1)
    27. Slopes and Angles (2)
    28. Parallel Lines and Slope: Open Middle
    29. Perpendicular Lines and Slope: Open Middle
    30. Writing Equations of Parallel and Perpendicular Lines
    31. Exploring Parallel and Perpendicular Lines in the Coordinate Plane
    32. Equations of Perpendicular Lines (Open Middle) - V1
    33. Equations of Perpendicular Lines (Open Middle) - V2
    34. Creating 3 Lines to Form a Right Triangle
    35. Position the Line Tangent to the Circle
    36. Open Middle: Circle Tangent to Line (1)
  7. Partitioning Segments
    1. Open Middle Midpoint Exercise (V1)
    2. Midpoint and Endpoint Coordinates Action!
    3. Midpoint Coordinates (II)
    4. Animation 173
    5. Animation 174
    6. Endpoint (x,y) of a Segment Given Midpoint & Other Endpoint
    7. Midpoint (x,y) of a Segment in the Coordinate Plane (Quiz)
    8. Partition Line Segments
  8. Distance in the Coordinate Plane
    1. Coordinate Plane Distance: Some Insight
    2. Distance in the Coordinate Plane (With Hints)
    3. Distance in the Coordinate Plane (II)
    4. Distance in the Coordinate Plane (III)
    5. Quiz: Distance in the Coordinate Plane (V1)
    6. Quiz: Distance in the Coordinate Plane (V2)
    7. Open Middle (Revamped): Creating Distances in 3 Space
    8. Distance in the CP: Conceptual Discovery Lesson
    9. Finding Distance in the Coordinate Plane: Quick Conceptual Practice
    10. Plotting Points via Distance in the Coordinate Plane
    11. Open Middle: Distance in the Coordinate Plane (2)
    12. Open Middle: Distance in the Coordinate Plane
    13. Open Middle: Distance in the Coordinate Plane (3)
  9. Distance, Midpoint, Slope
    1. Open Middle: Distance, Midpoint, Slope (3)
    2. Open Middle Midpoint Exercise (V2)
    3. Open Middle: Distance and Midpoint Exercise
  10. Other Geometry Resources
    1. Geometry Resources

1.

Conic Sections: 3D to 2D

  • 1. Exploring Conic Section Possibilities
  • 2. Conic Sections: Introduction
  • 3. Conic Sections
  • 4. Conic Section Explorations

1.1.

Exploring Conic Section Possibilities

STUDENTS:
In each of the apps below, a [color=#bf9000][b]plane [/b][/color]intersects a [b][color=#1e84cc]double napped cone[/color][/b].
In the app above:
Note the equation of the plane is z = some constant. [br][br]Change the equation of this plane to [math]z=2[/math]. [br]Then change it to [math]z=1[/math]. [br]Then change it to [math]z=4[/math]. [br][br]How would you describe the intersection of this [b][color=#bf9000]plane[/color] [/b]and [color=#1e84cc][b]double-napped cone[/b][/color]?
In the app below, use the tools of GeoGebra to prove your assertion is correct.
Change the equation in the app below to create an entirely different cross section.
How would you describe this cross section in your own words? What does it look like?
Can you create an another cross section that's different from the other 2 examples above? Try to do so.
How would you describe this cross section in your own words? What does it look like?
Can you create a cross section different from the other three? Try to do so!
How would you describe this cross section in your own words? What does it look like?
Is it possible to create a cross section that is not curvy? How could we do it here?
What do you think?
When a plane intersects a double-napped circular cone, what could possible cross sections look like? [br]How would you describe them?
– Tim Brzezinski

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2.

Equations of Circles

  • 1. Conic Sections
  • 2. Circle Equation: Center (0,0)
  • 3. Circle Equation: Center NOT (0,0)
  • 4. Animation 200
  • 5. Circle Equation: Center (h,k)
  • 6. Quiz: Circle Graphing - EQ in Standard Form
  • 7. Quiz: Circle Graphing - EQ in General Form
  • 8. Sphere Investigation
  • 9. Animation 156

2.1.

– Tim Brzezinski

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3.

Parabolas

  • 1. Parabola: Cause and Effect
  • 2. Locus Problem (1)
  • 3. Locus Construction 1
  • 4. Animation 85
  • 5. Parabola as a Locus
  • 6. Animation 4
  • 7. Locus Constructions (via Paper Folding)
  • 8. Parabola (Locus Construction)
  • 9. Parabola (Graph & Equation Anatomy)
  • 10. Parabolas: Quick Anatomy Exploration
  • 11. Reflectors, Cookers, and Other Phenomena
  • 12. Parabola (Locus): Quick Prompt
  • 13. Graphing Parabolas: Focus and Vertex Placement
  • 14. Quick Locus Exploration

3.1.

Parabola: Cause and Effect

We know that the intersection of a plane and double-napped cone is a conic section. [br][br]One particular conic section shown below is a [b]PARABOLA[/b]. [br][br]Interact with the applet below for a few minutes. Then answer the questions that follow. [br][br][b][color=#0000ff]Note:[/color][/b][br][b]LARGE POINTS [/b]are moveable [b]AT ANY TIME[/b]. [br][br][b][color=#1e84cc]To explore in Augmented Reality, see the directions below the applet. [/color][/b]
1.
How would you describe the inclination of the [b][color=#38761d]green plane[/color][/b] with respect to the [color=#bf9000][b]cones[/b][/color]? Explain.
2.
How would you describe the action that happens towards the end of the animation?
TO EXPLORE IN AUGMENTED REALITY:
1) Open up GeoGebra 3D app on your device. [br][br]2) Go to the MENU (horizontal bars) in the upper left corner. Select OPEN. [br] In the Search GeoGebra Resources input box, type [b]ztM2Yffm[/b][br] (Note this is the resource ID = last 8 digits of the URL for this resource.)[br][br][b]Note: [/b][br][br]The [b]i[/b] slider controls the animation. You can also move the [b][color=#bf9000]LARGE YELLOW POINT[/color][/b]. Then press AR again to explore in Augmented Reality.
– Tim Brzezinski

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4.

Ellipses and Hyperbolas

  • 1. Locus Problem (2)
  • 2. Locus Construction 2
  • 3. Ellipse (Locus Construction)
  • 4. Animation 86
  • 5. Ellipse: Magic Sum = ?
  • 6. Ellipse: Magic Sum = ? (V2)
  • 7. Animation 10
  • 8. Quiz: Graphing Ellipses (EQ in Standard Form)
  • 9. Locus Problem (3)
  • 10. Locus Construction 3
  • 11. Animation 87
  • 12. Hyperbola: Difference = ?
  • 13. Animation 30
  • 14. Locus Constructions (via Paper Folding)
  • 15. Conic Sections
  • 16. Hyperbola (Locus Construction)

4.1.

Locus Problem (2)

In the applet below, [br][br][color=#1e84cc][b][i]O[/i] is the center [/b][/color]of the circle shown. [br][b]Point [i]D[/i] is a point that lies ON this circle. [/b][br][color=#1e84cc][b]Point [i]A[/i] is a point that ALWAYS LIES INSIDE[/b] [/color]the circle. (You can move it anywhere you'd like). [br]The [color=#ff00ff][b]pink line[/b][/color] is the [color=#ff00ff][b]perpendicular bisector[/b][/color] of the segment with endpoints [i]A[/i] and [i]D[/i]. [br][br]Drag [b]point [i]D[/i][/b] around the circle a few times. What do you see? Describe in detail! [br] [br]Feel free to alter the locations of [i][color=#1e84cc][b]A[/b][/color][/i] and [i][color=#1e84cc][b]R[/b][/color]. [/i]Then clear the trace and drag [b]point [i]D[/i][/b] around again. [br][br]Why does this occur?
Please go to the [url=https://www.geogebra.org/m/TZu6tRwE]Locus Construction 2 Task[/url] & begin!
– Tim Brzezinski

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5.

Coordinate Geometry Proofs

  • 1. Parallelogram Creation Exercises (I)
  • 2. Parallelogram Creation Exercise (II)
  • 3. Parallelogram Creation Exercise (III)
  • 4. Parallelogram: Coordinate Geometry Setup
  • 5. Parallelogram Diagonals
  • 6. Isosceles Triangle Medians
  • 7. Rectangle Diagonals
  • 8. Isosceles Trapezoid Diagonals
  • 9. Isosceles Trapezoid Midpoints
  • 10. Rhombus Diagonals
  • 11. Inscribed Angle Intercepts Semicircle
  • 12. 3 Special Points!
  • 13. Finsler-Hadwiger Exercise (VA)
  • 14. Finsler-Hadwiger Exercise (VB)
  • 15. Slope to Angle Measure Calculator (I)
  • 16. Slope to Angle Measure Calculator (II)

5.1.

Parallelogram Creation Exercises (I)

[color=#000000]In the applets below, 3 vertices of a parallelogram are shown.[br]The coordinates (x,y) of these vertices are also displayed as well. [br][br]In each task below, determine the coordinates of each parallelogram's 4th vertex. [br]Plot this point in the coordinate plane. [br]Feel free to use any of the tools of the limited toolbar when doing so. [br]Then, use the Polygon tool to construct the parallelogram.[br][br]Afterwards, use any of the tools of each applet's limited toolbar to clearly show that the quadrilateral you've constructed is indeed a parallelogram. [/color]
– Tim Brzezinski

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6.

Slopes of Parallel and Perpendicular Lines

  • 1. Slope: Intuitive Introduction
  • 2. Slope: Intuitive Introduction (II)
  • 3. Quick Slope Types Illustrator
  • 4. Slope: Intuitive Intro and Exploration
  • 5. Quiz: Building Lines with a Given Slope
  • 6. Rise & Run (Revamped)
  • 7. Rise & Run
  • 8. Parallel Lines and Slope: Quick Investigation
  • 9. Coordinate Geometry: Quick Investigation
  • 10. Parallel Consequence? (VA)
  • 11. Parallel Consequence? (VB)
  • 12. Parallel Lines in the Coordinate Plane (VA)
  • 13. Parallel Lines in the Coordinate Plane (VB)
  • 14. Special Pair of Lines in the Coordinate Plane
  • 15. Parallel Consequence: Quick Reminder
  • 16. Parallel and Perpendicular Lines in the Coordinate Plane
  • 17. Parallel and Perpendicular Lines in the Coordinate Plane
  • 18. Perpendicular Lines in the CP: Quick Investigation
  • 19. Perpendicular Lines Investigation (VA)
  • 20. Perpendicular Lines Investigation (VB)
  • 21. Slope Triangle Rotation (VA)
  • 22. Animation 103
  • 23. 90 Degree Rotation about ANY POINT: Quick Exploration
  • 24. Parallel & Perpendicular Consequence
  • 25. Open Middle: Parallel and Perpendicular Lines
  • 26. Slopes and Angles (1)
  • 27. Slopes and Angles (2)
  • 28. Parallel Lines and Slope: Open Middle
  • 29. Perpendicular Lines and Slope: Open Middle
  • 30. Writing Equations of Parallel and Perpendicular Lines
  • 31. Exploring Parallel and Perpendicular Lines in the Coordinate Plane
  • 32. Equations of Perpendicular Lines (Open Middle) - V1
  • 33. Equations of Perpendicular Lines (Open Middle) - V2
  • 34. Creating 3 Lines to Form a Right Triangle
  • 35. Position the Line Tangent to the Circle
  • 36. Open Middle: Circle Tangent to Line (1)

6.1.

Slope: Intuitive Introduction

[color=#000000]Discovery Lesson Activity: [br][/color][color=#980000][br][/color][b][color=#0000ff][url=https://docs.google.com/document/d/1W5fggl-QmnwOoT1Yjs2MfZnwVzIF49pflGX33xe0PN0/edit?usp=sharing]Slope: Intuitive Introduction Investigation [/url][/color][/b]
– Tim Brzezinski

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7.

Partitioning Segments

  • 1. Open Middle Midpoint Exercise (V1)
  • 2. Midpoint and Endpoint Coordinates Action!
  • 3. Midpoint Coordinates (II)
  • 4. Animation 173
  • 5. Animation 174
  • 6. Endpoint (x,y) of a Segment Given Midpoint & Other Endpoint
  • 7. Midpoint (x,y) of a Segment in the Coordinate Plane (Quiz)
  • 8. Partition Line Segments

7.1.

Open Middle Midpoint Exercise (V1)

Creation of this resource was inspired by an Open Middle problem posted by Dane Ehlert.
Find locations for A and B so that M = (3,4) is the midpoint of the segment with endpoints A and B.
Find other locations for A and B so that M = (3,4) is the midpoint of the segment with endpoints A and B. Make sure your setup is different from the one you created above.
– Tim Brzezinski

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8.

Distance in the Coordinate Plane

  • 1. Coordinate Plane Distance: Some Insight
  • 2. Distance in the Coordinate Plane (With Hints)
  • 3. Distance in the Coordinate Plane (II)
  • 4. Distance in the Coordinate Plane (III)
  • 5. Quiz: Distance in the Coordinate Plane (V1)
  • 6. Quiz: Distance in the Coordinate Plane (V2)
  • 7. Open Middle (Revamped): Creating Distances in 3 Space
  • 8. Distance in the CP: Conceptual Discovery Lesson
  • 9. Finding Distance in the Coordinate Plane: Quick Conceptual Practice
  • 10. Plotting Points via Distance in the Coordinate Plane
  • 11. Open Middle: Distance in the Coordinate Plane (2)
  • 12. Open Middle: Distance in the Coordinate Plane
  • 13. Open Middle: Distance in the Coordinate Plane (3)

8.1.

Coordinate Plane Distance: Some Insight

Move the 2 points wherever you'd like! Press MOVE. Then CONTINUE.
What do you notice? What do you wonder?
Does what you see here look like anything else we may have learned in the past? If so, what?
In this app, position the two points to create a right triangle whose legs measure 3 units and 4 units.
How can we find the diagonal distance [math]d[/math] above? Show how below.
In this app, position the two points to create a right triangle whose legs measure 6 units and 8 units.
How can we find the diagonal distance [math]d[/math] above? Show how below.
Here, place the runner's initial point at (1,-5). Place the other point at (-6,3). Then press the buttons (like before) to create a right triangle.
How can we find the diagonal distance [math]d[/math] above? Show how below.
– Tim Brzezinski

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9.

Distance, Midpoint, Slope

  • 1. Open Middle: Distance, Midpoint, Slope (3)
  • 2. Open Middle Midpoint Exercise (V2)
  • 3. Open Middle: Distance and Midpoint Exercise

9.1.

Open Middle: Distance, Midpoint, Slope (3)

[size=150][b][url=https://www.geogebra.org/m/c6c396pd]The full Open Middle activity can be found here[/url][/b][/size].
The problem:
Quick (silent) demo
– Tim Brzezinski

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10.

Other Geometry Resources

  • 1. Geometry Resources

10.1.

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