Sullivan Algebra and
Trigonometry: Section 2.1
Rectangular Coordinates
Objectives
• Use the Distance Formula
• Use the Midpoint Formula
We define the rectangular or Cartesian coordinate
system as the plane formed by two number lines,
the x-axis and y-axis, intersecting at a right angle.
Points in the xy-plane are located and labeled
using ordered pairs (x,y), called the coordinates of
a point.
(4,5)
6
5
4
3
2
1
0
-6
-5
-4
-3
-2
-1
-1
-2
(-2,-4)
-3
-4
-5
-6
0
1
2
3
4
5
6
The four regions in the x-y plane are
known as quadrants, labeled as follows:
y
Quadrant II
Quadrant I
x < 0, y > 0
x > 0, y > 0
Quadrant III
x
Quadrant IV
x < 0, y < 0
x > 0, y < 0
Theorem: Distance Formula
The distance between two points
P1 = (x1 , y1 ) and P2 = (x 2 , y 2 ), denoted
by d ( P1, P2 ) is
d ( P1 , P2 ) =
( x 2  x1 )
2
 ( y 2  y1 )
2
Example: Find the distance between the
points (3,8) and (-1,2)
P1 = ( 3,8), P2 = (  1,2)
d ( P1, P2 ) =
( x2  x1)  ( y2  y1)
2
d ( P1 , P2 ) =
( 1  3)  (2  8)
2
d ( P1 , P2 ) =
d ( P1 , P2 ) =
2
(  4)
2
 (  6)
2
2
16  36 = 52 = 2 13
Theorem: Midpoint Formula
The midpoint (x, y) of the line
segment from P1 = ( x1 , y1 ) to
P2 = ( x2 , y2 ) is
x1  x 2 y1  y 2 

,

( x , y ) = 
2
2 
Find the midpoint of a line segment
from P1 = ( 3,8) to P2 = (  1,2) .
x1  x2 y1  y2 

,

( x , y ) = 
2
2 
8  2 10
3 (1) 2
=
=5
x=
= =1 y =
2
2
2
2
M = (1,5 )