Points, lines, and planes

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

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.

$\cong$

Points

Example of a point is period

This is the definition of a point

A point is something

Wait, hey, that doesn't make any sense! Let's 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. In other words, a point is a dimensionless object! An example of this would be an intersection of two lines. It has neither a length, nor a breadth, nor a height. That's why it is called dimensionless. 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.

Hands-On Examples of Points

Take a very small ball/sphere (grain of sand, small marble) and a pencil or other straight object that is thicker than the ball and demonstrate how the pencil can take an infinite number of orientations and still intersect ALL of the ball. This demonstrates that a point is dimensionless since any object can take any orientation (i.e. can have any set of values for it's dimensions) and still intersect the ENTIRE point. In this example, a pencil represents a line with only one dimension (length).

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 (so, it is also called a straight line). Lines also have a lot of points on them, in fact an infinite number of points, to be precise.

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.

If we don't go into mathematics, a line can also be defined in plain English as a collection of an infinite number of points, all placed side by side. Since a point has no dimensions, we would expect a line to have no dimensions, since it is a mere collection of points. However, when a number of points lie side by side, they do give rise to one dimension - that of length. Thus we see, line is a one-dimensional object. A line is denoted using small letters.

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).

So far, we have discussed only straight lines. However, lines can also be curved. Such lines are called curved lines.

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

Hands-On Examples of Lines

1. Since a straight line is defined by two points, you can prove that it takes exactly 2 points to make a line. --J.foster.davis 17:38, 20 February 2010 (UTC)
   1. On a sheet of paper, make a dot (representing a single point)
   2. Ask the students to draw a straight line through the dot.
   3. Ask the students to draw ANOTHER straight line. Since this is possible, you will now have 2 lines on the page. You can also repeat this step any number of times. This demonstrates that a single point cannot uniquely define/identify a single line.
   4. Now, on another sheet of paper, ask the students to place 2 small dots anywhere on the page.
   5. Ask the students to draw as many straight lines that intersect both dots as they can. Since you can only have 1 line, they will not be able to.
   6. Ask the students to draw 3 or more dots on a clean sheet of paper.
   7. Ask them to draw as many straight lines that intersect all 3 dots as they can. Since you cannot, this demonstrates that 3 or more dots cannot define a line. Since the case above using 2 dots is the only case that resulted in exatly one line, this demonstrates proves the definition and why that particular phrasing is used to define a line.
2. Demonstrate that an infinite number of points are contained within a line --J.foster.davis 17:38, 20 February 2010 (UTC)
   1. Draw a line segment and have each student place a dot on the line. Since dots (points) have no length, an unlimited number of dots can be drawn without any student touching any other student's dot.
   2. Repeat as desired using a line segment of any length. All students will still be able to place a dot.

Rays (Half-Lines)

A ray (or half-line) is a line that terminates at one of its points. It travels to infinity on one side of its two defining points, and terminates on the other.

Ray

In the picture above, points A and C define a line that terminates at point A. Therefore the ray is defined by points A and C.

Like a straight line, a ray is one-dimensional.