CCSS High School: Geometry (Similarity, Right T, Trig.)
[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]
Table of Contents
- HSG.SRT.A.1, A.1.A, A.1.B
- Dilation Introduction: Transformers!
- Dilating a Point (Intro)
- Dilating a Line: HSG.SRT.A.1.A
- Dilating a Segment: HSG.SRT.A.1.B
- HSG.SRT.A.2
- Similar Figures: Dynamic Illustration
- Animation 243
- Properties of Dilations
- Properties of Dilations (II)
- Proving Triangles Similar (1)
- Proving Triangles Similar (2)
- Proving Triangles Similar (3)
- Proving Triangles Similar (4)
- HSG.SRT.A.3
- Angle -Angle: Triangle Investigation
- AA Similarity Theorem
- Animation 97
- AAA: Similar Quads?
- HSG.SRT.B.4 & B.5
- SAS ~ Theorem
- SSS ~ Theorem (V1)
- Animation 63
- SSS ~ Theorem (V2)
- SSSS: Similar Quads?
- Similar Right Triangles (V1)
- Animation 13
- Pythagorean and similar triangles.
- Effortless Trisections
- Animation 134
- Parallel Lines Congruent Segments Theorem
- Parallel Lines Proportionality Theorem
- Triangle-Angle Bisector Theorem
- Animation 81
- Proof Without Words
- Wordless Pythagorean Theorem based on Translations
- Proof Without Words
- Proof Without Words
- Proof Without Words
- Proof Without Words
- Proof Without Words
- Proof Without Words
- Proof Without Words
- Proof Without Words
- Proof Without Words
- HSG.SRT.C.6 & C.7 & C.8
- Practice: Pythagorean Theorem (1)
- Quiz: Pythagorean Theorem (2B)
- Angles in Standard Position
- Animation 176
- Quiz: Evaluating Trigonometric Functions of Angles Given a Point on its Terminal Ray
- Evaluating Trig Functions Given a Point on the Terminal Ray
- Trig Ratio Possibilities (Hints)
- Right Triangle Trig: Solving for Sides (2)
- Quiz: Special Right Triangles
- Special Angles in Standard Position: Intro (V2)
- Special Angles in Standard Position: Intro (V3)
- Special Angles in Standard Position: Intro (V1)
- Animation 177
- Animation 178
- Right Triangle Trigonometry: Intro
- True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
- Trig Ratio Estimations (Quiz: V1)
- Trig Ratio Estimations (Quiz: V2)
- Trigonometric Ratios (Right Triangle Context)
- Discovering Sine, Cosine, and Tangent in Right Triangles
- What "CO" in COsine Means
- Quiz: Expressing Angles Using Inverse Trigonometric Function
- Animation 98
- Animation 99
- Animation 100
- YOUR Linear Speed?
- How Fast are You Spinning?
- HSG.SRT.D.9 & D.10 & D.11
- Law of Sines (& Area)
- Law of Sines (Illustrated)
- Law of Cosines: Discovery
- Bearing Illustrator (1)
- Direction Angle Illustrator
- Bearing Illustrator (2)
- Bearings Summary
- Bearing: Dynamic Illustration
- Bearings: Dynamic Illustrator
- Links to Other CCSS High School: Geometry Standards
- Geometry Resources
- Additional - Links to CCSS High School: Functions Resources
- Functions Resources
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?
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
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]
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]
Practice: Pythagorean Theorem (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?
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.)
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]