Rectangular (Two-dimensional, Cartesian) Coordinate System
Coordinate axes, x-axis and y-axis, origin, quadrants
Points in the Coordinate plane

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

Coordinate axes, x-axis and y-axis, origin, quadrants
The Cartesian coordinate system is defined by two axes at right angles to each other, forming a plane.
The horizontal axis is labeled x, and the vertical axis is labeled y
The point of intersection, where the axes meet, is called the origin labeled O
Given each axis, choose a unit length, and mark off each unit along the axis, forming a grid.
The position of each point in a plane is identified with an ordered pair of real numbers, in the form (x, y), called the coordinates of the point.
The x-coordinate, called the abscissa, equal to the distance of the point from the y-axis measured parallel to the x-axis.
The y-coordinate, called the ordinate, is the distance of the point from the x-axis measured parallel to the y-axis.
The origin O has coordinates (0, 0).
The intersection of the two axes creates four quadrants indicated by numerals I, II, III, and IV.
The quadrants are labeled counterclockwise starting from that in which both coordinates are positive.
Midpoint of a line segment
The point on a line segment that is equidistant from its endpoints is called the midpoint.
The coordinates of the midpoint M (xM, yM) of the line segment P1P2 where, P1(x1, y1) and P2(x2, y2) are endpoints,

Example:  Find the midpoint of the line segment AB where the endpoints, A(-5, 3) and B(-1, -1).
The distance formula
The distance between two given points in a coordinate (Cartesian) plane.
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 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.

As       l = AP : BP
and shown triangles are similar, then

which, with l negative, gives
the coordinates of the point P.

Example:  The line segment, with endpoints A(-3, 5) and B(6, -1), is divided by a point P internally in the ratio l = AP : BP = 1 : 2. Find the coordinates of the dividing point P.
 Solution:

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.
College algebra contents A