Ren

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"I think, therefore I am."
René Descartes
Founder of Analytic Geometry
Descartes lived during the early 17th century.
Descartes found a way to describe curves in an
arithmetic way. He developed a new method called
coordinate geometry, which was basic for the
future development of science.
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René Des cartes
Cartesian Co Ordinate System
Geometry and the Fly
One morning Descartes noticed a fly walking
across the ceiling of his bedroom. As he
watched the fly, Descartes began to think of how
the fly's path could be described without
actually tracing its path. His further
reflections about describing a path by means of
mathematics led to La Géometrie  and
Descartes's invention of coordinate geometry.
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Algebraic Equation in Geometry x
2y 1
Line
Geogebra
X2y 1 is a line in Geometry
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Turning point in the History of Mathematics
After 2000 years of Euclidean Geometry This
was the FIRST significant development by RENE
DESCARTES ( French) in 17th Century, Part of
the credit goes to Pierre Fermats (French)
pioneering work in analytic geometry. Sir Isaac
Newton (16401727) developed ten different
coordinate systems. It was Swiss
mathematician Jakob Bernoulli (16541705) who
first used a polar co-ordinate system for
calculus Newton and Leibnitz used the
polar coordinate system  
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• Two intersecting line determine a plane.

• Two intersecting Number lines determine
• a Co-ordinate Plane/system.
• or
• Cartesian Plane.
• or
• Rectangular Co-ordinate system.
• or
• Two Dimensional orthogonal
• Co-ordinate System or XY-Plane

GRID
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Use of Co-ordinate Geometry
Cell Address is (D,3) or D3
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Use of Co-ordinate Geometry
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Use of Co-ordinate Geometry
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Use of Co-ordinate Geometry
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Use of Co-ordinate Geometry
R A D A R MAP
R A D A R
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Use of Co-ordinate Geometry
Each Pixel uses x-y co-ordinates
Pixels in Digital Photos
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The screen you are looking at is a grid of
thousands of tiny dots called pixels that
together make up the image
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Practical Application
All computer programs written in
Java language, uses distance between two points.
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Lettering with Grid
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Frame of reference
Vertical





Half Plane
II
I
Above X-Axis
Horizontal
Left of Y-axis
origin

Right of Y-axis
Terms
Abscissa Ordinate Ordered Pair Quadrants Sign
Convention
IV
Below X-Axis
III
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Dimensions

• 1-D
• 2-D
• 3-D

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1-D
Distance Formula

• b-a
• or
• a-b

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2-D THE Distance formula
B
A
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2-D THE Distance formula
B
A
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From 3D to 2D
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Distance between two points.In general,
y
B(x2,y2)
AB2 (y2-y1)2 (x2-x1)2
y2
Hence, the formula for Length of AB or Distance
between A and B is
Length y2 y1
y1
A(x1,y1)
Length x2 x1
x
x1
x2
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Distance between two points.
X2 - x1 18-5
A ( 5 , 3 ) , B ( 18, 17 ) A ( x1 , y1
) B ( x2 , y2 )
y2 - y1 17-3
y
Using Pythagoras Theorem,
AB2 (18 - 5)2 (17 - 3)2
B(18,17)
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AB2 132 142
17 3 14 units
3
A(5,3)
18 5 13 units
x
5
18
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Distance formula is nothing but Pythagoras
Theorem
B
A
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The mid-point of two points.
Look at its horizontal length
y
B(18,17)
Mid-point of AB
y2
Look at its vertical length
Formula for mid-point is
y1
A(5,3)
x
x1
x2
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The mid-point of two points.
Look at its horizontal length
y
Mid-point of AB
B(18,17)
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11.5
Look at its vertical length
(11.5,
10)
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A(5,3)
(18,3)
10
x
5
18
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Find the distance between the points (-1,3) and
(2,-6)
y2y1 -6-3 -9 x2x12--(--1) 3
(-1, 3) (2, -6)
(x1 , y1 ) (x2 ,y2 )
AB 9.49 units (3 sig. fig)