4. 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
5. 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.
6. 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.
7. 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.
8. CARTESIAN PLANECARTESIAN PLANE
• x-axis (horizontal axis)
where the x values are
• y-axis (vertical axis)
where the y values are
• origin, symbolized by 0,
marks the value of 0 of
• coordinates are given in
the form (x,y) and is
used to represent
different points on the
13. 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.
15. SLOPE OF A LINESLOPE OF A LINE
•the tangent of the inclination
m = tan θ
16. 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
17. THEOREMS ON SLOPETHEOREMS ON SLOPE
•Two non-vertical lines are parallel
if, and only if, their slopes are
•Two slant lines are perpendicular
if, and only if, the slope of one is
the negative reciprocal of the
slope of the other.
19. ANGLE BETWEEN TWOANGLE BETWEEN TWO
• If θ is angle, measured counterclockwise, between two lines,
• where m2 is the slope of the terminal side and m1 is the slope
of the initial side