Geometry

Dr. Edgar, Mar 21, 2014

Table of Contents

  1. Transformations
    1. Reflections x-axis
    2. What's My Rule #2?
    3. 2 - What's My Rule
    4. What's My Rule #3?
    5. 4 - What's My Rule
    6. 5 - What's My Rule?
    7. What's My Rule #6?
    8. What's My Rule #2: Point Style!
    9. What's My Rule #3: Point Style!
    10. What's My Rule #1: Point Style!
    11. Rotating a Triangle Around a Coordinate Axis
    12. Double Reflection
    13. Reflection Activity
    14. Transformations
    15. Rigid Motions of the Plane
    16. Translations and Reflections, Hearne, Erik
    17. Kaleidoscope
    18. Reflections, Rotations, Dilations, and Translations
    19. Experimenting with reflection
  2. Angles
    1. Exterior Angles of Polygons
    2. Angle sum theorem
    3. Investigation of the sum of angles in a triangle
    4. Exterior angle of a triangle
  3. Area
    1. Cut Triangles into Rectangles & Dudeney's Puzzle Generalized
    2. Area of a Circle - Wedge/Sector Demonstration
    3. Area of Circles
    4. Trapezoid: Area (I)
    5. Area of Trapezoid
  4. Volume
    1. Patterns of a cube
    2. Net of a Cone
    3. Surface Area of Spheres
    4. Curved Surface Area of Cones (Parametric Curved Surface)
    5. Cavalieri's Principle
    6. Cavalieri's Principle
    7. Archimedes and the Volume of a Sphere
    8. cube cross section
    9. Volume of rectangular prisms
    10. Adjustable Prisms
    11. 12.1a Lateral Area of a Cone
    12. Right Circular Cylinder!
    13. Surface Area & Volume of Rectangular Prisms
    14. Rectangular Pyramid
  5. SIn, Cos, Tan
    1. Trigonometry- Finding A side Lenght in A Right Triangle
    2. Trigonometric Ratios (Right Triangle Context)
    3. Right Triangle Ratios
    4. Falling tree
    5. Ferris Wheel Unit Circle
    6. Law of Sines
  6. Fun
    1. Water Triangle
    2. Pythagorean Polygons
    3. Escherized Tessellation
    4. Relative Velocity: Boat Crossing a River
    5. Analog Clock
  7. Proofs
    1. Proof Without Words
    2. SAS: Dynamic Proof!
    3. SSS: Dynamic Proof!
    4. Is "SSA" Legit? What Do You Think?
    5. Animation 97
  8. Similarity
    1. Similar Right Triangles (II)
    2. AA Similarity Theorem
  9. Circles
    1. Congruent Chords (I)
    2. Inscribed Angle Theorem: Corollary 1
    3. Angle Formed by 2 Chords (II)
    4. Angle Formed by Chord & Tangent (II)

1.

Transformations

  • 1. Reflections x-axis
  • 2. What's My Rule #2?
  • 3. 2 - What's My Rule
  • 4. What's My Rule #3?
  • 5. 4 - What's My Rule
  • 6. 5 - What's My Rule?
  • 7. What's My Rule #6?
  • 8. What's My Rule #2: Point Style!
  • 9. What's My Rule #3: Point Style!
  • 10. What's My Rule #1: Point Style!
  • 11. Rotating a Triangle Around a Coordinate Axis
  • 12. Double Reflection
  • 13. Reflection Activity
  • 14. Transformations
  • 15. Rigid Motions of the Plane
  • 16. Translations and Reflections, Hearne, Erik
  • 17. Kaleidoscope
  • 18. Reflections, Rotations, Dilations, and Translations
  • 19. Experimenting with reflection

1.1.

Reflections x-axis

– Dr. Edgar

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1.11.

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1.16.

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1.17.

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1.19.

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2.

Angles

  • 1. Exterior Angles of Polygons
  • 2. Angle sum theorem
  • 3. Investigation of the sum of angles in a triangle
  • 4. Exterior angle of a triangle

2.1.

Exterior Angles of Polygons

[b]Click the "▶" button. Observe the angles turned by the car at the corners.[br]Drag the green points. Check the "Show Measurement" box to see the measurements of the angles.[br][/b]中文版：[url=https://ggbm.at/551]https://ggbm.at/551[/url]
[b]1. What is a + b + c + d + e, the total angles turned by the car?[br]2. a, b, c, d, e are called the exterior angles of the convex pentagon.[br]　　In general, what is the sum of all exterior angles of a convex polygon?[br]3. How would you modify the result if the pentagon is NOT convex?[br][/b][br]Anthony Or. GeoGebra Institute of Hong Kong.
– Anthony OR 柯志明

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3.

Area

  • 1. Cut Triangles into Rectangles & Dudeney's Puzzle Generalized
  • 2. Area of a Circle - Wedge/Sector Demonstration
  • 3. Area of Circles
  • 4. Trapezoid: Area (I)
  • 5. Area of Trapezoid

3.1.

Cut Triangles into Rectangles & Dudeney's Puzzle Generalized

A quadrilateral can be dissected into a rectangle by a line joining the mid-points of one pair of opposite sides, and the perpendiculars to this line from the mid-points of the other pair of opposite sides (see https://www.geogebra.org/student/m3459 ).[br][br]In this applet, a triangle is dissected into 4 pieces by the 3 lines. [br][br]Rotate the dissections, drag the red points to get a rectangle.[br][b]CLICK[/b] any black [b]SEGMENT[/b] to show/hide its [b]LENGTH[/b]. [br][br]Drag the Q to investigate its position at which the rectangle becomes a square (the generalized Dudeney's Puzzle).
Anthony Or. GeoGebra Institute of Hong Kong.
– Anthony OR 柯志明

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4.

Volume

  • 1. Patterns of a cube
  • 2. Net of a Cone
  • 3. Surface Area of Spheres
  • 4. Curved Surface Area of Cones (Parametric Curved Surface)
  • 5. Cavalieri's Principle
  • 6. Cavalieri's Principle
  • 7. Archimedes and the Volume of a Sphere
  • 8. cube cross section
  • 9. Volume of rectangular prisms
  • 10. Adjustable Prisms
  • 11. 12.1a Lateral Area of a Cone
  • 12. Right Circular Cylinder!
  • 13. Surface Area & Volume of Rectangular Prisms
  • 14. Rectangular Pyramid

4.1.

Patterns of a cube

Move the sliders and points.
Try to recreate some of the patterns with paper and fold the cube!
– GeoGebra Materials Team, R. Herzog

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4.11.

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4.13.

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4.14.

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5.

SIn, Cos, Tan

  • 1. Trigonometry- Finding A side Lenght in A Right Triangle
  • 2. Trigonometric Ratios (Right Triangle Context)
  • 3. Right Triangle Ratios
  • 4. Falling tree
  • 5. Ferris Wheel Unit Circle
  • 6. Law of Sines

5.1.

Trigonometry- Finding A side Lenght in A Right Triangle

Trigonometry- Finding A side Lenght in A Right Triangle
– melikeyigit5555

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6.

Fun

  • 1. Water Triangle
  • 2. Pythagorean Polygons
  • 3. Escherized Tessellation
  • 4. Relative Velocity: Boat Crossing a River
  • 5. Analog Clock

6.1.

Water Triangle

[br][br][br]Rotate a representation of a water triangle in 2 dimensions. Select from 3 different scenarios, including a constant water area, a constant distance along the left leg of the triangle, or a constant water height. Show the water area, the water height, and the distances along both legs of the triangle.[br][size=100][br][br]Contact us at [url=mailto:ptierneyfife@edc.org]ptierneyfife@edc.org[/url] if this resource should be updated, or with questions or suggestions. We appreciate your help maintaining the quality of this resource.[/size]
– EDC in Maine

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7.

Proofs

  • 1. Proof Without Words
  • 2. SAS: Dynamic Proof!
  • 3. SSS: Dynamic Proof!
  • 4. Is "SSA" Legit? What Do You Think?
  • 5. Animation 97

7.1.

Proof Without Words

Drag the points in the sketch below.[br]What do you notice? What does this prove? How does this prove it?
– Steve Phelps

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8.

Similarity

  • 1. Similar Right Triangles (II)
  • 2. AA Similarity Theorem

8.1.

Similar Right Triangles (II)

Interact with the applet below for a few minutes, then answer the questions that follow.  
[b]Questions:[/b]  [br][br]1) What is the sum of the measures of the [color=#ff0000]red[/color] and [color=#6aa84f]green[/color] angles?  [br]    How do you know this to be true?  [br] [br]2) The segment that was drawn as you dragged the slider is called an [b]altitude.[br][/b]    This [b]altitude [/b]was [b]drawn to the hypotenuse[/b].  [b]  [br][/b]    How many right triangles did this [b]altitude[/b] split the original right triangle into?[br][br]3) What does the the special movement of the red and green angles imply about[br]    these 2 smaller right triangle?  What previously learned postulate or theorem justifies[br]    your answer?  [br][br]4) Does your response for (3) also hold true for the relationship between the ORIGINAL[br]    BIG RIGHT TRIANGLE and either one of the smaller right triangles? If so, how/why[br]    do you know this?  
– Tim Brzezinski

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9.

Circles

  • 1. Congruent Chords (I)
  • 2. Inscribed Angle Theorem: Corollary 1
  • 3. Angle Formed by 2 Chords (II)
  • 4. Angle Formed by Chord & Tangent (II)

9.1.

Congruent Chords (I)

[color=#980000]Interact with the following applet below for a few minutes, then answer the questions that follow. [br][br][/color][color=#ff00ff][i]Be sure to change the locations of the BIG POINTS each time before re-sliding the slider! [/i][/color][br][br]
[color=#980000][b]Questions:[/b][/color][color=#000000][br][br][/color][color=#000000]1) Explain why the 2 circles shown are congruent. How do you know? [br][br]2) Notice how there are 3 congruent chords displayed. The red arc is said to be the arc "determined" (or "made") by these chords. Complete the following statement by filling in the blanks to make a true statement: [br][br][/color][color=#0000ff][b]In the same circle or _________________________ circles, _________________ chords [br][br]determine ________________________ arcs. [br][/b][b][br][br][/b]3) Write a formal, 2-column proof of this statement. [br][/color]
Quick Demo: 0:58 sec to 1:28 sec (BGM: Andy Hunter)
– Tim Brzezinski

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Information