Unit 6: Properties and Attributes of Triangles
Table of Contents
- 5.1 Perpendicular and Angle Bisectors
- Perpendicular Bisector Definition
- Angle Bisector Definition (I)
- Angle Bisector: Construction
- Perpendicular Bisector: Construction
- 5.2 Bisectors of Triangles
- Points of Concurrency: Investigation
- Circumcenter (& Questions)
- Circumcenter & Circumcircle Action!
- Circumcircle: Construction Exercise (VA)
- Incenter Exploration (A)
- Incenter Exploration (B)
- Incenter & Incircle Action!
- Incircle: Construction Exercise (VA)
- 5.3 Medians and Altitudes of Triangles
- Medians and Centroid Dance
- Orthocenter Exploration
- Medians & Equal Areas!
- Animation 25
- Where in NYC? (VA)
- 5.4 Triangle Midsegment Theorem
- Special Line through Triangle V2 (Theorem Discovery)
- Triangle Midsegment Action!
- 5.5 Inequalities in one triangle
- Triangle Inequality: Discovery Lesson
- Triangle Side Possibilities? (V2)
- Triangle Inequality (II)
- Triangle Inequality
- Inequalities in 1 Triangle
- 5.6 Inequalitites in 2 Triangles
- Inequalities in Two Triangles-Hinge Theorem
- Hinge Theorem
- 5.7 Pythagorean Theorem
- Acute, Right, or Obtuse?
- Acute, Right, Obtuse (III)
- Proof Without Words
- Proof Without Words
- Proof Without Words
- Proof Without Words
- 5.8 Applying Special Right Triangles
- Special Right Triangle (I)
- Another Special Right Triangle (Guided Discovery)
- Special Right Triangle (II)
- 12.6 Tessellations
Perpendicular Bisector Definition
[color=#000000]In the applet below, [/color][b][color=#cc0000]line p[/color][/b][color=#000000] is said to be the [/color][b][color=#cc0000]perpendicular bisector[/color][/b][color=#000000] of the segment with [/color][i][b]A[/b][/i][color=#000000] and [/color][i][b]B[/b][/i][color=#000000] as [/color][b]endpoints[/b][color=#000000]. [br][br]Interact with this applet for a few minutes, then answer the questions that follow. [br][/color][i]Be sure to change the locations of points [b]A[/b] and [b]B[/b] each time before you re-slide the slider. [/i]
[color=#000000]Reset the applet and re-slide the slider just one more time. [br][br][b]Questions: [/b] [br][br]1) What can you conclude about the white point you see in the applet above? [br] How do you know this? [br][br]2) What is the measure of the [/color][b]gray angle[/b][color=#000000]? How do you know this to be true? [br][br]3) Given your responses for (1) and (2) above, write your own definition for the term[br] [/color][i][color=#cc0000]perpendicular bisector[/color][color=#000000]of a segment[/color][/i][color=#000000]. In essence, complete the following sentence definition: [br][br] A [/color][b][color=#cc0000]perpendicular bisector[/color][/b][color=#000000] [b]of a segment[/b] is...[/color]
Points of Concurrency: Investigation
[color=#000000]You can use the applet below to serve as the template for your[br][/color][color=#0000ff][b]Altitudes, Angle Bisectors, Perpendicular Bisectors of Triangles[/b][/color][color=#000000] Investigation. [br][/color][b][color=#cc0000]The PDF file of [/color][color=#0000ff]this investigation[/color][color=#cc0000] can be found BELOW THE APPLET.[/color][/b]
Orthocenter, Circumcenter, Incenter GeoGebra Lab
Medians and Centroid Dance
[color=#000000]Recall that a [/color][b][color=#980000]median of a triangle[/color] [/b][color=#000000]is a[/color][color=#980000] [b]segment that connects any vertex to the midpoint of the side opposite that vertex. [/b] [/color][color=#000000]Since a triangle has 3 vertices, it has 3 medians. [/color][br][br][color=#000000]This applet will illustrate 2 very special properties about a triangle's 3 medians. Interact with it for a few minutes, then answer the questions that follow. [/color][br][br][color=#000000]Note: The[/color][color=#ff7700] [b]BIG ORANGE POINT[/b] [/color][color=#000000]that will appear is known as the[/color][color=#ff7700] [b]CENTROID[/b] [/color][color=#000000]of the triangle.[/color][br][br][i][color=#9900ff]Have fun with this![/color][/i] [color=#000000]Be sure to change the locations of the triangle's BIG WHITE VERTICES each time before re-sliding the slider. [/color]
1.
What word can you use to describe the intersection of a triangle's 3 medians? How do they intersect?
2.
[color=#000000]Suppose the [/color][color=#9900ff]entire purple median[/color] [color=#000000]of the triangle above measures[/color] [color=#9900ff]18 inches[/color]. [color=#000000]What would the distance[/color] [color=#9900ff][i]BG[/i] [/color][color=#000000]be? What would the distance[/color] [i][color=#9900ff]GF[/color][/i] [color=#000000]be? [/color]
3.
[color=#000000]Suppose the[/color] [color=#1e84cc]entire blue median[/color] [color=#000000]of the triangle above measures[/color] [color=#1e84cc]12 inches[/color]. [color=#000000]What would the distance[/color] [i][color=#1e84cc]AG [/color][/i][color=#000000]be? What would the distance [/color][i][color=#1e84cc]GE[/color][/i] [color=#000000]be?[/color]
4.
What is the exact value of the ratio AG/AE? [br][br]What is the exact value of the ratio CG/CD? [br][br]What is the exact value of the ratio BG/BF?
5.
What do you notice about your results for (4) above?
5.
[color=#000000]Suppose you have a triangle with only 1 median drawn. Without constructing its other 2 medians, explain how you can locate the [/color][color=#ff7700][b]centroid[/b][/color] [color=#000000]of the triangle. [/color]
Quick (Silent) Demo
Special Line through Triangle V2 (Theorem Discovery)
[b][color=#0000ff][url=https://docs.google.com/document/d/1VrXxqnAj27JwP2Pl5UaK-iY-_ST4cOtJD5r4ylttTLk/edit?usp=sharing]Special Line Through Triangle: Discovery Lesson (Activity)[/url][/color][/b]
Triangle Inequality: Discovery Lesson
[b]Students:[/b][br][br]Use this app to complete this [b][url=https://docs.google.com/document/d/0B9yLyXTmQG2UY1JGcjlNMFUtdk0/edit?usp=sharing&ouid=114160765969083014223&resourcekey=0-Xrf4YVQ7gm54ghu0gNq-Sw&rtpof=true&sd=true]Triangle Inequality investigation[/url][/b] given to you at the beginning of class. [br]
Be sure to move the endpoints of the segments shown as you try to build a triangle with 3 given side lengths. You can also show circles when you need.
Inequalities in Two Triangles-Hinge Theorem
Hinge Theorem
Acute, Right, or Obtuse?
[color=#000000]In the applet below, you'll see a triangle with a colored square built off each side. [br]You can change the size and shape of this triangle by moving its[/color] [b]BIG GRAY VERTICES[/b] [color=#000000]around.[/color][br][color=#000000]You can also use the[/color] [color=#ff0000][b]red slider. [br][/b][/color][br][color=#000000]Interact with the applet below for a few minutes. Then, answer the questions that follow.[/color]
[b][color=#980000]Questions: [br][/color][/b][br][color=#000000]1) Is it at all possible for the [/color][b][color=#000000]sum of the areas of the 2 smaller squares[/color][/b][color=#000000] to [/color][b][color=#0000ff]be EQUAL TO[/color][color=#000000] the area of the largest square? [/color][/b][color=#000000] If this is possible, [/color][i][b][color=#980000]how would you classify such a triangle (for which you observe this to be true) [/color][/b][/i][i][b][color=#980000]by its angles? [/color][/b][color=#000000][br][/color][br][/i][color=#000000]2) Is it at all possible for the [b]sum of the areas of the 2 smaller squares[/b] to [/color][color=#0000ff][b]be GREATER THAN[/b][/color][color=#000000] [b]the area of the largest square?[/b] If this is possible, [/color][color=#980000][b][i]how would you classify such a triangle (for which you observe this to be true) by its angles? [br][br][/i][/b][/color][color=#000000]3[/color][color=#000000]) Is it at all possible for the [b]sum of the areas of the 2 smaller squares[/b] to [/color][color=#0000ff][b]be LESS THAN[/b][/color][color=#000000] [b]the area of the largest square?[/b] If this is possible, [/color][color=#980000][b]how would you classify such a triangle (for which you observe this to be true) [i]by its angles? [/i][/b][/color]
Special Right Triangle (I)
[color=#000000]Interact with the applet below for a few minutes. [br][br]As you do, be sure to change the locations of the LARGE WHITE POINTS each time before re-sliding the slider. [br][br]Then, answer the questions that follow. [/color]
1.
How would you classify the triangle above [i]by its sides? [/i]
2.
[color=#000000]What is the measure of the [/color][b][color=#666666]gray angle[/color][/b]? [color=#000000]Explain how you know this to be true. [br][/color][color=#000000][br][/color]
3.
[color=#000000]What is the measure of each[/color] [color=#ff00ff]acute pink angle[/color]? [color=#000000]Explain how you know this to be true. [br][/color][color=#000000][br][/color]
4.
What are the measures of this triangle's interior angles? (List from least to greatest.) [br]
5.
Suppose the [b]thick black segment[/b] in the triangle above [b]measures 3 inches. [/b][br]Algebraically determine the length of the longest side of this triangle in simple radical form.
6.
Suppose the [b]thick black segment[/b] in the triangle above [b]measures 4 inches. [/b][br]Algebraically determine the length of the longest side of this triangle in simple radical form.
7.
Suppose the [b]thick black segment[/b] in the triangle above [b]measures 5 inches. [/b][br]Algebraically determine the length of the longest side of this triangle in simple radical form.
8.
Suppose the [b]thick black segment[/b] in the triangle above [b]measures 6 inches. [/b][br]Algebraically determine the length of the longest side of this triangle in simple radical form.
9.
Do you notice any patterns in your answers for questions (5) - (8) above? Explain.
10.
Suppose we call [b]thick black segment[/b] in the triangle above as [i][b]LEG. [/b][/i][br]What would the length of this triangle's longest side be (in terms of [b][i]"LEG"[/i][/b]?)
11.
Algebraically prove your response to (10) true.