8.01 Points, lines, and planes

Undefined notions

Where does the idea of geometry come from?  How do we talk about "things" in that "big thing" we call space?

One point

Let's start with a very humble concept-the point.  A point is a precise location in space, like the one below:

There are many ways to represent a location.  In geometric diagrams, we can represent a point by a dot and a capital letter to say "I mean that location right there!"

Points are dimensionless. We can't measure a point because it has no length, width, or depth.  No matter how much we might "zoom in" to a point, it's location will not change.

Point

Point A 

A precise location in space with zero length, zero width, and zero depth.

Points can be represented in diagrams by a dot and capital letter.

Reflection questions

Question 1

What sorts of things could be represented by a point?

Question 2

What makes a point different than a dot on a piece of paper? 

Question 3

What are some other ways to describe points that you've seen before?

Two points

Now imagine connecting any two points in a straight path that extends forever.  What does the path look like?

We can use the applet below to help explore what it might look like.  Color the path between and beyond points A and B with the blue triangle.

We can call this path a line.  A line is a geometrical object with zero thickness and zero width that is infinitely long.

We can name lines by any two points that made them.  The line through points $A$A and $B$B would be called line $AB$AB.

A line has zero thickness, zero width, and is infinitely long.  Because it is infinitely long, it also contains an infinite number of points. Unlike a line drawn from a pencil or a computer, a geometric line never ends!

When points are on the same line, they are said to be collinear.  Since we can always draw a line through any two points, it is also true that any two points are collinear.

Reflection questions

Question 4

Is it possible to draw more than one line through the same two points? (Try it out with a piece of paper, a straight edge or ruler, and a pencil).

Question 5

Is it always possible to draw a line connecting three points? (Try drawing a counterexample if you can.)

Question 6

What are some other ways to describe lines that you've seen before?

Three points

Now imagine what it might look like if we connect our first two points to another point that's not on the line.  What should we call this space?

We can use the applet below to help us imagine the space created by three points.  Move points $A$A, $B$B, and $C$C around and see how the space that includes all $3$3 points changes.

The space that includes all $3$3 points above is called a plane.  A plane is a flat surface with zero thickness that is infinitely long and infinitely wide.

Plane $ABC$ABC

A point is said to have $0$0 dimensions, and a line is said to have $1$1 dimension, a plane is said to have $2$2 dimensions.  

Just as a line contains an infinite number of points, a plane also contains an infinite number of points.  There's an unlimited amount of locations to pick from!

And since we can always create a plane that includes any $3$3 locations, any $3$3 points will always be coplanar (on the same plane).

Reflection questions

Question 7

What sorts of things around you resemble a plane?

Question 8

What makes those things different from a real geometric plane?

Question 9

What do you imagine is the "thing" connecting 4 noncoplanar points (4 points that aren't on the same plane)?

Intersections

After exploring straight paths such as lines, line segments, and rays, and flat spaces such as planes, let's explore what happens when any of those paths or spaces overlap (or don't overlap) one another.

In mathematics, an intersection is defined as the set of all points two (or more) entities share in common.

We can think of an intersection as the location (or locations) where two things overlap or meet - like a street intersection!

Let's look at some special types of intersections.

Intersecting lines

Consider two lines represented on a piece of paper. If they cross one another at all, how many times will they cross?

Two straight paths cross exactly once (left); curved paths may cross multiple times (right)

The two lines will only be able to cross one another once (if at all). Any paths that cross more than once cannot both be lines on the piece of paper (one might be a curve instead). 

We can formalize our observations in a postulate, or rule, concerning the intersections of lines.

Non-intersecting lines

Is it possible that two lines never meet? 

Yes! Let's explore what that might look like.  Rotate the cube in the applet below to view the two different types of non-intersecting lines.

In the applet above, line $FG$FG and line $BC$BC never meet, and they are on the same plane (one of the faces of the cube).  We can refer to them as parallel lines.

Line $HD$HD and line $FG$FG also never meet.  However, it's not possible to draw a plane that contains both of them.  Line $HD$HD and line $FG$FG are skew lines.

Communicating parallel lines

It's difficult to tell from a diagram whether two lines really intersect since lines extend forever in both directions.  Look at the diagram below.  Do you think the lines are parallel?

Are these lines parallel?

Actually, they're not.  Their intersection is so far away from the view in the diagram that we aren't able to tell.

The same lines intersect far to the right.

In order to communicate with someone reading a diagram, we can use special markings to show that lines are parallel.  Generally, parallel lines will be drawn with the same number of arrows to note that they are perfectly parallel.

$\overline{AB}\parallel\overline{CD}$AB∥CD

We can also use the symbol $\parallel$∥ to say "is parallel to".  For example $\overline{AB}\parallel\overline{CD}$AB∥CD means "segment $AB$AB is parallel to segment $CD$CD".

Rays and line segments can also be intersecting, parallel, or skew in relation to one another; they would have the same relationship as the lines that contain them.

Practice questions

Question 10

Question 11

Question 12