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Analytic GeometryAnalytic Geometry
Basic ConceptsBasic Concepts
Analytic GeometryAnalytic Geometry
a branch of mathematics
which uses algebraic
equations to describe the size
and position of geometric
figures on a coordinate
system.
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.
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.
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.
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.
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Analytic geometry basic 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
  • 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