0

CCSS High School: Geometry (Similarity, Right T, Trig.)

Tim Brzezinski, Emily Reed, Jul 13, 2017

[b]This GeoGebra Book contains lots of discovery-based learning activities, investigations, and meaningful remediation worksheets that were designed to help enhance students' learning of geometry concepts both inside and outside of the mathematics classroom. [color=#1551b5]This GeoGebra Book also contains contributions from the following individuals: Jennifer Silverman ([url]https://www.geogebra.org/jennifer+silverman[/url]) Steve Phelps ([url]https://www.geogebra.org/stevephelps[/url]) Dr. Ted Coe ([url]https://www.geogebra.org/tedcoe[/url]) Without their amazing talent and creativity, this resource would not have been possible. [/b][/color]

1.

HSG.SRT.A.1, A.1.A, A.1.B

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.

HSG.SRT.A.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.

HSG.SRT.A.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.

HSG.SRT.B.4 & B.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]

4.2.

4.3.

4.4.

4.5.

4.6.

4.7.

4.8.

4.9.

4.10.

4.11.

4.12.

4.13.

4.14.

4.15.

4.16.

4.17.

4.18.

4.19.

4.20.

4.21.

4.22.

4.23.

4.24.

4.25.

5.

HSG.SRT.C.6 & C.7 & C.8

1. Angles in Standard Position
2. Animation 176
3. Special Angles in Standard Position: Intro (V2)
4. Special Angles in Standard Position: Intro (V3)
5. Special Angles in Standard Position: Intro (V1)
6. Animation 177
7. Animation 178
8. Right Triangle Trigonometry: Intro
9. True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
10. Trig Ratio Estimations (Quiz: V1)
11. Trig Ratio Estimations (Quiz: V2)
12. Trigonometric Ratios (Right Triangle Context)
13. Discovering Sine, Cosine, and Tangent in Right Triangles
14. What "CO" in COsine Means
15. Animation 98
16. Animation 99
17. Animation 100
18. How Fast are You Spinning?

5.1.

Angles in Standard Position

The angle drawn below in the coordinate plane is classified as being drawn in [b]STANDARD POSITION. [br][br][/b]Interact with the applet for a minute.[br]Then answer the question that follows.

ANGLE IN STANDARD POSITION:

1.

What does it mean for an angle drawn in the coordinate plane to be drawn in [b]STANDARD POSITION? [br][br][/b](Your definition should list 2 criteria.)

5.2.

5.3.

5.4.

5.5.

5.6.

5.7.

5.8.

5.9.

5.10.

5.11.

5.12.

5.13.

5.14.

5.15.

5.16.

5.17.

5.18.

6.

HSG.SRT.D.9 & D.10 & D.11

1. Law of Sines (& Area)
2. Law of Sines (Illustrated)
3. Law of Cosines: Discovery

6.1.

Law of Sines (& Area)

Interact with the applet below for a minute. [br]Then, answer the questions that follow. [br](Please don't slide the 2nd slider until prompted to in the directions below.)

1) Take a look at the yellow right triangle on the left.[br]Write an equation that expresses the relationship among angle [i]B[/i], the triangle's height, and side [i]c[/i].

2) Rewrite this equation so that [i]height [/i]is written in terms of side [i]c[/i] and angle [i]B[/i].

3) Now consider the pink right triangle on the right. Write an equation that expresses the relationship among angle [i]C[/i], side [i]b[/i], and the triangle's height. [br]

4) Rewrite this equation so that [i]height[/i] is written in terms of side [i]b[/i] and angle [i]C[/i].

5) Take your responses to questions (2) and (4) to write a new equation that expresses the relationship among [i]C[/i], [i]B[/i], [i]c[/i], and [i]b[/i]. Write this equation so that [i]C[/i] and [i]c[/i] appear on one side of the equation and that [i]B[/i] and [i]b[/i] appear on the other.

6) Now drag the slider in the upper right hand corner. Now, given the fact that the length of segment [i]BC[/i] would be denoted as [i]a [/i](it's just not drawn in the applet above), write an expression for the area of this original triangle in terms of [i]a[/i], [i]b[/i], and [i]C[/i].

7) Same question as in (6) above, but this time write the area of the triangle in terms of [i]a[/i], [i]c[/i], and [i]B[/i].

8) Suppose that dragging the first slider dropped a height from point [i]C[/i] instead of point [i]A[/i]. Answer questions (1) - (5) again, this time letting [i]c[/i] serve as the base of this triangle (vs. side [i]a[/i]). Notice anything interesting in your results?

6.2.

6.3.

7.

Links to Other CCSS High School: Geometry Standards

1. Geometry Resources

7.1.

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.)

8.

Additional - Links to CCSS High School: Functions Resources

1. Functions Resources

8.1.

Functions Resources

[list][*][b][url=https://www.geogebra.org/m/k6Dvu9f3]Interpreting Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/uTddJKRC]Building Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/GMvvpwrm]Linear, Quadratic, and Exponential Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/aWuJMDas]Trigonometric Functions[/url][/b][/*][/list]

0