Rectangular (Two-dimensional, Cartesian) Coordinate System
The area of a triangle
The coordinates of the centroid of a triangle

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.
Functions contents B