Analytic Foundations of Geometry

1. Equations of Lines

Robert S. Wilson

Definitions

 

A point is an ordered pair of real numbers.

 

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

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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

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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.

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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.

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