Үшбұрыштар теоремалары

[color=#000000]Төмендегі сызбаны жылжытыңыз.[br]Содан кейін келесі сұрақтарға жауап беріңіз.[br][br][/color][i][color=#980000]Жүгірткіні сүйреп апармас бұрын, үшбұрыштың төбелерінің орындарын өзгерткеніне көзіңізді жеткізіңіз!!![/color][/i]
[b][color=#980000]Questions:[/color][/b][br][br][color=#000000]1) Жоғарыдағы сызбада қандай геометриялық түрлендірулер болды?[/color][br][color=#000000]  [/color][br]2) Үшбұрыштың ішкі бұрыштарымен жұмыс істегенде, көк немесе жасыл бұрыштардың өлшемдері өзгерді ма?[color=#000000][br][br]3) Сіздің бақылауларыңыз бойынша кез келген үшбұрыштың ішкі бұрыштарының өлшемдерінің қосындысы неге тең?[br]4) Сіздің зертеуіңіз бойынша үшбұрыштың сыртқы бұрыштарының қосындысы неге тең?[/color]

Special Line through Triangle V1 (Theorem Discovery)

Interact with the following applet for a few minutes. [br][br]As you do, be sure to change the locations of the triangle's vertices each time before (and even after) sliding the slider. [br][br]Then, answer the questions that follow.
1.
How would you describe the FIRST small white point that appeared as you animated the diagram?
2.
What can you conclude about the gray line and the gray side of the triangle? Why can you conclude this?
3.
What can you conclude about a line that passes through the midpoint of one side of a triangle and is parallel to another side of the triangle?
4.
Suppose you know that a line that passes through the midpoint of one side of a triangle and is parallel to another side of the triangle. Write a formal argument that proves this line [b]MUST[/b] pass through the [color=#9900ff][b]midpoint of the 3rd side[/b][/color] of the triangle.

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]

NCTM Calendar Problem (11-2-2016)

Creation of this applet was inspired by the the calendar problem (11/2/2016) that appeared in [url=http://www.nctm.org/]NCTM[/url]'s November 2016 issue of the [url=http://www.nctm.org/publications/mathematics-teacher/][i]Mathematics Teacher[/i] magazine[/url]. [br][br]"In right triangle [i]ABC[/i],[color=#0000ff][b] [i]CD[/i] is the bisector[/b][/color] of [color=#666666][b]right angle [i]ACB[/i][/b][/color], [color=#9900ff][b][i]CM[/i] is the median to the hypotenuse[/b][/color], and [color=#980000][b][i]CP [/i]is the altitude to the hypotenuse[/b][/color]. Prove [i][color=#0000ff][b]CD[/b][/color][/i] [color=#ff00ff][b]bisects[/b][/color] angle [i]PCM[/i]. " [br][br]This applet informally illustrates what this calendar problem is asking you to prove. [br][br]Can you formally prove what this applet informally illustrates?

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]

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