Coordinate Geometry
“Now for something completely different!”
We know that there is a correspondence between number
and geometry.
Given a unit measure, we can determine distance, area, and
volume; but there is much more.
Remember:
There is a one to one correspondence between the point on
the number line between 0 and 1 and the real numbers
between 0 and 1.
In Euclidean geometry, if we then take a unit square, there
is a one to one correspondence between points in the square
and the ordered pair, (x,y) where x and y are real numbers
between 0 and 1.
This is the Cartesian coordinate system where the first
number is in the horizontal direction and the second is in
the vertical.
We can go further to the unit cube and beyond, but let us
just think about things in two dimensions.
In this context the simplest object is a point.
Point: Position only, no length, width, or thickness.
Represented by a dot or coordinates (a,b) in the plane (or
(a,b,c) in space.)
What is the next simplest concept in Euclidean geometry in
the plane?
It is a line which has length but no width or thinkness.
In geometric terms, we would think of two points
determining a line by being the set of points consisting of
the shortest distance between the points.
How would we express the same concept in terms of the
Cartesian coordinate system?
How about the points, (x,y) which are solutions to ax+by=c
where a, b, and c are real constant numbers with a and b
both not equal to 0.
Coordinate geometry then deals with the relationship
between the kind of Euclidean geometry that we have been
discussing and the standard (x,y) coordinate system. In
algebra then when ones talks about y = ax + b being a line;
it is a coordinate geometry question. Will show this in
more detail later.
The concept is to show geometric ideas using equations and
algebra.
Let us discuss y = ax. We have said both that they are lines
and that they are functions. Let us review these ideas.
Generally, lines are written as: ax + by = c with a and b not
both zero or a  b  0 .
2
2
Remember a function is a rule that assigns to a value in its
domain one and only one value in its range.
Are all straight lines in the plane functions?
Yes, if the line can be written as y=mx + c so in the general
equation if b is not equal to zero. Lines of the form x = c
are not functions, they assign either no value or an infinite
number of values to a given input.
We have been discussion lines and ax + by = c as the same
thing. Let us show why that is true.
Let us show that the solutions of y = ax + b are lines. To
do so, let us examine the solutions of y = mx.
( x0 , y0 )
Let (x,y) be a point on the graph of y = mx and let (s,t) be
any other point on the line connecting (x,y) to the origin,
(0,0). After dropping a perpendicular to the x axis from
both points, one gets two similar triangles, the green one
and the red one, because they are both right triangles with a
common angle. Therefore, the sides are in proportion so
the ratio of s/t = x/y or s = (x/y)t so the point (s,t) is on the
graph of y =mx. If there are other points on the graph that
are not on the line, it violates the fact that y = mx is a
function so y = mx is a line.
The next question is whether or not all lines have the form
y = ax + b. Using the same diagram as above, if we
examine a non-vertical line through the origin and any
point on the line (x,y), then any other point because of the
similar triangles satisfies t = (y/x) s.
Now we can ask some questions about lines in the plane.
Given two lines in the plane, they can either intersect or be
parallel. Note that perpendicular is a special case of
intersect and that by definition, lines are parallel to
themselves. If they intersect, they intersect at only one
point which is the solution of the system of two linear
equations in two unknown.
Let us see the characteristics of parallel lines.
Let us examine: x + y = 2 and 2x + 2y = 8.
Then the first equation can be written as: y = -x + 2
The second can be written as: y = -x + 4
so they have the same slope -1 and different y intersepts.
If we examine: x + y = 2 and 2x + 2y =4 which can both be
written as y = -x + 2, they yield the same line.
If two line are non-vertical and the product of their slopes
are -1, they are perpendicular.
Let us examine a simpler case while remembering the other
examples can be obtained by rotation and translation.
Let us examine the lines, y=x and y=-x.
Given the two lines, one creates two triangles, red and
green, that are congruent due to SSS. This means that the
red and green angles are equal; and since they add up to
180 degrees, each must be 90 degrees.
Finally, let us examine a concept where it would be very
difficult to understand geometrically but easy to understand
using coordinate geometry.
Center of Means of Three Points: Starting with three points,
A, B, and C, one bisects the line segment joining A and B
yielding a point G. Joining C to G to form a new line
segment, one finds the point H which is 1/3 of the way
from G to C. This is the Center of Means. We want to
show that this point is independent of how A, B, and C are
labeled.
where once again we use similar triangle to determine what
it means to both bisect and cut a line into three parts by
translating these concepts into distances in the x direction
and y direction and then by adding these distances to
determine the coordinates of the new point.
For example, the coordinates of the point halfway between
A and B is the x value of the point A plus ½ the distance
from A to B in the x direction. The same is true of the y
value. To find the center of means, you do the same thing
but with 1/3 of the distance starting at the new bisection
point.
The final answer shows that it is 1/3 of the x values and 1/3
of the y values independent of the order of the points.
I must also point out that the geometry software programs
like Cabri Geometry or Geometers Sketchpad either on a
computer or graphing calculator show coordinate geometry
in very interesting ways. One can examine geometry
figures that depending on certain parameters, for example,
conic sections like parabolas, ellipses, or hyperbolas, and
by moving the geometric shape, the program computes the
changing parameters.
This then is what coordinate geometry is all about, the
relationship between geometry and algebra. Algebraic
Geometry which is one of the most important areas in
advanced mathematical research started with coordinate
geometry so you are getting in on the ground floor of an
exciting discipline.