Points, lines, and planes
Geometry builds everything from a few simple ideas. Points, lines, and planes are left undefined, and all other figures are described in terms of them.
Undefined terms
A point marks a location and has no size; it is named with a capital letter, such as A. A line is straight, extends forever in both directions, and has no thickness; it is named by two of its points, such as line AB. A plane is a flat surface extending forever, named by three points not on one line or by a single script letter. These are undefined terms: we describe them but do not define them using simpler ideas.
Euclid's approach
Around 300 BCE, the Greek mathematician Euclid wrote the Elements, which built geometry from a short list of definitions and postulates (statements accepted without proof). Every other result, called a theorem, was proved from these. The Elements was used as a textbook for over two thousand years and is the model for the proofs studied in this course.
Segments and rays
A line segment is part of a line with two endpoints, such as segment AB. A ray has one endpoint and extends forever in one direction; ray AB starts at A and passes through B. Opposite rays share an endpoint and point in opposite directions, together forming a line.
Collinear and coplanar
Points on the same line are collinear; points in the same plane are coplanar. Any two points are collinear, and any three points are coplanar, but four points may not be. Two distinct lines intersect in at most one point, and two distinct planes intersect in a line, like two walls meeting at a corner.
Measuring segments
The length of segment AB, written AB, is the distance between A and B. The segment addition postulate says that if B lies between A and C, then \(AB + BC = AC\). If AC = 15 and AB = 6, then BC = 9. Segments with equal lengths are congruent.
Midpoints and bisectors
The midpoint of a segment divides it into two congruent halves. A segment bisector is any line, ray, or segment through the midpoint. On a number line, the midpoint of points at -4 and 10 is \(\frac{-4 + 10}{2} = 3\).
Postulates in everyday terms
Some basic postulates: through any two points there is exactly one line; through any three non-collinear points there is exactly one plane; and if two points lie in a plane, the line through them lies in the plane. A three-legged stool never wobbles because its three feet determine a single plane, while a four-legged chair can rock.
Worked example
Points A, B, and C are collinear with B between A and C. If AB = 2x + 3, BC = 4x - 1, and AC = 32, then \(2x + 3 + 4x - 1 = 32\), so \(6x = 30\) and x = 5. That makes AB = 13 and BC = 19, which add to 32 as required.
Key takeaways
Points, lines, and planes are undefined terms from which geometry is built, following Euclid's Elements (about 300 BCE), where postulates are accepted and theorems are proved. Segments have two endpoints; rays have one. Collinear points share a line and coplanar points share a plane. The segment addition postulate says AB + BC = AC when B is between A and C. A midpoint splits a segment into congruent halves. Two points determine a line and three non-collinear points determine a plane.
Flashcard
A location with no size
Check your understanding
B lies between A and C with AB = 6 and AC = 15. What is BC?