** 2.3 Distance and Ruler Axioms
***Printout*

*Whereas
at the outset geometry is reported to have concerned herself with measurement
of muddy land, she now handles celestial as well as terrestrial problems: she
has extended her domain to the furthest bounds of space.
—W. B. Frankland, *The Story of

** **

For most common day-to-day measurements
of length, we use rulers, meter sticks, or tape measures. The distance and
ruler postulates formalize our basic assumptions of these items into a general
geometric axiomatic system. The SMSG Ruler Postulate defines a correspondence
between the points on a line (markings on a meter stick)
and the real numbers (units
of measurement) in such a manner that the absolute value of the difference
between the real numbers is equal to the distance (measurement of the length of
an object by a
meter stick matches our usual Euclidean distance). The Ruler
Placement Postulate basically says that it does not matter how we place a meter
stick to measure the distance between two points; that is, the origin (end of
the meter stick) does not need to be at one of the two given points.

** Postulate 2.**
(

** Postulate 3.**
(

i. To every point of the line there corresponds exactly one real number.

ii. To every real number there corresponds exactly one point of the line.

iii. The distance between two distinct points is the absolute value of the difference of the corresponding real numbers.

** Postulate 4.**
(

By Proposition 2.1 and the accompanying exercises, the Euclidean plane, Taxicab plane, Max-distance plane, Missing Strip plane, Poincaré Half-plane, Modified Riemann Sphere, and discrete planes all satisfy the Distance Postulate. Tools for working with rulers in Geometer's Sketchpad are available in the Appendix B Prepared Geometer's Sketchpad and GeoGebra Sketches.

** Definition.**
A

Note the first and second conditions of the Ruler Postulate imply that

** Definition.** A function

** Proposition 2.4. The Euclidean Plane satisfies the Ruler Postulate.** Before we begin
the proof, we do some scratch work to find the correct form for the rulers for
the lines. We need a relationship between the distance and a ruler, so we begin
with the distance function. First, consider a vertical line

This motivates the definition for the standard ruler of a
vertical line *l _{a}* to be

which motivates the definition for the standard ruler of a
nonvertical line *l _{m,b}* to
be .

For an example, consider the simplest nonvertical line,

In the "real-world" sense, the standard ruler (coordinate system) is the placement of a meter stick such that the zero end is at the

*Proof.* Let *l*
be a line in the Euclidean Plane. Then *l * is either a vertical line or a nonvertical
line.

Case 1. Assume *l* = *l _{a}*
a vertical line. Define by

Case 2. Assume *l* =
*l _{m,b}* is a nonvertical
line. Define by .
We first show

.

Hence *f* is onto. Finally,
we show the distance condition. Let *P*(*x*_{1}*, y*_{1}) and *Q*(*x*_{2}*, y*_{2}) be points on line
*l _{m,b}*.

Therefore, by Cases 1 and 2, an arbitrary line in the Euclidean plane has a ruler (coordinate system).//

As was discussed in Chapter 1,
axioms need not be independent, which is the case with the Ruler Placement
Postulate.

*Theorem
2.5. The Ruler Placement Postulate is not independent of the other axioms.
*

Assume is a ruler.

Set .

Define by

Show

Show

*Definitions.*

A point *B is between
points A and C*, denoted *A-B-C*,
if {*A, B, C*} is a collinear
set of three distinct points and *AB* +*
BC* = *AC*. (Here, *AB* represents the distance from *A* to *B*,
i.e. *d*(*A, B*) = *AB.*)

A *line segment* is the union of two
distinct points and all points between those two points, denoted either as
segment *AB* or .
The points *A* and *B* are called the *endpoints* of segment *AB*.

Two segments are *congruent*
if they have the same measure, denoted .

A point *M* is the *midpoint of
segment AB* if *AM* = *MB* and {*A, M, B*} is
collinear.

A
*bisector* of a segment is a line that contains the midpoint of
the segment.

A *ray AB* is the union of the
segment *AB* and the set of all points *C* such that *B* is between *A* and *C, *denoted either as ray *AB* or .
The point *A* is called the *endpoint*
of the ray *AB*. *(Note ray AB and ray BA are different rays.)
*A

** Exercise
2.17.** Find the axioms from a high school geometry book that
correspond to SMSG Postulates 2, 3, and 4.

Prove that is an equivalence relation for the set of all segments.

*
Don't
measure yourself by what you have accomplished, but by what you should have
accomplished with your ability.
—*

© Copyright 2005, 2006 - Timothy Peil |