

Rectangular
(Twodimensional, Cartesian) Coordinate System 
Coordinate axes, xaxis
and yaxis,
origin, quadrants

Points in the
Coordinate plane 
Midpoint of a line segment

The distance formula

Dividing
a line segment in a given ratio 
The
area
of a triangle 
The
coordinates of the centroid of a triangle 
Lines
parallel to the axes, horizontal and vertical lines






Coordinate axes, xaxis
and yaxis,
origin, quadrants

The Cartesian coordinate system is defined by two
axes at right angles to each other, forming a plane. 
The horizontal axis is
labeled x, and the vertical axis is labeled
y. 
The point of intersection,
where the axes meet, is called the origin labeled
O. 
Given each axis,
choose a unit length, and mark off each unit along the axis, forming
a grid. 
The position of each point in a plane is identified with an
ordered pair of real numbers, in the form (x,
y), called the
coordinates of the point. 
The
xcoordinate, called the
abscissa, equal to the distance of the point from the yaxis measured parallel
to the xaxis.

The
ycoordinate, called the
ordinate, is the distance of the point from the xaxis measured parallel to the
yaxis.

The
origin O
has coordinates (0,
0). 
The intersection of the two axes creates four quadrants
indicated by numerals I, II, III, and IV. 
The quadrants are labeled
counterclockwise starting from that in which both coordinates are
positive. 


Midpoint of a line segment

The
point on a line segment that is equidistant from its endpoints
is called the midpoint. 
The coordinates of the midpoint
M (x_{M},
y_{M})
of the line segment P_{1}P_{2
}
where, P_{1}(x_{1},
y_{1})
and P_{2}(x_{2},
y_{2})
are endpoints, 



Example:
Find the midpoint of the
line segment AB
where the endpoints, A(5,
3)
and B(1,
1). 


The distance formula

The
distance between two given points in a coordinate
(Cartesian) plane. 


Dividing a line segment in a given ratio

A
given line segment AB
in a Cartesian plane can be divided by a point P
in a fixed ratio, internally or externally. 
If
P
lies between endpoints then it divides AB
internally. If P
lies beyond the endpoints A
and B
it divides the segment AB
externally. 
The
ratio of the directed segments l
=
AP
:
BP 
is
negative in the case of the internal division since the segments
AP
and BP
have opposite sense, while in the external division, the ratio l
is positive. 


As
l
=
AP
:
BP 
and
shown triangles are similar, then 

which,
with l
negative, gives 

the coordinates of
the point P. 




Example:
The line segment, with
endpoints A(3,
5)
and B(6, 1),
is divided by a point P
internally in the ratio l
= AP
: BP
= 1 : 2. Find the coordinates
of the dividing point P. 
Solution: 













The
area
of a triangle 
The
rectangular coordinates of three points in a coordinate plane
describe a triangle. Using given coordinates we derive the
formula for the area of the triangle, as is shown in the diagram
below. 

The
area of the given triangle P_{1}P_{2}P_{3}
equals 
the
area of the trapezium P_{1}MNP_{3}_{
}minus
the 
sum
of the areas of the right triangles, P_{1}MP_{2}
and 
P_{2}NP_{3},
that is 
P_{D} =
1/2·[(y_{1 }
y_{2})
+ (y_{3 }
y_{2})]
· (x_{3 }
x_{1})
 

1/2·[(y_{1
} y_{2})·(x_{2
} x_{1})
+ (y_{3 }
y_{2})·(x_{3
} x_{2})] 
which
after simplifying and rearranging gives 
P_{D}=1/2·[x_{1}(y_{2
} y_{3})
+ x_{2}(y_{3 }
y_{1})
+ x_{3}(y_{1 }
y_{2})] 



Example:
Two vertices of a triangle
lie at points A(4,
0)
and B(5, 2)
while third vertex C
lies on the yaxis.
Find the coordinates of the point C
if the area of the triangle is 31 square units. 
Solution: 










The
coordinates of the centroid of a triangle 
The
point of coincidence of the medians of a triangle is called the centroid. 
The
median is a straight line joining one vertex of a triangle to
the midpoint of the opposite side and divides the triangle into
two equal areas. 
The
centroid cuts every median in the ratio 2
: 1 from a vertex to the
midpoint of the opposite side. 
The
coordinates of the centroid of a triangle given its three
points, P_{1}, P_{2} and
P_{3
} in a coordinate plane: 
The
centroid M(x,
y),
where 
x
= 1/3 · (x_{1 }+ x_{2}
+ x_{3}), y =
1/3 · (y_{1 }
+ y_{2}
+ y_{3}) 



Lines
parallel to the axes, horizontal and vertical lines

If
the y
value never changes, i.e., if it takes the same constant value
y =
c
a line is parallel to the
xaxis
and is called a horizontal line (or constant). 
If
the x
value never changes, i.e., if it takes the same constant value
x
=
c a line is parallel to the yaxis
and is called a vertical line. 










College
algebra contents A




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