

Rectangular
(Twodimensional, Cartesian) Coordinate System 
The
area
of a triangle 
The
coordinates of the centroid of a triangle 
Lines
parallel to the axes, horizontal and vertical lines


Polar Coordinate System

Polar
and Cartesian coordinates relations 
Conversion from polar
to rectangular coordinates 
Conversion from
rectangular to polar coordinates 
Polar coordinates of a point 





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. 



Polar coordinate system

The
polar coordinate system is a twodimensional coordinate system
in which each point P
on a plane is determined by the length of its position vector r
and the angle q
between it and the positive direction of the xaxis,
where 0 <
r
< + oo
and 0
<
q
< 2p. 


Polar
and Cartesian coordinates relations, 

Note,
since the inverse tangent function (arctan or tan^{}^{1})
returns values in the range p/2
< q <
p/2, then 
for points lying in the 2nd or 3rd quadrant 


and for points lying in the 4th quadrant 



Example:
Convert Cartesian coordinates
(1,
Ö3)
to polar coordinates. 
Solution: 

and since the point lies in the 3rd quadrant,
then 

















College
algebra contents A




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