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 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)]
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)
Beginning Algebra Contents E
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