Expressing Geometric Properties with Equations (GPE) Cluster

Tim Brzezinski, JohnCMorris, Nov 14, 2017

MAFS.912.G-GPE.1.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. MAFS.912.G-GPE.2.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). MAFS.912.G-GPE.2.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). MAFS.912.G-GPE.2.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. MAFS.912.G-GPE.2.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Table of Contents

  1. MAFS.912.G-GPE.1.1
    1. Circle Equation: Center (0,0)
    2. Circle Equation: Center NOT (0,0)
    3. Animation 200
    4. Circle Equation: Center (h,k)
    5. Equation & Graph of a Circle
  2. MAFS.912.G-GPE.2.4
    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)
  3. MAFS.912.G-GPE.2.5
    1. Rise & Run (Revamped)
    2. Rise & Run
    3. Slope: Intuitive Introduction
    4. Slope: Intuitive Introduction (II)
    5. Slope: Intuitive Intro and Exploration
    6. Parallel Consequence? (VA)
    7. Parallel Consequence? (VB)
    8. Parallel Lines in the Coordinate Plane (VA)
    9. Parallel Lines in the Coordinate Plane (VB)
    10. Coordinate Geometry: Quick Investigation
    11. Parallel and Perpendicular Lines in the Coordinate Plane
    12. Perpendicular Lines Investigation (VA)
    13. Perpendicular Lines Investigation (VB)
    14. Slope Triangle Rotation (VA)
    15. Animation 103
    16. Parallel & Perpendicular Consequence
    17. Parallel Lines Definition
    18. Perpendicular Lines Definition (& Consequence)
    19. Writing Equations of Parallel and Perpendicular Lines
  4. MAFS.912.G-GPE.2.6
    1. Midpoint and Endpoint Coordinates Action!
    2. Midpoint Coordinates (II)
    3. Animation 173
    4. Animation 174
    5. Endpoint (x,y) of a Segment Given Midpoint & Other Endpoint
    6. Midpoint (x,y) of a Segment in the Coordinate Plane (Quiz)
    7. Partition Line Segments
  5. MAFS.912.G-GPE.2.7
    1. Distance in the Coordinate Plane (With Hints)
    2. Distance in the Coordinate Plane (II)
    3. Distance in the Coordinate Plane (III)
    4. Distance in the Coordinate Plane (Quiz without Grid)
    5. Distance in the Coordinate Plane (Quiz With Grid)

1.

MAFS.912.G-GPE.1.1

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

  • 1. Circle Equation: Center (0,0)
  • 2. Circle Equation: Center NOT (0,0)
  • 3. Animation 200
  • 4. Circle Equation: Center (h,k)
  • 5. Equation & Graph of a Circle

1.1.

Circle Equation: Center (0,0)

For the questions below, be sure to zoom out if you need to!
1.
Suppose [i]P(x,y)[/i] = any point that lies on a circle with center (0,0) and radius 5. [br]Use what you've observed to write an equation that expresses the relationship among [i]x[/i], [i]y[/i], and [i]r[/i].
2.
What is the equation of a circle with center (0,0) and radius [i]r[/i] = 9?
3.
Suppose another circle has center (0,0). Suppose this circle also passes through the point (12, -5).[br]Write the equation of this circle. [br]
4. FINAL QUESTION:
Suppose [i]P(x,y)[/i] = any point that lies on a circle with center (0,0) and radius [i]r[/i], where [i]r[/i] > 0. [br]Use what you've observed to write an equation that expresses the relationship among [i]x[/i], [i]y[/i], and [i]r[/i].
Quick (Silent) Demo
– Tim Brzezinski

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1.5.

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2.

MAFS.912.G-GPE.2.4

Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2).

  • 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)

2.1.

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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2.11.

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3.

MAFS.912.G-GPE.2.5

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

  • 1. Rise & Run (Revamped)
  • 2. Rise & Run
  • 3. Slope: Intuitive Introduction
  • 4. Slope: Intuitive Introduction (II)
  • 5. Slope: Intuitive Intro and Exploration
  • 6. Parallel Consequence? (VA)
  • 7. Parallel Consequence? (VB)
  • 8. Parallel Lines in the Coordinate Plane (VA)
  • 9. Parallel Lines in the Coordinate Plane (VB)
  • 10. Coordinate Geometry: Quick Investigation
  • 11. Parallel and Perpendicular Lines in the Coordinate Plane
  • 12. Perpendicular Lines Investigation (VA)
  • 13. Perpendicular Lines Investigation (VB)
  • 14. Slope Triangle Rotation (VA)
  • 15. Animation 103
  • 16. Parallel & Perpendicular Consequence
  • 17. Parallel Lines Definition
  • 18. Perpendicular Lines Definition (& Consequence)
  • 19. Writing Equations of Parallel and Perpendicular Lines

3.1.

Rise & Run (Revamped)

[color=#9900ff][b]Link to Accompanying Lesson Activity:[/b] [/color][br][br][b][color=#0000ff][url=https://docs.google.com/document/d/1JxXYo0xWbWfqmFjScOvexHhdZ1Zmui-ju0NuXwQKSIc/edit?usp=sharing]Rise & Run: Investigation[/url] [/color][/b]
– Tim Brzezinski

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3.10.

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3.11.

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3.13.

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3.15.

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3.16.

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3.17.

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3.18.

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3.19.

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4.

MAFS.912.G-GPE.2.6

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

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

4.1.

Midpoint and Endpoint Coordinates Action!

Here, point A is an endpoint and point M is a midpoint. Move A and M where you'd like and slide the slider to find the other endpoint (B).
– Tim Brzezinski

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5.

MAFS.912.G-GPE.2.7

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

  • 1. Distance in the Coordinate Plane (With Hints)
  • 2. Distance in the Coordinate Plane (II)
  • 3. Distance in the Coordinate Plane (III)
  • 4. Distance in the Coordinate Plane (Quiz without Grid)
  • 5. Distance in the Coordinate Plane (Quiz With Grid)

5.1.

Distance in the Coordinate Plane (With Hints)

Discovery Lesson (Can be used without interactive figure below)
– Tim Brzezinski

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