Analytic geometry introduced in the 1630s by Descartes and Fermat uses algebraic equations to describe geometric figures on a coordinate system. It connects algebra and geometry by plotting points using a coordinate system with real number coordinates. This allows geometric shapes to be represented by algebraic equations which can be graphed. Key concepts include the Cartesian plane, slope, distance and midpoint formulas, and relationships between lines such as parallel, perpendicular and angles between lines based on their slopes.

1. Analytic GeometryAnalytic Geometry
Basic ConceptsBasic Concepts

2. Analytic GeometryAnalytic Geometry
a branch of mathematics
which uses algebraic
equations to describe the size
and position of geometric
figures on a coordinate
system.

3. Analytic GeometryAnalytic Geometry
It was introduced in the 1630s, an
important mathematical development,
for it laid the foundations for
modern mathematics as well as aided
the development of calculus.
Rene Descartes (1596-1650) and
Pierre de Fermat (1601-1665),
French mathematicians,
independently developed the
foundations for analytic geometry.

4. Analytic GeometryAnalytic Geometry
the link between algebra and
geometry was made possible by the
development of a coordinate
system which allowed geometric
ideas, such as point and line, to be
described in algebraic terms like
real numbers and equations.
also known as Cartesian geometry
or coordinate geometry.

5. Analytic GeometryAnalytic Geometry
 the use of a coordinate system to
relate geometric points to real
numbers is the central idea of analytic
geometry.
 by defining each point with a unique
set of real numbers, geometric figures
such as lines, circles, and conics can be
described with algebraic equations.

6. Cartesian PlaneCartesian Plane
 The Cartesian plane, the basis of analytic
geometry, allows algebraic equations to be
graphically represented, in a process called
graphing.
 It is actually the graphical representation
of an algebraic equation, of any form --
graphs of polynomials, rational functions,
conic sections, hyperbolas, exponential and
logarithmic functions, trigonometric
functions, and even vectors.

7. Cartesian PlaneCartesian Plane
 x-axis (horizontal axis)
where the x values are
plotted along.
 y-axis (vertical axis)
where the y values are
plotted along.
 origin, symbolized by 0,
marks the value of 0 of
both axes
 coordinates are given in
the form (x,y) and is
used to represent
different points on the
plane.

8. Cartesian Coordinate SystemCartesian Coordinate System
y
5
4
3
(-, +) 2 (+, +)
1
x
-5 -4 -3 -2 -1 0 1 2 3 4 5
-1
-2
-3
(-, -) (+, -)
-4
-5
III
III IV

9. Cartesian CoordinateCartesian Coordinate
SystemSystem
O x
y

10. Distance between Two PointsDistance between Two Points

11. Midpoint between Two PointsMidpoint between Two Points

12. Inclination of a LineInclination of a Line
The smallest angle θ, greater
than or equal to 0°, that the line
makes with the positive direction
of the x-axis (0° ≤ θ < 180°)
Inclination of a horizontal line is
0.

13. Inclination of a LineInclination of a Line
O M
θ
x
y
L
O M
θ
x
y
L

14. Slope of a LineSlope of a Line
the tangent of the inclination
m = tan θ

15. Slope of a LineSlope of a Line
passing through two given points,
P1(x1, y1) and P2 (x2,y2) is equal to
the difference of the ordinates
divided by the differences of the
abscissas taken in the same order

16. Theorems on SlopeTheorems on Slope
Two non-vertical lines are parallel
if, and only if, their slopes are
equal.
Two slant lines are perpendicular
if, and only if, the slope of one is
the negative reciprocal of the
slope of the other.

17. Angle between Two LinesAngle between Two Lines

18. Angle between Two LinesAngle between Two Lines
 If θ is angle, measured counterclockwise,
between two lines, then
 where m2 is the slope of the terminal
side and m1 is the slope of the initial side