Points, Lines, and Segments
What's the smallest thing you can think of? A grain of sand? An atom? Rhode Island?
Actually, a point is even smaller. Even though it's one of the most fundamental objects in geometry, the point is just barely there. It's the tiniest object imaginable. It has no size, no mass, no nothing. All it has is a location.
For example, the center of this circle is a point. We typically draw points with little dots, but a point itself is even smaller than a dot (and smaller than Rhode Island…probably).
Even though some points are known for being breathtaking national parks or just being pleasant, mathematicians usually name their points a single, capital letter. How creative. So in the above picture, we'd simply call the point P. Hi, P.
Imagine now that you're standing on point A, and your friend on point B, and you're both tugging on a rope (don't ask us what's so exciting about this rope—you're the one tugging on it!). The rope will end up going straight between the two of you every time you repeat this process.
You and your friend have ingeniously shown that connecting two points gives a line segment. That's because there's only one way to join two points without curves or corners. Give yourselves a round of applause.
If you can't stop yourself and extend the segment in both directions forever, off the page, off the planet, and to the ends of the universe, we'll get a line. Actually drawing a line would use up a lot of ink, so we'll use little arrows to show that it goes on forever instead.
Finally, if you extend the segment in only one direction, you end up with a ray, which looks like this:
Think "ray of sunshine," not "stingray."
The thing to remember in all this is that any two points make exactly one line segment and exactly one line. They give two rays, depending on which endpoint you extend and which you don't. We don't really have a preference, so we'll leave that up to you.
That means we can write lines using names for the points involved, like so.
Sometimes we instead use single, lowercase letters to name lines, so we can call this guy XY or YX or just l. Don't call him Twiggy, though. He hates that.
Even though lines contain several points (in fact, they contain infinitely many), we get tired after drawing about ten or so. Instead, we'll use 2 points to name a line. That's it.
So this line can go by RS or ST or TS, but NOT RST. Putting in extra points is redundant, and if there's one thing that mathematicians could get rid of in this world, it's redundancy. Either that or the absurdly long lines at Disneyland.
Here's a video on graphing points in a grid: