
Coordinate Geometry or
Analytic Geometry 


Plane Analytic Geometry 
Coordinate
plane, points, line segments and lines

The distance formula

The midpoint formula

Dividing
a line segment in a given ratio 
Area
of a triangle 
The
coordinates of the centroid of a triangle 




Coordinate
plane, points, line segments and lines

The distance formula

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

The midpoint formula

The
point on a line segment that is equidistant from its endpoints
is called the midpoint. 


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 lengths 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. 




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})] 



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}) 










Beginning
Algebra Contents E 



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