Geometry: Triangle Theorems
内容目录
- Theorems Involving Angles
- Triangle Angle Theorems
- Triangle Angle Theorems (V2)
- Triangle Angle Theorems (V3)
- Triangle Angle Sum Theorem
- Exterior Angles of a Triangle
- Triangle Exterior Angle
- Angles of a Right Triangle
- Triangle Theorems (General)
- Triangle Midsegment Action!
- A Special Triangle & Its Properties (I)
- Converse of IST (V1)
- Another Special Triangle and its Properties (II)
- Another Special Right Triangle (Guided Discovery)
- Special Right Triangle (I)
- Acute, Right, or Obtuse?
- Acute, Right, Obtuse (III)
- Isosceles Triangle Medians
- Special Right Triangle (II)
- SAS: Dynamic Proof!
- Isosceles Triangle Theorem: Discovery Lab
- Points of Concurrency
- Incenter Exploration (A)
- Incenter Exploration (B)
- Incenter & Incircle Action!
- Orthocenter Exploration
- Circumcenter (& Questions)
- Circumcenter & Circumcircle Action!
- Medians and Centroid Dance
- Medians Centroid Theorem (Proof without Words)
- Midpoint of HYP
- Points of Concurrency: Investigation
- Morley Action!
- Points of Concurrency - Extension Activities
- 3 Special Points!
- 9 Point Circle Action
- 9-Point Circle Action (Part 2)
- 9-Point Circle Action (Part 3A)
- 9-Point Circle Action (Part 3B)
- 9-Point Circle Action (Part 4)
- Euler Line Generator
- Older (Earlier) Applets
- Midsegment of a Triangle
- Isosceles Triangle (Properties)
- Euler Line (Informal Investigation)
- 9-Point Circle (Informal Investigation)
Triangle Angle Theorems
Interact with the app below for a few minutes. [br]Then, answer the questions that follow. [br][br]Be sure to change the locations of this triangle's vertices each time [i]before[/i] you drag the slider!
What is the [b]sum of the measures of the interior angles of this triangle? [/b]
What is the [b]sum of the measures of the exterior angles [/b]of this triangle?
Triangle Midsegment Action!
[b][color=#0000ff][url=https://docs.google.com/document/d/1HsFCBz3suKP-AhU50diM9G9zdtdswAsaR-m4jCJ8omI/copy]Triangle Midsegment Action! (Lesson Activity)[/url][/color][/b]
Quick (Silent) Demo
Incenter Exploration (A)
[color=#000000]Recall that 3 or more lines are said to be concurrent if and only if they intersect at exactly one point. [br][br]The angle bisectors of a triangle's 3 interior angles are all concurrent. [br]Their point of concurrency is called the I[b]NCENTER[/b] of the triangle. [br][br]In the applet below, [b]point I [/b]is the triangle's [b]INCENTER[/b]. [br]Use the tools of GeoGebra in the applet below to complete the activity below the applet. [br][i]Be sure to answer each question fully as you proceed. [/i] [/color]
[color=#000000][b]Directions: [/b][br][br][/color][left][color=#000000]1) In the applet above [/color][color=#38761d]construct a line passing through I and is perpendicular to [i]AB[/i][/color][color=#000000]. [br]2) Use the [/color][b][color=#000000]Intersect[/color][/b][color=#000000] tool to plot and label a point [/color][i]G[/i][color=#000000] where [/color][color=#38761d]the line you constructed in (1)[/color][color=#000000] intersects [/color][i][/i][color=#000000][i]AB[/i].[/color][color=#000000][br]3) [/color][color=#38761d]Construct a line that passes through I and is perpendicular to [i]BC[/i].[/color][color=#000000] [br][/color][color=#000000]4) Plot and label a point [i]H[/i] where [/color][color=#38761d]the line you constructed in (3)[/color][color=#000000] intersects [i]BC[/i].[/color][br][color=#000000]5) [/color][color=#38761d]Construct a line that passes through I and is perpendicular to [i]A[/i][i]C[/i]. [/color][color=#000000][br][/color][color=#000000]6) Plot and label a point [/color][i]J[/i][color=#000000] where [/color][color=#38761d]the line you constructed in (5)[/color][color=#000000] intersects [/color][i][color=#000000]AC[/color][/i][color=#000000]. [br]7) Now, use the [b]Distance[/b] tool to measure and display the lengths [i]IG[/i], [i]IH[/i], and [i]IJ[/i]. What do you notice?[br][br][br]8) Experiment a bit by moving any one (or more) of the triangle's vertices around[br] Does your initial observation in (7) still hold true? [br] Why is this? (If you need a hint, refer back to the worksheet found [url=https://tube.geogebra.org/m/tU3ZqhjN]here[/url]. [/color][/left][color=#000000][br]9) Construct a circle centered at I that passes through [i]G[/i]. What else do you notice? [br] Experiment by moving any one (or more) of the triangle's vertices around. [br] This circle is said to be the triangle's [i]incircle[/i], or [i]inscribed circle[/i]. [br] It is the largest possible circle one can draw [i]inside[/i] this triangle. [br] Why, according to your results from (7) is this possible? [br][br]10) Do the angle bisectors of a triangle's interior angles also bisect the sides opposite theses angles? [br] Use the [b]Distance[/b] tool to help you answer this question. [/color][br][br][color=#000000]11) Is it ever possible for a triangle's [b]INCENTER[/b] to lie OUTSIDE the triangle?[br] If so, under what condition(s) will this occur? [br][br]12) Is it ever possible for a triangle's [b]INCENTER[/b] to lie ON the triangle itself?[br] If so, under what condition(s) will this occur? [/color]
3 Special Points!
[color=#000000]Recall the following: [/color][br][br][color=#ff7700]1) The lines that contain a triangle's 3 altitudes are concurrent (intersect at exactly one point.) [br] This point of concurrency is called the orthocenter of the triangle. [/color][br][br][color=#cc0000]2) A triangle's 3 perpendicular bisectors are concurrent at a point called the circumcenter of the triangle.[/color][br][br][color=#38761d]3) A triangle's 3 medians are concurrent at a point called the centroid of the triangle. [br][br][/color][color=#000000]Interact with the applet below for a few minutes. Then answer the discussion questions that follow. [/color]
[color=#000000][b]Questions: [/b][br][br]1) What conclusion can you make about the positioning of a triangle's[/color] [color=#ff7700]orthocenter,[/color] [color=#cc0000]circumcenter,[/color] [color=#000000]and[/color] [br] [color=#38761d]centroid[/color][color=#000000]? Explain how you can use the toolbar to illustrate this. [/color][br][br][color=#000000]2) How does the sliding the slider also informally show that your response to (1) is true? [/color][br][br][color=#000000]3) Let's denote the [/color][color=#ff7700]orthocenter as [/color][i][color=#ff7700]O[/color], [/i][color=#000000]the [/color][color=#cc0000]circumcenter as [i]C[/i][/color], [color=#000000]and the[/color] [color=#38761d]centroid as [i]G[/i][/color][color=#000000].[/color] [br] [color=#000000]What is the exact value of the ratio [i]CG/CO[/i]? What is the exact value of the ratio [i]CG[/i]/[i]GO[/i]?[/color] [br][br][color=#000000]4) Prove your assertion for (1) true using a coordinate geometry format. [br] For simplicity's sake, position the triangle so its vertices have coordinates (0,0), (6a, 0), and (6b, 6c). [br][br]5) Prove your responses to (3) are true using the same coordinate geometry setup you used in (4) above. [br][br]6) Research information about the Euler Line of a triangle. [br] How does the Euler Line relate to the context of the above applet? [/color]
Midsegment of a Triangle
Definition: A [b]midsegment of a triangle[/b] is a segment that connects the midpoints of any 2 sides of that triangle. [br]Question: How many midsegments does a triangle have? [br][br]Let's proceed:[br][br]In the applet below, points [color=#1551b5]D[/color] and [color=#c51414]E[/color] are midpoints of 2 sides of triangle ABC. One [color=#0a971e]midsegment[/color] of Triangle ABC is shown in [color=#0a971e]green[/color]. [br]Move the vertices A, B, and C of Triangle ABC around. As you do, observe the two comments off to the right side. [br]Then, answer the questions below the applet.
Questions: [br][br]1) What do you notice about the slopes of segments [color=#0a971e]DE[/color] and AB? What does this imply about these 2 segments? [br]2) What does the ratio of [color=#0a971e]DE[/color] to AB tell us about the [color=#0a971e]midsegment [/color]of any triangle? [br][br]3) If we refer to the black side of the triangle as the triangle's "3rd side", complete the following statement. Be sure to use the phrase "3rd side" in each blank below. [br][br] [b]The [color=#0a971e]MIDSEGMENT of a triangle[/color] is ALWAYS [br][br] i) ________________________________________________________________________, and[br][br] ii) ________________________________________________________________________. [/b]