It was invented in the 17th century by Renè Descartes. The Cartesian Plane It was invented in the 17th century by Renè Descartes.

Cartesian plane: Is a plane consisting of a set of two lines intersecting each other at right angles. The horizontal line is the x-axis. The vertical line is the y-axis.

The origin is located at ordered pair (0, 0) The x-axis and y-axis intersect at the origin. The origin is located at ordered pair (0, 0)

The cartesian plane is divided into 4 quadrants. Quadrant one (QI): there are only positive coordinates Quadrant two (QII): negative x and positive y Quadrant three (QIII): negative x and negative y Quadrant two (QIV): positive x and negative y

B (-4, 3) A (2, 3) A: QI B: QII C: QIII D: QIV C (-3, -4) D (4, -6)

Graphing points in the cartesian plane The points are identified by coordinates: P (x, y) The x-coordinate represents a value on the horizontal axis. The y coordinate represents a value on the vertical axis.

Example: to graph P (6, -8) we move right 6 from the origin, then down 8

Example 2: to graph P (0, -9) we move right 0 from the origin, then down 9

Distance between two points The lenght of the segment between the two points is the distance between them How can we find it?

Let's do some examples d = |y2 - y1| The distance between C and D is 4 +4 The distance between C and D is 4 +3 The distance between A and B is 2 B(1,3) +2 +1 C(-4,1) A(1,1) -5 -4 -3 -2 -1 +1 +2 +3 +4 +5 -1 d = |y2 - y1| -2 -3 D(-4, -3) -4

d = |x2 - x1| The distance between C and D is 3 +4 The distance between C and D is 3 +3 d = |x2 - x1| +2 +1 C(-4,1) D(-1, 1) -5 -4 -3 -2 -1 +1 +2 +3 +4 +5 -1 -2 B(1,-2) The distance between A and B is 4 A(-3, -2) -3 -4

Can you find the general formula? +4 B(3,3) +3 The distance between A and B is 5 +2 +1 -5 -4 -3 -2 -1 +1 +2 +3 +4 +5 -1 C(-3,-1) A(0, -1) The distance between C and D is 5 -2 -3 -4 D(1, -4)

P(x2,y2) d = ? d 90° Q(x1,y1)

Can you find the general formula? The midpoint +4 Can you find the general formula? B(3,3) +3 +2 M(2, 2) +1 A(1, 1) -5 -4 -3 -2 -1 +1 +2 +3 +4 +5 -1 C(-4, -1) -2 N(-1, -2) -3 D(2,-3) -4

The midpoint is the point halfway between each of the points. P(x1,y1) y1 yM M(xM=?, yM=?) Q(x2,y2) y2 x2 xM x1

M

Symmetry in the cartesian plane Two points A and A' are symmetric with respect to a line if this line is the axis of the segment AA'

Symmetry with respect the x axis +4 A' is the symmetric of A with respect the x axis Can you find the general formula? +3 +2 A'(1,2) +1 B(-4,1) -5 -4 -3 -2 -1 +1 +2 +3 +4 +5 -1 B'(-4,-1) B' is the symmetric of B with respect the x axis -2 A(1,-2) -3 -4

Symmetry with respect the x axis P(x,y) P'(x,-y)

Symmetry with respect the y axis +4 Can you find the general formula? A' is the symmetric of A with respect the y axis +3 +2 A(-1,2) A'(1,2) +1 -5 -4 -3 -2 -1 +1 +2 +3 +4 +5 -1 B(4,-1) B'(-4,-1) B' is the symmetric of B with respect the y axis -2 -3 -4

Symmetry with respect the y axis P(x,y) P'(-x,y)

Symmetry with respect the origin +4 Can you find the general formula? B(4,3) +3 +2 A(-1,2) +1 -5 -4 -3 -2 -1 +1 +2 +3 +4 +5 -1 B' is the symmetric of B A'(1,-2) -2 A' is the symmetric of A with respect the origin -3 B'(-4,-3) -4

Symmetry with respect the origin P'(-x,-y) P(x,y)

Polygons in the cartesian plane +4 Draw the polygon of vertex A(-2,-3) B(3,-4) and C (4,0) +3 +2 +1 C(4,0) -5 -4 -3 -2 -1 +1 +2 +3 +4 +5 -1 -2 Convex polygon -3 A(-2,-3) -4 B(3,-4)

Polygons in the cartesian plane +4 Draw the polygon of vertex A(-3,1) B(1,2) C (-1,-2) and D(-1,1) +3 B(1,2) +2 +1 A(-3,1) Concave polygon D(-1,1) -5 -4 -3 -2 -1 +1 +2 +3 +4 +5 -1 -2 C(-1,-2) -3 -4