Similarity & Right Triangle Trigonometry

Tim Brzezinski, Sep 30, 2016

Table of Contents

  1. Exploring Dilations
    1. Dilations Part 1: What Do You Notice?
    2. Dilation Exploration
    3. Dilation Introduction: Transformers!
    4. Dilating a Point (Intro)
    5. Dilating a Line: HSG.SRT.A.1.A
    6. Dilating a Segment: HSG.SRT.A.1.B
    7. Building Similar Rectangular Prisms
  2. Proving Figures Similar
    1. Similar Figures: Dynamic Illustration
    2. Animation 243
    3. Properties of Dilations
    4. Properties of Dilations (II)
    5. Proving Triangles Similar (1)
    6. Proving Triangles Similar (2)
    7. Proving Triangles Similar (3)
    8. Proving Triangles Similar (4)
    9. Building Similar Cones
    10. Similar Solids?
  3. AA~ Theorem
    1. Angle-Angle (AA) Similarity Theorem: Quick Investigation
    2. Angle -Angle: Triangle Investigation
    3. AA Similarity Theorem
    4. Animation 97
    5. AAA: Similar Quads?
  4. Prove Similarity Theorems
    1. Side-Angle-Side: Quick Investigation
    2. SAS ~ Theorem
    3. Side-Side-Side: Quick Investigation
    4. SSS ~ Theorem (V1)
    5. Animation 63
    6. SSS ~ Theorem (V2)
    7. SSSS: Similar Quads?
    8. Similar Right Triangles (V1)
    9. Similar Right Triangles (V2)
    10. Animation 13
    11. Pythagorean and similar triangles.
    12. Effortless Trisections
    13. Animation 134
    14. Parallel Lines Congruent Segments Theorem
    15. Parallel Lines Proportionality Theorem
    16. Triangle-Angle Bisector Theorem
    17. Animation 81
    18. Scaling Figures: Effects on Areas
    19. Proof Without Words
    20. Wordless Pythagorean Theorem based on Translations
    21. Proof Without Words
    22. Proof Without Words
    23. Proof Without Words
    24. Proof Without Words
    25. Proof Without Words
    26. Proof Without Words
    27. Proof Without Words
    28. Proof Without Words
    29. Proof Without Words
  5. Pythagorean Theorem
    1. Practice: Pythagorean Theorem (1)
    2. Open Middle: Pythagorean Theorem (1)
    3. Open Middle: Pythagorean Theorem (2)
    4. Open Middle: Diagonal of a Rectangular Prism
    5. Creating Distances in 3 Space
    6. Quiz: Pythagorean Theorem (2A)
    7. Quiz: Pythagorean Theorem (2B)
  6. Right Triangle Trig
    1. Right Triangle Generator for Right Triangle Trigonometry
    2. Right Triangle Trigonometry: Intro
    3. Right Triangles: Identifying Sides WRT Acute Angles
    4. Identifying Trig Ratios: Quick Formative Assessment
    5. Finding Exact Trig Ratios: Quiz Question Generator
    6. True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
    7. Trig Ratio Estimations (Right Triangle Context)
    8. Trig Ratio Estimations (Quiz: V1)
    9. Trig Ratio Estimations (Quiz: V2)
    10. Trigonometric Ratios (Right Triangle Context)
    11. Right Triangle Solver: Formative Assessment
    12. Right Triangle Trig: Solving for Sides
    13. Right Triangle Trig: Solving for Sides (2)
    14. Finding Acute Angles of Right Triangles
    15. Special Right Triangles: Basic Questions
    16. Quiz: Expressing Angles Using Inverse Trigonometric Function
    17. What "CO" in COsine Means
    18. Special Trigonometric Identities: Quick Exploration
    19. Quiz: Special Right Triangles
    20. YOUR Linear Speed?
    21. Angles in Standard Position
    22. Animation 176
    23. Does R Matter?
    24. Quiz: Evaluating Trigonometric Functions of Angles Given a Point on its Terminal Ray
    25. Evaluating Trig Functions Given a Point on the Terminal Ray
    26. Trig Ratio Possibilities (Hints)
    27. Special Angles in Standard Position: Intro (V2)
    28. Special Angles in Standard Position: Intro (V3)
    29. Special Angles in Standard Position: Intro (V1)
  7. Trig Extensions
    1. Triangle Investigation
    2. Law of Sines (& Area)
    3. Law of Sines (Illustrated)
    4. Law of Cosines: Discovery
    5. Bearing Illustrator (1)
    6. Direction Angle Illustrator
    7. Bearing Illustrator (2)
    8. Bearings Summary
    9. Bearing: Dynamic Illustration
    10. Bearings: Dynamic Illustrator
    11. Law of Cosines: Open Middle Problem (V1)
    12. Open Middle Problem: Law of Cosines (V2)
  8. Other Geometry Resources
    1. Geometry Resources

1.

Exploring Dilations

  • 1. Dilations Part 1: What Do You Notice?
  • 2. Dilation Exploration
  • 3. Dilation Introduction: Transformers!
  • 4. Dilating a Point (Intro)
  • 5. Dilating a Line: HSG.SRT.A.1.A
  • 6. Dilating a Segment: HSG.SRT.A.1.B
  • 7. Building Similar Rectangular Prisms

1.1.

Dilations Part 1: What Do You Notice?

Interact with the app below for a few minutes. Have fun exploring! (LARGE POINTS, the slider, and Lisa's pic are moveable.)
In the app above, Lisa's pic is said to be [b]dilated[/b] about [b][color=#ff00ff]point A[/color][/b] by a [b]scale factor [i]k[/i][/b]. What does a dilation seem to do to Lisa's original pic? What can it do? Describe.
In the app below, use the [b]line tool [icon]/images/ggb/toolbar/mode_join.png[/icon] [/b]to construct [math]\overline{BC}[/math] and [math]\overline{B'C'}[/math] Then move [color=#ff00ff][b][i]A[/i][/b][/color], [b][i]B[/i][/b], and [b][i]C[/i][/b] around a bit. Move the slider as well. What seems to be true about the two lines?
Use the tool(s) of GeoGebra to prove your assertion is true. Then explain how what you did shows your claim is indeed true.
So when we dilate a line about a point (with a scale factor [math]k\ne1[/math]) , its image is another line that [br]__________ the original (pre-image) line.
So when we dilate a line about a point with a scale factor [math]k=1[/math], its image is another line that [br]__________ the original (pre-image) line.
– Tim Brzezinski

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2.

Proving Figures Similar

  • 1. Similar Figures: Dynamic Illustration
  • 2. Animation 243
  • 3. Properties of Dilations
  • 4. Properties of Dilations (II)
  • 5. Proving Triangles Similar (1)
  • 6. Proving Triangles Similar (2)
  • 7. Proving Triangles Similar (3)
  • 8. Proving Triangles Similar (4)
  • 9. Building Similar Cones
  • 10. Similar Solids?

2.1.

Similar Figures: Dynamic Illustration

SIMILAR FIGURES
[b]DEFINITION:[br][br]ANY 2 figures are said to be SIMILAR FIGURES if and only if one can be mapped perfectly onto the other under a single transformation OR a composition of 2 or more transformations. [br][br][/b]The applet below dynamically illustrates what it means, by definition, for any 2 triangles to be similar. [br]Feel free to move any of the white vertices anywhere you'd like. [color=#38761d][b]You can also change the size of the green triangle by moving the green slider. [/b][/color]
Quick (Silent) Demo
– Tim Brzezinski

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3.

AA~ Theorem

  • 1. Angle-Angle (AA) Similarity Theorem: Quick Investigation
  • 2. Angle -Angle: Triangle Investigation
  • 3. AA Similarity Theorem
  • 4. Animation 97
  • 5. AAA: Similar Quads?

3.1.

Angle-Angle (AA) Similarity Theorem: Quick Investigation

Interact with this app for a few minutes. You can move the colored sliders (lower left) to change the sizes of the colored angles. The LARGE POINTS (vertices) of these triangles are also moveable.
In the app above, what do you notice? What do you wonder?
Make a conjecture about the two triangles you see above.
Interact with this app for a few minutes. You can move the colored sliders (lower left) to change the sizes of the colored angles. The LARGE POINTS (vertices) of these triangles are also moveable.
Suppose two angles of one triangle are congruent to two angles of another triangle. What can we conclude about these two triangles? Why can we conclude this?
– Tim Brzezinski

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4.

Prove Similarity Theorems

  • 1. Side-Angle-Side: Quick Investigation
  • 2. SAS ~ Theorem
  • 3. Side-Side-Side: Quick Investigation
  • 4. SSS ~ Theorem (V1)
  • 5. Animation 63
  • 6. SSS ~ Theorem (V2)
  • 7. SSSS: Similar Quads?
  • 8. Similar Right Triangles (V1)
  • 9. Similar Right Triangles (V2)
  • 10. Animation 13
  • 11. Pythagorean and similar triangles.
  • 12. Effortless Trisections
  • 13. Animation 134
  • 14. Parallel Lines Congruent Segments Theorem
  • 15. Parallel Lines Proportionality Theorem
  • 16. Triangle-Angle Bisector Theorem
  • 17. Animation 81
  • 18. Scaling Figures: Effects on Areas
  • 19. Proof Without Words
  • 20. Wordless Pythagorean Theorem based on Translations
  • 21. Proof Without Words
  • 22. Proof Without Words
  • 23. Proof Without Words
  • 24. Proof Without Words
  • 25. Proof Without Words
  • 26. Proof Without Words
  • 27. Proof Without Words
  • 28. Proof Without Words
  • 29. Proof Without Words

4.1.

Side-Angle-Side: Quick Investigation

Interact with this app below for a few minutes. Be sure to move each slider to see what it does. Be sure to also move the LARGE POINTS (vertices) of these triangles.
In the app above, what do you notice? What do you wonder?
Make a conjecture about the two triangles you see above.
In the app below here, use any one (or more) of the GeoGebra tools to informally prove your conjecture is true.
Suppose two sides of one triangle are drawn in proportion to two sides of another triangle, and suppose their included angles are also congruent. What can we conclude about these two triangles? Why can we conclude this?
– Tim Brzezinski

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5.

Pythagorean Theorem

  • 1. Practice: Pythagorean Theorem (1)
  • 2. Open Middle: Pythagorean Theorem (1)
  • 3. Open Middle: Pythagorean Theorem (2)
  • 4. Open Middle: Diagonal of a Rectangular Prism
  • 5. Creating Distances in 3 Space
  • 6. Quiz: Pythagorean Theorem (2A)
  • 7. Quiz: Pythagorean Theorem (2B)

5.1.

Practice: Pythagorean Theorem (1)

– Tim Brzezinski

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6.

Right Triangle Trig

  • 1. Right Triangle Generator for Right Triangle Trigonometry
  • 2. Right Triangle Trigonometry: Intro
  • 3. Right Triangles: Identifying Sides WRT Acute Angles
  • 4. Identifying Trig Ratios: Quick Formative Assessment
  • 5. Finding Exact Trig Ratios: Quiz Question Generator
  • 6. True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
  • 7. Trig Ratio Estimations (Right Triangle Context)
  • 8. Trig Ratio Estimations (Quiz: V1)
  • 9. Trig Ratio Estimations (Quiz: V2)
  • 10. Trigonometric Ratios (Right Triangle Context)
  • 11. Right Triangle Solver: Formative Assessment
  • 12. Right Triangle Trig: Solving for Sides
  • 13. Right Triangle Trig: Solving for Sides (2)
  • 14. Finding Acute Angles of Right Triangles
  • 15. Special Right Triangles: Basic Questions
  • 16. Quiz: Expressing Angles Using Inverse Trigonometric Function
  • 17. What "CO" in COsine Means
  • 18. Special Trigonometric Identities: Quick Exploration
  • 19. Quiz: Special Right Triangles
  • 20. YOUR Linear Speed?
  • 21. Angles in Standard Position
  • 22. Animation 176
  • 23. Does R Matter?
  • 24. Quiz: Evaluating Trigonometric Functions of Angles Given a Point on its Terminal Ray
  • 25. Evaluating Trig Functions Given a Point on the Terminal Ray
  • 26. Trig Ratio Possibilities (Hints)
  • 27. Special Angles in Standard Position: Intro (V2)
  • 28. Special Angles in Standard Position: Intro (V3)
  • 29. Special Angles in Standard Position: Intro (V1)

6.1.

Right Triangle Generator for Right Triangle Trigonometry

Math Teachers and Students:
Here, we have a custom tool (far right) that lets you quickly construct a right triangle by simply plotting 2 points and THEN entering the measure of one of its acute interior angles. [br][br][b]Note: [/b][br]If you select the RightTriangle tool (far right), simply plot 2 points. Then, enter in the measure of any acute angle. (You can also, if you choose, enter [math]\alpha[/math] = name of slider) if you wish to quickly change the size of this acute angle.
Quick (Silent) Demo: How to Use
– Tim Brzezinski

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7.

Trig Extensions

  • 1. Triangle Investigation
  • 2. Law of Sines (& Area)
  • 3. Law of Sines (Illustrated)
  • 4. Law of Cosines: Discovery
  • 5. Bearing Illustrator (1)
  • 6. Direction Angle Illustrator
  • 7. Bearing Illustrator (2)
  • 8. Bearings Summary
  • 9. Bearing: Dynamic Illustration
  • 10. Bearings: Dynamic Illustrator
  • 11. Law of Cosines: Open Middle Problem (V1)
  • 12. Open Middle Problem: Law of Cosines (V2)

7.1.

Triangle Investigation

[b]Students:[/b][br][br]Please use this applet in conjunction with the activity you received at the beginning of today's class.
– Tim Brzezinski

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8.

Other Geometry Resources

  • 1. Geometry Resources

8.1.

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