Points, lines, and planes

From Wikiversity

Hello, and welcome to the first lesson of this geometry course!

[edit] Congruency

Two things are congruent if they are the same. Okay, onto next topic...

Wait a second; we didn't define that very well! That's okay. Congruency is a thing that is defined for everything that is used in geometry. For example, congruency can be defined for peanut butter jars

Two jars of peanut butter are congruent if they have jars of the same shape, are filled to the same height with peanut butter, and the peanut butter tastes the same for both jars.

You could do a similar thing for gumballs, but that wouldn't be relevant to geometry, unlike peanut butter jars. Every element of geometry has its own definition of congruence.

This is the symbol of congruence.

[edit] Points

This is the definition of a point

A point is something

Wait, hey, that doesn't make any sense! Lets try again.

A point is a point that is somewhere

Yes, a point is a point. Third time is the charm?

A point is an infinitesimally small point that is somewhere

Yes, we know. A point is a point! Where is this thing's definition?

Actually, points have a definition.

A point is an infinitesimally small location; something having position but no spatial extent. An example of this would be an intersection of two lines. Sometimes points are represented graphically with a dot, which of course is a gross over-representation of its infinite smallness.

Every point is represented with the dot, and the letter that labels it. All labels are uppercase for points. This is important because the opposite is true for lines.

[edit] Lines

Let's not attempt to define the line, for it is impossible. However, it has some properties. A line is infinitely long; it goes forever in both directions. A line is infinitely thin, and also infinitely straight. Lines also have a lot of points on them.

Actually, on the other hand, let's define the line, for it is possible. A straight line can be defined in a number of ways, mathematically - but this will depend on the coordinate system being used. However, no matter which coordinate system is chosen, a straight line can be defined by specifying two distinct points through which it passes.

On a cartesian coordinate system, a line is defined as a set of all points whose range (i.e. y-coordinates) is the solutions of a function involving one domain (i.e. set of x-coordinates) that can be simplified to a first-degree trinomial with two variables (x and y).

Similarly a plane can be defined by three distinct points in any coordinate system involving more than one dimension.