

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






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. 










Functions
contents B




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