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

2. 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.

3. 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)

4. 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

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

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.

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

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

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

11. 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

12. 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

13. 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)

14. P(x2,y2)
d = ? d
90°
Q(x1,y1)

16. 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

17. 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

19. M

20. 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'

21. 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

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

23. 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

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

25. 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

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

27. 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)

28. 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