Analytic Foundations of Geometry

1. Equations of Lines

Robert S. Wilson



A point is an ordered pair of real numbers.


The plane is the set of all ordered pairs of real numbers.


The distance between   (x1, y1)   and   (x2, y2)   is denoted by

|(x1, y1)(x2, y2)|

and is defined to be

Two points are vertical if they have the same   x-coordinates.


Two points are horizontal if they have the same   y-coordinates.


A rational expression with a zero denominator and a nonzero numerator has a value of infinity.1


The slope between   (x1, y1)   and   (x2, y2)   is

A line is the set of all points whose coordinates are solutions to a linear equation in two unknowns.2

The slope of a line is the slope between any two points on the line.3

A vertical line is one where all of the points are vertical

A horizontal line is one where all the points are horizontal.


The point-slope form of the equation of the line through   (x1, y1)   with slope   m   is

y - y1 = m(x - x1)

Two lines are parallel if their slopes are the same.


Two lines are perpendicular if their slopes are negative reciprocals of each other.4

Given a line and a point, the point where the line through the given point perpendicular to the given line intersects the given line is called the foot of the point in the line.