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Similarity, Right Triangles, and Trigonometry (SRT) Cluster

Tim Brzezinski, JohnCMorris, Nov 14, 2017

MAFS.912.G-SRT.1.1 Verify experimentally the properties of dilations given by a center and a scale factor: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.The dilation of a line segment is longer or shorter in the ratio given by the scale factor. MAFS.912.G-SRT.1.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. MAFS.912.G-SRT.1.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. MAFS.912.G-SRT.2.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. MAFS.912.G-SRT.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. MAFS.912.G-SRT.3.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. MAFS.912.G-SRT.3.7 Explain and use the relationship between the sine and cosine of complementary angles. MAFS.912.G-SRT.3.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

1.

MAFS.912.G-SRT.1.1

1. Dilation Introduction: Transformers!
2. Dilating a Point (Intro)
3. Dilating a Line: HSG.SRT.A.1.A
4. Dilating a Segment: HSG.SRT.A.1.B

1.1.

Dilation Introduction: Transformers!

Explore this applet to see how the center of dilation and the scale factor affect the image.

Turn off the images and explore the way the lines change. What do you notice?

1.2.

1.3.

1.4.

2.

MAFS.912.G-SRT.1.2

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)

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

2.2.

2.3.

2.4.

2.5.

2.6.

2.7.

2.8.

3.

MAFS.912.G-SRT.1.3

1. Angle -Angle: Triangle Investigation
2. AA Similarity Theorem
3. Animation 97
4. AAA: Similar Quads?

3.1.

Angle -Angle: Triangle Investigation

[b]Students:[br][/b][br]Please use the GeoGebra task applet below to complete the [b]Angle-Angle (Investigation) [/b]given to you at the beginning of class. [br][br][b]LINK:[/b] [b][color=#0000ff][url=https://docs.google.com/document/d/10gizSV-6UM_E6amlcEYLmVmnWgqpW_jWti6MYAyoSwk/edit?usp=sharing]Angle-Angle (Investigation)[/url][/color] [/b]

3.2.

3.3.

3.4.

4.

MAFS.912.G-SRT.2.4, 2.5

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

4.1.

SAS ~ Theorem

[color=#000000]In the applet below, you'll find two triangles. [br][br]The [b]black angle[/b] in the [/color][color=#38761d][b]green triangle[/b][/color] [b][color=#000000]is congruent to[/color][/b][color=#000000] the [/color][b][color=#000000]black angle[/color][/b][color=#000000] in the [/color][b][color=#ff00ff]pink triangle[/color][/b][color=#000000]. [/color][br][br][color=#000000]In the [/color][color=#38761d][b]green triangle[/b][/color][color=#000000], the [b]black angle is the included angle between sides [/b][/color][b][i][color=#000000]a[/color][/i][color=#000000] and [/color][i][color=#000000]b[/color][/i][/b][color=#000000]. [/color][br][color=#000000]In the [/color][b][color=#ff00ff]pink triangle[/color][/b][color=#000000], the [b]black angle is the included angle between sides [i]ka[/i] and [i]kb[/i][/b]. [/color][br][br][color=#000000]Interact with the applet below for a few minutes. [/color][color=#000000]As you do, be sure to move the locations of the [/color][color=#38761d][b]green triangle's[/b][/color][color=#000000] [b]BIG BLACK VERTICES[/b] and the location of the [b]BIG X[/b].[br][/color][color=#000000]You can also adjust the value of [/color][i][color=#000000]k[/color][/i][color=#000000] by using the slider or by entering a value between 0 & 1. [/color][color=#000000] [br][/color][color=#000000] [/color][br]

[color=#000000]Notice how these two triangles have 2 pairs of corresponding sides that are in proportion. (After all, as long as [br]a > 0 & b > 0, ka/a = k and kb/b = k, right? ) [br][br]The [b]BLACK ANGLES INCLUDED[/b] between these two sides [b]ARE CONGRUENT[/b] as well. [/color][br][br][b][color=#0000ff]From your observations, what can you conclude about the two triangles? Why can you conclude this?[br]Clearly justify your response! [/color][/b]

MAFS.912.G-SRT.3.6, 3.7, 3.8

1. Practice: Pythagorean Theorem (1)
2. Quiz: Pythagorean Theorem (2B)
3. Angles in Standard Position
4. Animation 176
5. Quiz: Evaluating Trigonometric Functions of Angles Given a Point on its Terminal Ray
6. Evaluating Trig Functions Given a Point on the Terminal Ray
7. Trig Ratio Possibilities (Hints)
8. Right Triangle Trig: Solving for Sides (2)
9. Quiz: Special Right Triangles
10. Special Angles in Standard Position: Intro (V2)
11. Special Angles in Standard Position: Intro (V3)
12. Special Angles in Standard Position: Intro (V1)
13. Animation 177
14. Animation 178
15. Right Triangle Trigonometry: Intro
16. True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
17. Trig Ratio Estimations (Quiz: V1)
18. Trig Ratio Estimations (Quiz: V2)
19. Trigonometric Ratios (Right Triangle Context)
20. Discovering Sine, Cosine, and Tangent in Right Triangles
21. What "CO" in COsine Means
22. Quiz: Expressing Angles Using Inverse Trigonometric Function
23. Animation 98
24. Animation 99
25. Animation 100
26. YOUR Linear Speed?
27. How Fast are You Spinning?

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