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Rectangular
(Two-dimensional, Cartesian) Coordinate System |
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
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Dividing a line segment in a given ratio
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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. |
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As
l
=
AP
:
BP |
and
shown triangles are similar, then |
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which,
with l
negative, gives |
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the coordinates of
the point P. |
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Example:
The line segment, with
endpoints A(-3,
5)
and B(6, -1),
is divided by a point P
internally in the |
ratio
l
= AB
: BP
= 1 : 2. Find the coordinates
of the dividing point P. |
Solution: |
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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. |
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The
area of the given triangle P1P2P3
equals |
the
area of the trapezium P1MNP3
minus
the |
sum
of the areas of the right triangles, P1MP2
and |
P2NP3,
that is |
PD =
1/2·[(y1 -
y2)
+ (y3 -
y2)]
· (x3 -
x1)
- |
-
1/2·[(y1
- y2)·(x2
- x1)
+ (y3 -
y2)·(x3
- x2)] |
which
after simplifying and rearranging gives |
PD=1/2·[x1(y2
- y3)
+ x2(y3 -
y1)
+ x3(y1 -
y2)] |
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Example:
Two vertices of a triangle
lie at points A(-4,
0)
and B(5, -2)
while third vertex C
lies on the y-axis.
Find the coordinates of the point C
if the area of the triangle is 31 square units. |
Solution: |
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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, P1, P2 and
P3
in a coordinate plane: |
The
centroid M(x,
y),
where |
x
= 1/3 · (x1 + x2
+ x3), y =
1/3 · (y1
+ y2
+ y3) |
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Lines
parallel to the axes, horizontal and vertical lines
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If
the y
value never changes, i.e., if it takes the same constant value
y =
c
a line is parallel to the
x-axis
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 y-axis
and is called a vertical line. |
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Pre-calculus
contents B
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