∑Richard Hummel Math

Points, Lines & Planes

The undefined building blocks of geometry — zero-dimensional locations, one-dimensional paths, and two-dimensional flat surfaces.

The basic objects geometry builds from
Point
A
A point marks a location. The dot is only a drawing of it.
Line
AB
A line continues forever in both directions. Two distinct points determine one line.
Plane
PQR
A plane is flat and extends without edge; three non-collinear points determine one.
Segment
AB
A segment is the finite part of a line between endpoints.
Ray
C
A ray starts at one endpoint and continues forever in one direction.
Relations
intersectparallel
Coplanar lines either intersect once or stay parallel. In space, skew lines can miss too.
Definition

Geometry starts with three ideas so basic they can't really be defined — only described:

  • A point is a location in space. It has no size, no width, no height. It's just a place. We draw it as a tiny dot and label it with a capital letter, like AA or BB.
  • A line is a perfectly straight path that goes on forever in both directions. It has no thickness — only length. Two points determine exactly one line.
  • A plane is a perfectly flat, two-dimensional surface that extends forever in all directions. Think of an infinite table top. Three points that aren't all on the same line determine exactly one plane.

These three form the foundation of all of geometry. Everything else — triangles, circles, angles, shapes — is built from them.

Definition

Some important relationships:

  • Two lines in the same plane either intersect at exactly one point or are parallel (they never meet).
  • A line segment is the part of a line between two points — it has a definite length.
  • A ray starts at a point and goes on forever in one direction only, like a laser beam.
  • When two lines intersect, they form angles at the crossing point.
Points on a number line

A number line is a line where every point corresponds to a real number. The point labelled 00 is the origin. The point labelled 33 is 3 units to the right. These are all just points sitting on a line.

The distance between the points A=2A = 2 and B=7B = 7 on a number line is âˆĢ7−2âˆĢ=5|7 - 2| = 5 units.

Try it

How many lines can pass through a single point? How many lines can pass through two distinct points?

Solution

Infinitely many lines can pass through a single point — you can draw a line in any direction through it.

Exactly one line passes through two distinct points. The two points "pin down" the line completely.

Related concepts

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