Geometry
Geometry Knapp
Table of Contents
- Foundations: Points, Lines, Planes, Angles
- Parallel & Perpendicular Lines
- Triangle Centers & Inequality
- Constructing the Centroid
- Medians and Centroid Dance
- Triangle Inequality (II)
- Incenter Exploration (A)
- Incenter Exploration (B)
- Converse of the Pythagorean Theorem
- Converse of the Pythagorean Theorem
- Transformations and ∆ Congruence
- Proving Triangles Congruent (2): One Step Transformations
- Congruent Figures: Dynamic Illustration
- SAS: Dynamic Proof!
- Drawing Symmetrical Polygons
- Quadrilaterals & Polygons
- Exploring Polygon Angles: Triangle through Octagon
- Properties of Special Parallelograms Exploration
- Similarity
- Dilations Part 1: What Do You Notice?
- Properties of Dilations
- AA Similarity Theorem
- SAS Similarity Theorem: Exploration
- SSS Similarity Theorem: Exploration
- Trigonometry
- Exploring Special Right Triangles 30-60-90
- Special Right Triangle (2): Thinking Classroom Exploration
- Thin Slice Discovery: Special Right Triangle (1)
- Circles
- Inscribed Angles Investigation (Revamped)
- Central Angles, Inscribed Angles, and their Intercepted Arcs
- Circumscribed, Central, and Inscribed Angles
- Tangent-radius relationship
- Secant-Secant Investigation
- Angles from Secants and Tangents (V1)
- Surface Area & Volume
- Surface Area: Intuitive Introduction
- Statistics
- Color-Coded Linear Regression (Intro)
- Mystery Number: What Does it Tell Us?
Constructing the Centroid
Play through this construction of the centroid...
And now try it on your own.
Proving Triangles Congruent (2): One Step Transformations
Use any one (or more) transformation(s) to prove the colored triangle is congruent to the outlined triangle.
Use a single transformation to prove the colored triangle is congruent to the outlined triangle.
Use a single transformation to prove the colored triangle is congruent to the outlined triangle.
Use a single transformation to prove the colored triangle is congruent to the outlined triangle.
Use a single transformation to prove the colored triangle is congruent to the outlined triangle.
Use a single transformation to prove the colored triangle is congruent to the outlined triangle.
Use a single transformation to prove the colored triangle is congruent to the outlined triangle.
Use a single transformation to prove the colored triangle is congruent to the outlined triangle.
Exploring Polygon Angles: Triangle through Octagon
Move the vertices (corners) of this TRIANGLE anywhere you'd like. Explore!
Move the vertices (corners) of this QUADRILATERAL anywhere you'd like. Explore!
Pentagon
Hexagon
Heptagon
Octagon
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.
Exploring Special Right Triangles 30-60-90
1. Calculate the ratios: [br]- XZ/ZY[br]- XZ/XY[br]- ZY/XY[br][br]
Slide the "s" slider in corner to adjust side lengths to new lengths.[br]Use the new lengths to calculate ratios again:[br][br]- XZ/ZY[br]- XZ/XY[br]- ZY/XY[br]
Fill in the blank:[br][br]The blue hypotenuse is always equal to the length of the red short leg multiplied by __________________.
Try and write an equation that helps you find the blue hypotenuse if you know the short leg (red).
What do you get when you divide the green long leg by the red short leg? give 3 decimals
Change the length of S using the slider.[br][br][br]What do you get when you divide the green long leg by the red short leg now? give 3 decimals
Change the length of S using the slider one more time.[br][br][br]What do you get when you divide the green long leg by the red short leg now? give 3 decimals
Fill in the blank in this statement with a number to 3 decimals:[br][br]The green long leg always equals the red short let multiplied by __________________.
Try to write an equation that finds the length of the green long leg if you know the red short leg.
Using the triangle above,[br][br]If the short leg is 2.7, how long is the blue hypotenuse?
Using the triangle above,[br][br]If the short leg is 2.7, how long is the green long leg? Give 2 decimals.
Using the triangle above,[br][br]If the blue hypotenuse is 5, how long is the short leg? Give one decimal.
Once you find the red short leg, use that length to find the long green long leg. Give 2 decimals.
Inscribed Angles Investigation (Revamped)
The two colored angles you see are said to be [i]inscribed angles. [br][/i]You can move these angles by moving their vertices around the circle. [br]You can also move the endpoints of the same dashed arc they both intercept. [br][br]Slide the slider you see all the way to the right. Move (and rotate) these two angles around.[br]What do you notice? What do you wonder? Describe.
Surface Area: Intuitive Introduction
TEACHERS:
For an introductory class activity related to this, [url=https://www.geogebra.org/m/mgwejudc]click here[/url].
Color-Coded Linear Regression (Intro)
The applet below displays 10 points plotted in the coordinate plane.[br]It also displays (what's called) the [b]best-fit-line[/b] for these 10 points.[br]Shown also is a number we call the [b]correlation coefficient ([i]r[/i]). [/b][br][br]You can drag these points anywhere you'd like. [br]As you do, watch what happens. [br][br]Interact with this applet for a few minutes. [br]Then, answer the questions that follow.
1.
Reposition (rearrange) all 10 points so that the [b][color=#1c4587]correlation coefficient ([i]r[/i]) = +1[/color][/b]. How would you describe the position(s) of these points? Be specific!
2.
Reposition (rearrange) all 10 points so that the [color=#cc0000][b]correlation coefficient ([i]r[/i]) = -1[/b][/color]. How would you describe the position(s) of these points? Be specific!
3.
Can you drag the points around so the correlation coefficient of the best fit line is zero? Try it.
4.
Try to position the point(s) anywhere so that the correlation coefficient [i]r[/i] is between 0.90 and 1.00. Describe what you see.
5.
Repeat question (4) a few times, but this time try to make [i]r[/i] [br][br]a) between 0.80 and 0.90[br]b) between 0.50 and 0.60[br]c) between 0.20 and 0.30[br][br]Can you make any generalizations based upon what you see?
6.
Try to position the point(s) anywhere so that the correlation coefficient [i]r[/i] is between -1.00 and -0.90. Describe what you see.
7.
Repeat question (6) a few times, but this time try to make [i]r[/i] [br][br]a) between -0.90 and -0.80[br]b) between -0.60 and -0.50[br]c) between -0.30 and -0.20[br][br]Can you make any generalizations based upon what you see?