basic analytical geometry

DONE BY :SSM MASEKO201124386

ANALYTIC ANALYTIC GEOMETRYGEOMETRY BASIC BASIC CONCEPTS FOR CONCEPTS FOR GRADE 10-12GRADE 10-12

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.

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.

CARTESIAN COORDINATE CARTESIAN COORDINATE SYSTEMSYSTEM

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

CARTESIAN COORDINATE CARTESIAN COORDINATE SYSTEMSYSTEM

O x

y

DISTANCE BETWEEN TWO DISTANCE BETWEEN TWO POINTSPOINTS

MIDPOINT BETWEEN TWO MIDPOINT BETWEEN TWO POINTSPOINTS

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.

INCLINATION OF A LINEINCLINATION OF A LINE

O M

θx

y

L

O M

θx

y

L

SLOPE OF A LINESLOPE OF A LINE

• the tangent of the inclinationm = tan θ

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

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.

ANGLE BETWEEN TWO ANGLE BETWEEN TWO LINESLINES

ANGLE BETWEEN TWO ANGLE BETWEEN TWO LINESLINES

• 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

REFERENCE LISThttp://stochastix.wordpress.com/2009/07/28/analytical-geometry-with-pov-ray/

This work belongs to NancyFelipe And  Mustafa Demirdag and google images

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