Analytic Geometry deals with the properties, behaviors and solutions to points, lines, curves, angles, surfaces and solids by means of algebraic methods in relation to a coordinate system. Hence, this branch of Mathematics will tackle all about geometric figures as plotted on the rectangular coordinate system, otherwise known as the Cartesian coordinate system.

1. ORDERED PAIRS OF REAL
NUMBERS
Prepared by:
Estelita A. Villa

2. ANALYTIC GEOMETRY
Analytic Geometry deals with the properties,
behaviors and solutions to points, lines, curves,
angles, surfaces and solids by means of algebraic
methods in relation to a coordinate system.
Hence, this branch of Mathematics will tackle all
about geometric figures as plotted on the rectangular
coordinate system, otherwise known as the Cartesian
coordinate system.
Their solutions and proofs shall likewise be
resolved using the concepts of Algebra.

3. THE REAL NUMBER LINE
A basic concept of analytic geometry is the
representation of all real numbers by points on a
directed line. The real numbers consist of the positive
numbers, negative numbers, and zero.

4. THE REAL NUMBER LINE
Another basic concept is that of a line and a plane. Two
points determine a line
and three non-collinear points determine a plane.

5. To establish the desired representation, select a point O of the line,
and call it the origin, to represent the number zero. Then, mark
points at distances 1, 2, 3, and so on, units to the right of the origin.
These points represent the numbers 1, 2, 3, and so on. In the same
way, locate the points to the left of the origin to represent the
numbers -1, -2, -3, and so on. Points are now assigned to the
positive integers, the negative integers and the integer zero.
Numbers whose values are between two consecutive integers have
their corresponding points between the points associated with those
integers. Thus, the number 2¼ corresponds to the point 2¼ units to
the right of the origin. And, in general, any positive number p is
represented by the point p units to the right of the origin, and a
negative number q is represented by the point q units to the left of
the origin. Assume that every real number corresponds to one point
on the line and, conversely, every point on the line corresponds to
one real number. This relation of the set of real numbers and the set
of points on a directed line is called one-to-one correspondence.

6. The directed line with its points corresponding to real numbers, is
called a real number line. The number corresponding to a point on
the line is called the coordinate of the point. Since the positive
numbers correspond to points in the chosen positive direction from
the origin and the negative numbers correspond to points in the
opposite or negative direction from the origin, then the coordinate of
point on a number line is said to be the directed distance from the
origin.

7. DIRECTED LINE SEGMENTS
A line segment is measured in a definite sense
from one endpoint to the other. Its direction is
indicated by an arrowhead. The distance between two
distinct points on a directed line segment is called
directed distance.
On the other hand, the undirected distance (∞,
read as infinity) is the length of the segment in which
no definite distance has been defined and that it is
taken as positive.
In addition, if the segment is read from left to
right, then the segment has a positive magnitude; if
right to left, it has a negative magnitude.
Thus, the positive or negative sign indicates the
direction from which the segment is taken.

8. DIRECTED DISTANCE
(a) positive directed distance
(b) negative directed distance

9. UNDIRECTED DISTANCE
(c) positive undirected
distance
(d) negative undirected
distance

10. FUNDAMENTAL PROPERTY OF
DIRECTED LINE SEGMENT
If A, B and C are three distinct points on a line
(collinear), the following relation holds as a fundamental
property of a directed line segment.

11. EXAMPLE
If points A, B and C are collinear, find the values of the
following:

12. CARTESIAN COORDINATE SYSTEM
Cartesian coordinate system (also known as
rectangular coordinate system) consists of two
perpendicular lines which intersect at the origin.
x-axis and y-axis are
called coordinate axe
coordinate plane

13. DEFINITIONS
Definition 1.1: A pair of numbers (x, y) in which the
order of occurrence of the numbers is distinguished is
an ordered pair of numbers. Two ordered pairs, (x, y)
and (x’, y’), are equal if and only if x = x’ and y = y’.
Definition 1.2: The x-coordinate, or abscissa, of a
point P is the directed distance from the y-axis to the
point. The y-coordinate, or ordinate, of a point P is the
directed distance from the x-axis to the point. The
above definitions tell us that in an ordered pair, say
(x,y), the first number (x) is called the abscissa and the
second number (y) is called the ordinate of the point.
Thus, given an ordered pair (2, 7), 2 is the abscissa

14. POSITION OF A POINT ON A
SURFACE
If a point lies on a given surface, two magnitude, or
coordinates are necessary to determine its position, each
coordinate being measured in a definite sense.
To locate a point defined by an ordered pair, say (x,y)
on the Cartesian coordinate system, the abscissa
determines the number of units the point departs from the
origin horizontally.
If the abscissa is positive, then the point departs to
the right; if it is negative, then it departs to the left.
Consequently, the ordinate of the point determine the
number of units the point departs from the origin
vertically. If the ordinate is positive, then the point departs
upward; if negative, it departs downward.

15. EXAMPLE
Using the rectangular coordinate system, plot the following points:
a. (1, 4)
b. (-3, 2)
c. (-5, -3)
d. (5, -4)
Solution: The graph
is shown in the right
figure.
.
.
. .
x
y
-x
-y
a (1, 4)
d (5, -4)
b (-3, 2)
c (-5, -3)

16. TRY THESE
A. Draw the figures defined by the following points on
separate coordinate paper.