Geometry Notes 1.1: Points,Lines, and Planes
[color=#3c78d8]Undefined terms: [br]A [b]point[/b] is a location and is named by a capital letter.[br]A [b]line [/b]is made up of points and has no thickness or width. It is named with two letters representing points on the line, or a lowercase script letter.[br]A [b]plane[/b] is an infinite flat surface made up of points and is named by a capital script letter or by three letters representing non collinear points on a plane.[br][b]Collinear[/b] points can be contained on a line.[br][b]Coplanar [/b]points can be contained on a plane.[/color]
[size=150][size=200]FIRST DIAGRAM: Drag the diagram around by moving the points (you can even make it spin!), and then wait for the class.[/size][/size]
[size=200][color=#ff0000]BREAK OUT ROOMS: remember to leave your microphone on, and talk through the problems with your partner. Compare answers, and discuss what you see. Do #1 - #4 on GeoGebra, and #1-#4 on your [/color][color=#9900ff]printed notes[/color][color=#ff0000].[/color][/size]
1. Name 3 collinear points
B, A, C[br][br](note: did you write capital letters? did you use commas?)
2. Name 3 noncollinear points:
Many correct answers![br]Here is one: A, D, G
3. Name a plane.
Many answers! Here are two:[br][i][br]Q[/i] (note: this should be a SCRIPT capital letter, but GeoGebra won't let me write in script!)[br]plane AFD
4. Name [math]\text{\overleftrightarrow{AB}}[/math] in a different way. (Choose all that apply)
[size=150][size=200][color=#cc4125]Now do #1-#4 on your [/color][color=#9900ff]printed notes[/color][color=#cc4125] together, and then:[br]STOP! [br]LEAVE YOUR BREAK-OUT ROOM AND GO BACK TO THE MAIN ROOM! [br](Don't do #5 yet.)[/color][/size][/size]
[color=#a64d79]Next:[/color][br][br]With the teacher:[br]Draw [math]\text{\overleftrightarrow{BD}}[/math], ([icon]/images/ggb/toolbar/mode_join.png[/icon]) and then drag it. Notice how the part of the line that is behind the plane is dashed.[br]Draw plane CEA ([icon]/images/ggb/toolbar/mode_planethreepoint.png[/icon]), and then drag it.[br]Then delete the line and plane that you drew.[br][br]
[size=200][color=#ff0000]BREAK OUT ROOMS: remember to leave your microphone on, and talk through the problems with your partner. Compare answers, and discuss what you see. Do #5 - #12 on GeoGebra, and then do # 5 - #10 on your [/color][color=#9900ff]printed notes[/color][color=#ff0000].[/color][/size]
5. If possible, draw a line through E and C. Are E and C collinear?
6. If possible, draw a line through E, G, and C. Are E, G, and C collinear?
7. If possible, name two points that are non collinear, because you CANNOT draw a line through them.
Not possible.[br]We say:[br]Through any two points there exists a line.[br]or:[br]Two points determine a line.
8. If possible, draw a plane through D, B, and F. Are D, B, and F coplanar?
9. If possible, draw a plane through A, G, E, and B. Move the diagram around to see if the four points are on the plane. Are A, G, E, and B coplanar?
No. (not possible to draw a plane, because A, G, E, and B are not coplanar.)
10. If possible, name 3 points that are NOT coplanar, because you CANNOT draw a plane through them.
Not possible.[br]We say:[br]Through any 3 points exists a plane.[br]or:[br]Three points determine a plane.
SECOND DIAGRAM: Use points F, D, and E to drag the rectangular prism around. Then answer the following questions. Remember to discuss each problem with your partner-- microphones on, talk over the question.
11. Name the intersection of [math]\text{\overline{FA}}[/math] and [math]\text{\overline{AE}}[/math].
12. Name the intersection of [math]\text{\overleftrightarrow{FA}}[/math] and [math]\text{\overleftrightarrow{AE}}[/math] (the lines are not shown)
[color=#ff0000][size=200]Do #5 - #10 on your printed notes together, and then:[br]Stop! Leave the breakout room and go to the main room.[/size][/color][size=200][color=#ff0000](Don't do #16 yet.)[/color][/size]
Next we will practice drawing a 3 dimensional diagram in two dimensions, as though you are drawing on paper. How can you represent a plane? How can you represent a line or point that is NOT on that plane? [br][br]First, practice with the teacher:[br]a) Draw a plane that looks like the plane in the printed notes. ( using the polygon tool. ([icon]/images/ggb/toolbar/mode_polygon.png[/icon])[br]b) Draw two points on the plane, and draw the line through them. ([icon]/images/ggb/toolbar/mode_pointonobject.png[/icon] and [icon]/images/ggb/toolbar/mode_join.png[/icon])[br]c) Draw two points NOT on the plane, draw the line through them. ([icon]/images/ggb/toolbar/mode_point.png[/icon] and [icon]/images/ggb/toolbar/mode_join.png[/icon] )[br]d) Use dashed segments to make it clear that that line is not on the plane. ([icon]/images/ggb/toolbar/mode_segment.png[/icon] )[br][br]
16. Use the Applet below.[br]a) Draw two points, P and Q using the [icon]/images/ggb/toolbar/mode_point.png[/icon] tool.[br]b) Sketch [math]\text{\overleftrightarrow{PQ}}[/math] using the [icon]/images/ggb/toolbar/mode_join.png[/icon] tool.[br]c) Add a point R on the line so that Q is between P and R.
17. Use the Applet below. [br]a) Draw a plane using the [icon]/images/ggb/toolbar/mode_polygon.png[/icon] tool.[br]b) Put point Q on the plane using the [icon]/images/ggb/toolbar/mode_pointonobject.png[/icon] tool.[br]c) Draw [math]\text{\overleftrightarrow{QR}}[/math] so that it intersects the plane only once. [br]d) We can use the segment tool to do a pretty good job of creating a dashed section. [icon]/images/ggb/toolbar/mode_segment.png[/icon][br]
Now let's finish the [size=200][size=150][color=#9900ff]PRINTED NOTES[/color][/size][/size]. Don't log off GeoGebra yet, though.