Rectangular (Two-dimensional, 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 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)]
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:
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)
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 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.
Polar coordinate system
The polar coordinate system is a two-dimensional 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 x-axis, 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, -3) to polar coordinates.
Solution:
  and since the point lies in the 3rd quadrant, then
 
College algebra contents A
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