Points, Lines, and Planes
The fundamental building blocks of all geometric figures
Look around the room you are in right now. The corner where two walls meet the ceiling? That is a point. The edge where the wall meets the floor? That is a line. The wall itself? That is a plane. Before you even opened this lesson, you already had an intuitive understanding of these fundamental geometric concepts. Now we are going to give you the precise language and tools to work with them mathematically.
Here is something fascinating: unlike most of mathematics, geometry begins with ideas so basic that we cannot define them using simpler terms. A point, a line, and a plane are what mathematicians call undefined terms. We describe them, we understand them intuitively, but we do not define them in terms of other things. Everything else in geometry builds up from these three simple ideas.
Core Concepts
The Undefined Terms: Point, Line, and Plane
A point is a location in space. It has no size, no width, no length, no thickness. It is simply a position. We represent points with dots (though any dot you draw has actual size, while a true mathematical point does not) and name them with capital letters: point $A$, point $B$, point $C$.
Think of a point like the precise location where you are standing. You occupy space, but the exact spot on the ground directly beneath your feet is a point. GPS coordinates mark points. The tip of a needle approaches the idea of a point.
A line extends infinitely in two directions. It is perfectly straight, has no thickness, and goes on forever in both directions. We draw lines with arrows on both ends to show they never stop. A line can be named in two ways:
- Using any two points on the line: $\overleftrightarrow{AB}$ (read “line AB”)
- Using a single lowercase letter: line $m$
A laser beam shooting across a room gives you a sense of a line, except a mathematical line would extend infinitely in both directions. The horizon, if it were perfectly straight and went on forever, would be like a line.
A plane is a flat surface that extends infinitely in all directions. It has length and width but no thickness. Think of the surface of a perfectly calm, infinite lake, or a wall that stretches forever. We name planes in two ways:
- Using a capital letter: plane $P$
- Using three non-collinear points in the plane: plane $ABC$
Naming Conventions
Geometry has consistent rules for naming things, which helps everyone communicate clearly:
| Object | How to Name It | Example |
|---|---|---|
| Point | Capital letter | Point $A$ |
| Line | Two points with a double arrow, or lowercase letter | $\overleftrightarrow{AB}$ or line $m$ |
| Plane | Capital letter, or three non-collinear points | Plane $P$ or plane $ABC$ |
When you name a line using two points, either order works: $\overleftrightarrow{AB}$ and $\overleftrightarrow{BA}$ are the same line. The same is true for planes: plane $ABC$, plane $BCA$, and plane $CAB$ all refer to the same plane (as long as the three points are not collinear).
Collinear and Coplanar Points
Collinear points are points that lie on the same line. If you can draw a single straight line through all the points, they are collinear. Three or more points being collinear is a special situation. Any two points are automatically collinear because you can always draw exactly one line through any two points.
Coplanar points are points that lie on the same plane. If you can place a flat surface so that it contains all the points, they are coplanar. Any three points are always coplanar (though they might be collinear too). Four or more points being coplanar is something you need to check.
Think of it this way: if you have four thumbtacks and they all lie flat on a tabletop, they are coplanar. But if one thumbtack is floating in the air above the table, those four points are not coplanar.
Line Segments and Rays
While a line goes on forever in both directions, we often need to work with parts of lines.
A line segment is the part of a line between two points, including those endpoints. The segment from point $A$ to point $B$ is written as $\overline{AB}$ (notice the bar over the letters, not a double arrow). A line segment has a definite length that we can measure. The edge of your desk, a piece of string, the side of a triangle - these are all line segments.
A ray starts at one point (called the endpoint) and extends infinitely in one direction. We write the ray starting at point $A$ and passing through point $B$ as $\overrightarrow{AB}$. The first letter is always the endpoint.
A flashlight beam is a great example: it starts at the flashlight and shoots off into the distance. Sunlight reaching Earth behaves like a ray. The order of letters matters for rays: $\overrightarrow{AB}$ and $\overrightarrow{BA}$ are different rays pointing in opposite directions.
Opposite Rays
Opposite rays are two rays that share the same endpoint and together form a complete line. If point $B$ is between points $A$ and $C$ on a line, then $\overrightarrow{BA}$ and $\overrightarrow{BC}$ are opposite rays.
Imagine standing in a hallway that extends infinitely in both directions. You are at point $B$. One ray extends from you toward point $A$ (one end of the hallway), and the opposite ray extends from you toward point $C$ (the other end). Together, these opposite rays form the entire infinite line.
Intersections
When geometric figures share points, we say they intersect. The set of points they share is called the intersection.
Two lines can intersect in exactly one point (or not at all, if they are parallel). When two lines cross, they share exactly one point. Picture two roads crossing at an intersection - that single crossing point is where the lines meet.
A line and a plane can intersect in exactly one point (if the line passes through the plane at an angle) or in a line (if the line lies entirely within the plane), or not at all (if the line is parallel to the plane).
Two planes can intersect in exactly one line (or not at all, if they are parallel). Think of two walls meeting at a corner - the corner edge is a line where the two planes intersect. The floor and a wall meet along a line. Two pages of an open book meet along the binding, which is a line.
Notation and Terminology
| Term | Meaning | Example |
|---|---|---|
| Point | A location with no size | Point $A$ |
| Line | Extends infinitely in both directions | $\overleftrightarrow{AB}$ or line $m$ |
| Plane | A flat surface extending infinitely | Plane $P$ or plane $ABC$ |
| Collinear | Points on the same line | $A$, $B$, $C$ are collinear |
| Coplanar | Points on the same plane | Points in plane $P$ |
| Line segment | Part of a line with two endpoints | $\overline{AB}$ |
| Ray | Part of a line with one endpoint | $\overrightarrow{AB}$ |
| Opposite rays | Two rays sharing an endpoint, forming a line | $\overrightarrow{BA}$ and $\overrightarrow{BC}$ |
| Intersection | The set of points figures have in common | Lines $m$ and $n$ intersect at point $P$ |
Examples
Consider a triangular pyramid (like the kind you might see in Egypt). Identify examples of points, lines, and planes.
Solution:
Points: Each corner (vertex) of the pyramid represents a point. A pyramid with a triangular base has 4 vertices, so there are 4 points: the three corners of the base and the apex at the top.
Lines: Each edge of the pyramid lies on a line. The edges of the triangular base lie on three different lines. The three edges connecting the base to the apex lie on three more lines. In total, we can identify 6 lines (though the actual line extends beyond just the edges).
Planes: Each face of the pyramid lies in a plane. The triangular base is in one plane. Each of the three triangular sides is in its own plane. That gives us 4 planes.
Note that any two points on the pyramid determine a line, and any three non-collinear points determine a plane, so there are actually infinitely many lines and planes associated with the pyramid. We are just identifying the most obvious ones.
A line passes through points $P$, $Q$, and $R$ (with $Q$ between $P$ and $R$). List all the different ways to name this line using these points.
Solution:
Any two points on a line can be used to name it, and either order works. Using points $P$, $Q$, and $R$, we can name the line as:
- $\overleftrightarrow{PQ}$ or $\overleftrightarrow{QP}$
- $\overleftrightarrow{PR}$ or $\overleftrightarrow{RP}$
- $\overleftrightarrow{QR}$ or $\overleftrightarrow{RQ}$
That gives us 6 different names for the same line.
If we also use a lowercase letter, we could call it something like line $\ell$, giving us a 7th name. All of these refer to the exact same line.
Points $A$, $B$, $C$, and $D$ are positioned as follows: $A$, $B$, and $C$ lie on line $m$. Point $D$ is not on line $m$. Determine which sets of points are collinear: a) $A$, $B$, $C$ b) $A$, $B$, $D$ c) $B$, $C$, $D$ d) $A$, $C$, $D$
Solution:
a) $A$, $B$, $C$ are collinear. We are told they all lie on line $m$, so by definition they are collinear.
b) $A$, $B$, $D$ are not collinear. Points $A$ and $B$ lie on line $m$, but $D$ does not. Therefore, no single line contains all three points.
c) $B$, $C$, $D$ are not collinear. Points $B$ and $C$ lie on line $m$, but $D$ does not. Therefore, no single line contains all three points.
d) $A$, $C$, $D$ are not collinear. Points $A$ and $C$ lie on line $m$, but $D$ does not. Therefore, no single line contains all three points.
Key insight: Once you know three points lie on a specific line, any point not on that line cannot be collinear with any two of those three points.
Two walls of a room meet at a corner. What is the intersection of the two walls (considered as planes)?
Solution:
The intersection of two non-parallel planes is always a line.
In this case, the intersection is the vertical edge where the two walls meet - the corner of the room. This edge is a line segment (with endpoints at the floor and ceiling), but the planes of the walls extend infinitely, so their true intersection is an infinite line.
If you imagine the walls extending forever in all directions (through the floor, ceiling, and beyond), they would still meet along that same vertical line, extending infinitely upward and downward.
Real-world verification: Run your finger along the corner where two walls meet. You are tracing the intersection of those two planes.
Postulate (fundamental assumption): Through any two distinct points, there is exactly one line.
Using this postulate, answer the following: a) How many lines pass through two given points $A$ and $B$? b) Points $X$, $Y$, and $Z$ are non-collinear. How many distinct lines can be drawn using pairs of these three points? c) If four points are positioned so that no three are collinear, how many distinct lines can be drawn through pairs of these points?
Solution:
a) By the postulate, there is exactly one line through points $A$ and $B$. Not zero, not two - exactly one. This line can be named $\overleftrightarrow{AB}$.
b) With three non-collinear points $X$, $Y$, $Z$, we can form three pairs:
- Points $X$ and $Y$ determine line $\overleftrightarrow{XY}$
- Points $X$ and $Z$ determine line $\overleftrightarrow{XZ}$
- Points $Y$ and $Z$ determine line $\overleftrightarrow{YZ}$
Since the points are non-collinear, these are three distinct lines. (If the points were collinear, all three pairs would determine the same single line.)
c) With four points (no three collinear), we can form pairs using combinations. The number of ways to choose 2 points from 4 is: $$\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6$$
So we can draw 6 distinct lines. Label the points $A$, $B$, $C$, $D$. The six lines are: $\overleftrightarrow{AB}$, $\overleftrightarrow{AC}$, $\overleftrightarrow{AD}$, $\overleftrightarrow{BC}$, $\overleftrightarrow{BD}$, $\overleftrightarrow{CD}$
The condition “no three collinear” ensures that each pair of points gives us a truly different line.
Key Properties and Rules
Fundamental Postulates
These are statements we accept as true without proof. They form the foundation for all geometric reasoning:
Postulate 1: Two Points Determine a Line Through any two distinct points, there exists exactly one line.
Postulate 2: Three Non-Collinear Points Determine a Plane Through any three points that are not on the same line, there exists exactly one plane.
Postulate 3: Points on a Line in a Plane If two points lie in a plane, then the entire line containing those two points lies in that plane.
Postulate 4: Intersection of Two Planes If two distinct planes intersect, then their intersection is exactly one line.
Key Properties
About Points:
- Points have no dimension (no length, width, or height)
- Points are named with capital letters
- Any two points determine exactly one line
- Any three non-collinear points determine exactly one plane
About Lines:
- Lines have one dimension (length, but no width)
- Lines extend infinitely in both directions
- Two distinct lines intersect in at most one point
- Lines in the same plane either intersect in exactly one point or are parallel
About Planes:
- Planes have two dimensions (length and width, but no thickness)
- Planes extend infinitely in all directions
- Two distinct planes either intersect in exactly one line or are parallel
About Segments and Rays:
- A segment $\overline{AB}$ has a measurable length
- A ray $\overrightarrow{AB}$ has an endpoint at $A$ and extends through $B$ infinitely
- Two rays with the same endpoint that form a line are opposite rays
Real-World Applications
Architecture and Construction
Architects and builders think in terms of points, lines, and planes constantly. The corner where three walls meet is a point. The edge where a wall meets the ceiling is a line segment (part of a line). Each wall, floor, and ceiling is a plane.
Understanding these relationships is essential for:
- Ensuring walls are truly flat (planar)
- Making sure edges are straight (collinear points along an edge)
- Verifying corners are proper intersections
When a contractor uses a laser level, they are essentially creating a plane of light to ensure a surface is flat.
Navigation and Mapping
GPS systems locate your position as a point on Earth’s surface. Flight paths between two cities follow lines (or more precisely, great circles on a sphere). Maps represent the curved Earth as flat planes (with some distortion).
When navigators plot a course between two points, they rely on the postulate that two points determine exactly one line (the shortest path). Surveyors use the intersection of sight lines to pinpoint locations.
Computer Graphics
Every 3D model you see in video games, movies, or design software is built from points, lines, and planes:
- Vertices (points): The corners of every shape
- Edges (line segments): The connections between vertices
- Faces (portions of planes): The surfaces of objects
A simple cube in a 3D program has 8 vertices (points), 12 edges (line segments), and 6 faces (plane sections). Complex models in animated films can have millions of vertices, all organized using the same fundamental concepts.
Manufacturing and Engineering
Precision manufacturing relies on geometric fundamentals:
- Laser alignment uses the principle that light travels in straight lines
- Quality control checks that surfaces are planar (flat)
- Assembly requires understanding how parts intersect
When machinists verify that a surface is flat, they are checking that all points on the surface are coplanar. When they check that an edge is straight, they are verifying that points along the edge are collinear.
Art and Design
Artists use these concepts for perspective drawing:
- Vanishing points are where parallel lines appear to meet
- Horizon lines help create realistic depth
- Understanding planes helps represent 3D objects on 2D surfaces
Interior designers plan room layouts by thinking of floors as planes, walls as intersecting planes, and furniture placement using points and lines.
Self-Test Problems
Problem 1: Name all the undefined terms in geometry, and give a real-world example of each.
Show Answer
The three undefined terms in geometry are:
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Point - A location with no size. Example: The tip of a pin, a GPS coordinate, or the exact spot where two streets cross on a map.
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Line - Extends infinitely in both directions with no thickness. Example: A laser beam (imagined to extend forever), the path of light from a distant star.
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Plane - A flat surface extending infinitely in all directions. Example: The surface of a calm lake (imagined to extend forever), a wall (imagined to extend infinitely).
Problem 2: Points $W$, $X$, $Y$, and $Z$ lie in plane $M$. Points $W$, $X$, and $Y$ are collinear, but point $Z$ is not on line $\overleftrightarrow{WX}$. Are all four points coplanar? Explain.
Show Answer
Yes, all four points are coplanar.
We are told that all four points ($W$, $X$, $Y$, and $Z$) lie in plane $M$. By definition, points that lie in the same plane are coplanar.
The fact that $W$, $X$, and $Y$ are collinear (all on the same line) does not change this. Being collinear is a special case of being coplanar - if points are on the same line, they are certainly on the same plane (in fact, infinitely many planes contain any given line).
The key information is that all four points are in plane $M$, so they are coplanar.
Problem 3: How many lines can be drawn through 5 points if no three of the points are collinear?
Show Answer
We need to count how many pairs of points can be formed from 5 points, since each pair determines exactly one line.
The number of ways to choose 2 points from 5 is: $$\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10$$
So 10 distinct lines can be drawn.
The condition “no three collinear” is important. If three or more points were on the same line, different pairs would give the same line, reducing the total count.
Problem 4: Identify each of the following as a point, line, line segment, ray, or plane: a) The path of a shooting star across the sky b) The tip of a pencil c) The surface of a frozen pond d) The edge of a ruler e) A beam of light from a flashlight
Show Answer
a) Line segment - A shooting star has a visible beginning and end point as it streaks across the sky.
b) Point - The tip of a pencil represents a single location (though technically it has some size, it represents the concept of a point).
c) Plane - The surface of a frozen pond is flat and extends in all directions (in the mathematical ideal, infinitely). A portion of a plane, more precisely.
d) Line segment - A ruler has a definite start and end point, making its edge a segment rather than a full line.
e) Ray - The beam starts at the flashlight (the endpoint) and extends outward in one direction.
Problem 5: Two distinct planes intersect. What is true about their intersection? Can two planes intersect in exactly one point? Explain.
Show Answer
When two distinct planes intersect, their intersection is exactly one line.
No, two planes cannot intersect in exactly one point.
Here is why: If two planes share one point, then by the properties of planes, they must share an entire line through that point. Consider any line in one plane that passes through the shared point. If that line had only one point in common with the other plane, the rest of the line would be on only one side of the second plane. But a plane is infinite and flat, so any line passing through a point in the plane must lie entirely in the plane or cross it. If the two planes are different (not the same plane), the intersection must be exactly one line.
Think of it physically: two flat sheets of paper (extending infinitely) that touch at one point would have to touch along an entire edge. They cannot just “kiss” at a single point.
Problem 6: Ray $\overrightarrow{BA}$ and ray $\overrightarrow{BC}$ are opposite rays. Point $D$ is on ray $\overrightarrow{BA}$ but is not point $B$. a) Are points $A$, $B$, and $C$ collinear? b) Are points $D$, $B$, and $C$ collinear? c) Is point $D$ on line $\overleftrightarrow{AC}$?
Show Answer
a) Yes, $A$, $B$, and $C$ are collinear.
Opposite rays share an endpoint and form a complete line. Since $\overrightarrow{BA}$ and $\overrightarrow{BC}$ are opposite rays, they form line $\overleftrightarrow{AC}$. Points $A$, $B$, and $C$ all lie on this line.
b) Yes, $D$, $B$, and $C$ are collinear.
Point $D$ is on ray $\overrightarrow{BA}$, which is part of line $\overleftrightarrow{AC}$. Point $B$ is on this line (it is the shared endpoint). Point $C$ is on ray $\overrightarrow{BC}$, which is also part of line $\overleftrightarrow{AC}$. All three points lie on the same line.
c) Yes, point $D$ is on line $\overleftrightarrow{AC}$.
Ray $\overrightarrow{BA}$ is part of line $\overleftrightarrow{AC}$ (since opposite rays form a complete line). Since $D$ is on ray $\overrightarrow{BA}$, it is also on line $\overleftrightarrow{AC}$.
Problem 7: Draw and label a diagram showing:
- Plane $P$ containing points $A$, $B$, and $C$ (with $A$, $B$, $C$ non-collinear)
- Line $m$ passing through points $A$ and $B$
- Point $D$ not in plane $P$
Then answer: What is the intersection of line $m$ and plane $P$?
Show Answer
Diagram description: Draw a parallelogram shape to represent plane $P$ (we usually draw planes as parallelograms even though they extend infinitely). Place three non-collinear points labeled $A$, $B$, and $C$ inside this shape, forming a triangle. Draw a line through points $A$ and $B$, extending it with arrows on both ends, and label it $m$. Place point $D$ somewhere above or below the plane (floating in space outside the parallelogram shape).
The intersection of line $m$ and plane $P$ is line $m$ itself.
Since both points $A$ and $B$ are in plane $P$, and two points determine a line, the entire line $\overleftrightarrow{AB}$ (which is line $m$) lies in plane $P$. When a line lies entirely within a plane, the intersection of the line and the plane is the entire line.
This follows from Postulate 3: If two points lie in a plane, then the entire line containing those two points lies in that plane.
Summary
You have now learned the fundamental building blocks of all geometry:
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Points, lines, and planes are the undefined terms of geometry. Points are locations with no size, lines extend infinitely in both directions, and planes are flat surfaces extending infinitely in all directions.
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Naming conventions keep our communication clear: points get capital letters ($A$, $B$, $C$), lines get two points with a double arrow ($\overleftrightarrow{AB}$) or a lowercase letter ($m$), and planes get a capital letter ($P$) or three non-collinear points (plane $ABC$).
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Collinear points lie on the same line; coplanar points lie on the same plane. Any two points are collinear. Any three points are coplanar (but may also be collinear).
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Line segments ($\overline{AB}$) have two endpoints and measurable length. Rays ($\overrightarrow{AB}$) have one endpoint and extend infinitely in one direction.
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Opposite rays share an endpoint and form a complete line. Together, $\overrightarrow{BA}$ and $\overrightarrow{BC}$ form $\overleftrightarrow{AC}$.
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Intersections follow predictable patterns: two lines can intersect in at most one point, a line and a plane can intersect in a point or a line, and two planes intersect in exactly one line (if they intersect at all).
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Postulates are fundamental truths we accept: two points determine exactly one line, three non-collinear points determine exactly one plane, and if two points are in a plane, the line through them is in that plane.
These concepts might seem abstract, but they are the precise language for describing the physical world around you. Every corner, edge, and surface you see can be understood through points, lines, and planes. From here, we will build all of geometry: angles, shapes, measurements, and proofs - all resting on these simple foundations.