2. called its coordinate.
Each point on a flat, two-dimensional surface, called a
coordinate plane or xy-plane, is associated with an ordered pair
of numbers called coordinates of the point.
Ordered pairs are denoted by (a, b), where the real number
a is the x-coordinate or abscissa and the real number b is
the y-coordinate or ordinate.
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Cartesian Coordinate Systems
The coordinates of a point are determined by the point’s
position relative to a horizontal coordinate axis called the
x-axis and a vertical coordinate axis called the y-axis. The axes
intersect at the point (0, 0), called the origin.
In Figure 2.1, the axes are
labeled such that positive
numbers appear to the right
of the origin on the x-axis and
above the origin on the y-axis.
Figure 2.1
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3. 5
Cartesian Coordinate Systems
The four regions formed by the axes are called quadrants and
are numbered counterclockwise. This two-dimensional
coordinate system is referred to as a Cartesian coordinate
system in honor of René Descartes.
To plot a point P(a, b) means
to draw a dot at its location in
the coordinate plane.
In Figure 2.2, we have plotted
the points (4, 3), (–3, 1), (–2, –3),
(3, –2), (0, 1), (1, 3), and (3,1).
Figure 2.2
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Cartesian Coordinate Systems
The order in which the coordinates of an ordered pair are
listed is important. Figure 2.2 shows that (1, 3) and (3, 1)
do not denote the same point.
Data often are displayed in visual form as a set of points called
a scatter plot.
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4. 7
Cartesian Coordinate Systems
For instance, the scatter plot in Figure 2.3 shows the growth in
text messaging during the years 2005 to 2011, with each point
representing the data from a 12-month period ending in June.
Figure 2.3
Source: CTIA.
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Cartesian Coordinate Systems
In some instances, it is important to know when two ordered
pairs are equal.
Definition of the Equality of Ordered Pairs
The ordered pairs (a, b) and (c, d) are equal if and only if
a = c and b = d.
Example
If (3, y) = (x, –2), then x = 3 and y = –2.
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5. Distance and Midpoint Formulas
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Distance and Midpoint Formulas
The Cartesian coordinate system makes it possible to combine
the concepts of algebra and geometry into a branch of
mathematics called analytic geometry.
The distance between two points on a horizontal line is the
absolute value of the difference between the x-coordinates of
the two points.
The distance between two points on a vertical line is the
absolute value of the difference between the y-coordinates of
the two points.
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Distance and Midpoint Formulas
For example, as shown in Figure 2.4, the distance d between the
points with coordinates (1, 2) and (1, –3) is
d = | 2 – (–3) | = 5.
6. Figure 2.4
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Distance and Midpoint Formulas
If two points are not on a horizontal or vertical line, then a
distance formula for the distance between the two points can be
developed as follows.
The distance between the
points P1(x1, y1) and P2(x2, y2)
in Figure 2.5 is the length of the
hypotenuse of a right triangle
whose sides are horizontal and
vertical line segments that
measure | x2 – x1 | and | y2 – y1 |,
respectively.
Figure 2.5
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Distance and Midpoint Formulas
Applying the Pythagorean Theorem to this triangle produces
7. Use the square root procedure.
Because d is nonnegative,
the negative root is not listed.
| x2 – x1 |2 = (x2 – x1)2 and
| y2 – y1 |2 = (y2 – y1)2
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Distance and Midpoint Formulas
Thus we have established the following theorem.
Distance Formula
The distance d(P1, P2) between the points P1(x1, y1) and
P2(x2, y2) is
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Distance and Midpoint Formulas
Example
The distance between P1(–3, 4) and P2(7, 2) is given by
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Distance and Midpoint Formulas
The midpoint M of a line segment is
the point on the line segment that is
equidistant from the endpoints
P1(x1, y1) and P2(x2, y2) of the segment.
See Figure 2.6.
Midpoint Formula
The midpoint M of the line segment from P1(x1, y1) to
P2(x2, y2) is given by
Figure 2.6
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Distance and Midpoint Formulas
Example
The midpoint of the line segment between P1(–2, 6) and P2(3,
9. 4) is given by
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Distance and Midpoint Formulas
The midpoint formula states that the x-coordinate of the
midpoint of a line segment is the average of the
x-coordinates of the endpoints of the line segment and that
the y-coordinate of the midpoint of a line segment is the
average of the y-coordinates of the endpoints of the line
segment.
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Example 1 – Find the Midpoint and Length of a Line Segment
Find the midpoint and the length of the line segment connecting
the points whose coordinates are P1(– 4, 3) and P2(4, –2).
