Points, Lines, and Planes
The Building Blocks
Everything in geometry starts with three fundamental ideas: points, lines, and planes. These are the simplest objects we work with, and every other geometric figure is built from them. Triangles, circles, cubes, pyramidsâ€"all of these are made from combinations of points, lines, and planes.
What makes these three concepts special is that they're actually undefined terms. We can't define them using simpler ideas because nothing is simpler. Instead, we describe what they are and how they behave. Think of them like the atoms of geometry: you can't break them down further, but you can combine them to build everything else.
Points
A point represents a location in space. That's it. It has no size, no width, no length, no dimensionâ€"just position. You can't actually see a point because it takes up zero space, but we represent points with dots so we can work with them on paper.
We name points using capital letters. You might have point A, point B, point C, and so on. When you see a dot labeled with a letter, that letter is the name of the point at that location.
Points are everywhere once you start looking. The corner of your desk is a point. The tip of a pencil approximates a point. The place where two walls meet the ceiling is a point. In coordinate geometry, every pair of numbers like \((3, 5)\) represents a specific point on a grid.
Here's something important: two different points can't be in the exact same location. If they were, they'd be the same point. Each point has its own unique position in space.
Lines
A line is a straight path that extends infinitely in both directions. Unlike a point, which has no dimensions at all, a line is one-dimensional: it has length (infinite length, actually) but no width or thickness.
When we draw a line, we put arrows on both ends to show that it keeps going forever. In real life, nothing is actually infinite, but in geometry, lines don't stop. They go on and on in both directions.
We have a few ways to name lines. The most common method is to pick any two points on the line and write their names with a line symbol (â†") on top. If points A and B are on a line, we can call it \(\overleftrightarrow{AB}\) (read as "line AB"). We could just as easily call it \(\overleftrightarrow{BA}\)â€"the order doesn't matter.
Sometimes lines are named with a single lowercase letter, like "line m" or "line ℓ." This is handy when you don't want to reference specific points or when multiple lines are involved in a diagram.
A few key facts about lines: First, any two points determine exactly one line. If you have points A and B, there's only one straight line that goes through both of them. Second, a line contains infinitely many points. Even though we only mark a few points on a line when we draw it, the line passes through an unlimited number of locations.
Planes
A plane is a flat surface that extends infinitely in all directions. While a line is one-dimensional and a point is zero-dimensional, a plane is two-dimensional: it has length and width but no thickness.
Think of a plane as an infinitely large, perfectly flat sheet of paper. Or picture the surface of a calm lake extending forever in all directions. The floor of your room approximates a plane, except it stops at the walls. A true geometric plane doesn't stop.
We usually represent planes by drawing a parallelogram or some other four-sided shape. Of course, the actual plane extends way beyond the edges of our drawing. In the diagram above, plane P contains three points (A, B, and C) and a line passing through some of them. Of course, the actual plane extends way beyond the edges of our drawing. We name planes using a single capital letter placed in a corner of the figure, like "plane P" or "plane M."
You can also name a plane using three points that lie in it (and aren't all on the same line). For example, if points A, B, and C are in a plane but don't all fall on the same line, you could call it "plane ABC."
Just like with lines and points, any three points that aren't all on the same line determine exactly one plane. Think about a three-legged stool: those three points of contact with the floor define a plane, and the stool won't wobble (unlike a four-legged table, which can rock if the legs aren't perfectly even).
Line Segments and Rays
Once you understand lines, two related concepts come up constantly: line segments and rays.
Line Segments
A line segment is the part of a line between two points, including those two points. Unlike a line, which goes on forever, a line segment has a definite beginning and end. The two endpoints define where the segment starts and stops.
We write a line segment using its two endpoints with a bar on top: \(\overline{AB}\) (read as "segment AB" or "line segment AB"). The segment includes point A, point B, and every point between them on the straight path from A to B. The segment includes point A, point B, and every point between them on the straight path from A to B.
You use line segments all the time without realizing it. When you draw a rectangle, the sides are line segments. When you measure the distance from your house to school on a map using a straight line, you're thinking about a line segment.
The length of a line segment is the distance between its endpoints. This is a number we can actually measure. If \(\overline{AB}\) has a length of 5 inches, we might write \(AB = 5\) (note: no bar on top when we're talking about length, just the letters).
Rays
A ray is like a line that has been cut in half. It has one endpoint and then extends infinitely in one direction.
Think of a ray as a beam of light from a flashlight: it starts at the bulb and goes on forever in one direction. We name a ray using two points: the endpoint first, then any other point on the ray. The notation is \(\overrightarrow{AB}\) (read as "ray AB"), which means the ray starts at A, passes through B, and continues forever beyond B. We name a ray using two points: the endpoint first, then any other point on the ray. The notation is \(\overrightarrow{AB}\) (read as "ray AB"), which means the ray starts at A, passes through B, and continues forever beyond B.
The order matters here. \(\overrightarrow{AB}\) and \(\overrightarrow{BA}\) are different rays. They have different endpoints and go in opposite directions.
Opposite Rays
Two rays are called opposite rays if they share the same endpoint and together form a straight line. If point B is between points A and C on a line, then \(\overrightarrow{BA}\) and \(\overrightarrow{BC}\) are opposite rays.
Opposite rays form what we call a straight angle, which measures \(180°\). In the diagram, ray BA and ray BC share endpoint B (shown in red) and extend in opposite directions along the same line. More on angles in the next lesson.
Naming Conventions
Let's review the notation, because keeping these symbols straight is crucial:
Point A: Just write the letter. No symbols needed.
\(\overleftrightarrow{AB}\) (line AB): Two points with arrows on both ends of the symbol. The line goes through A and B and extends forever in both directions.
\(\overline{AB}\) (segment AB): Two points with a bar (no arrows). The segment starts at A and ends at B.
\(\overrightarrow{AB}\) (ray AB): Two points with an arrow on one end only. The ray starts at A, goes through B, and continues forever beyond B.
Length AB or \(AB\): When we write just the letters with no symbol on top, we're talking about the distance between them, which is a number.
Getting these notations right matters because \(\overleftrightarrow{AB}\), \(\overline{AB}\), and \(\overrightarrow{AB}\) all mean different things. If a problem asks you to name a line segment and you write the wrong symbol, you're technically wrong even if you meant the right thing.
Relationships Between Points
Now that we have the basic vocabulary, we can talk about how points relate to each other and to lines and planes.
Collinear Points
Points are collinear if they all lie on the same line. If you can draw a single straight line through all the points, they're collinear.
For example, if points A, B, and C all lie on \(\overleftrightarrow{AB}\), then A, B, and C are collinear (shown in the top part of the diagram). But if point D is off to the side and not on that line (like point F in the bottom diagram), then those points are not all collinear.
You need at least three points to talk about collinearity, since any two points are automatically collinear (remember, two points determine exactly one line).
Coplanar Points
Points are coplanar if they all lie in the same plane. If you can fit all the points on a single flat surface, they're coplanar.
Any two or three points are always coplanar because you can always find a plane that contains them. In the diagram, points A, B, C, and D all lie in plane P, so they're coplanar. But point E (shown in red) is outside the plane, so all five points together are not coplanar. But once you have four or more points, they might not all fit in the same plane. Imagine three points on your desk and a fourth point on the ceiling: those four points are not coplanar.
Intersections
When two lines meet, they intersect at a point. That point is called the point of intersection. Two lines can intersect at most at one pointâ€"if they intersect at two points, they must be the same line.
A line and a plane can intersect at a point (the line pokes through the plane) or the line can lie entirely within the plane. Two planes can intersect along a line. Think about two walls in a roomâ€"they meet along a line where the corner is.
Putting It Together
These terms might seem abstract, but they're the foundation of everything else in geometry. Once you understand points, lines, planes, segments, and rays, you can move on to angles, triangles, circles, and all the other shapes you'll study.
The key is to think about these as idealized versions of things you see every day. A star in the night sky looks like a point. The edge of a ruler is like a line segment. The surface of a table is like a plane. Geometry takes these everyday ideas and makes them precise so we can reason about them mathematically.
As you work through more geometry, you'll use this vocabulary constantly. You'll talk about whether points are collinear, whether lines intersect, whether figures lie in the same plane. You'll name line segments and rays and use the correct notation to communicate clearly. This foundational lesson gives you the language you need to describe the geometric world.
Make sure you can visualize each concept. When someone says "plane," you should picture an infinite flat surface. When you see \(\overline{AB}\), you should think "line segment from A to B." The more comfortable you become with this vocabulary, the easier everything else in geometry will be.
Certain diagrams created with Desmos Geometry.