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Coordinate System

Coordinate systems identify locations by making measurements on a framework of intersecting lines that resemble a complex net. On a map, the lines are straight and the measurements are done in terms of distance but on a circular surface (like a football or a sphere) the lines become circles and the measurements are done in terms of angle.
These measurements are based on Cartesian coordinates.

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Coordinate System

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Cartesian Coordinate System:
The following are some important concepts to know about the coordinate system.
A two dimensional surface on which a coordinate system has been set up, is known as Cartesian plane, or graph or coordinate grid.
The numbers in an ordered pair that locate a point in the coordinate plane is known as coordinates.
X – axis is the horizontal number line on the coordinate grid.
Y – axis is the vertical number line on the coordinate grid.
The four parts made by the above X and Y axes are called quadrants.
The intersection of the X and Y axis is known as the origin.
For plotting any point of the coordinate system, we have to write first the value of x and then y, like (2, 3), here x = 2 and y = 3.

If the coordinates of any group of points are given to us then we can:
1.    Find out the distance between those two points.
2.    Determine their mid point, their slopes and their equation of lines.
3.    Check if those lines are perpendicular lines or the parallel lines.
4.    Calculate the perimeter and determine the area of the polygon which is defined by its points.
5.    Do the transformation by either rotation, reflection or by moving it.
6.    Determine the equations of different conic sections.
7.    Determine the point of intersection of two lines etc.

Some Common Formulas:
The distance formula:
The distance D between any two points is given by the following equation:
D = $\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$, where the coordinates  are (x1, y1) and (x2, y2) respectively.
The mid point formula:
If a line segment has the end points ((x1, y1) and (x2, y2), the midpoint is given by the following formula: $[(\frac{x_{1}+x_{2}}{2}),(\frac{y_{1}+y_{2}}{2})]$.

Solved Examples:
Example1:  Find the point of intersection of the lines:
    2x + y = 3    (a)
and    y = 3x – 1.    (b)
Solution: To find the point of intersection we need to solve the equations (a) and (b) simultaneously. We can substitute (b) into equation (a):
    2x + (3x – 1) = 3
i.e.    5x – 1 = 3
i.e.    x = $\frac{4}{5}$
Substituting this into equation (b):
    y = 3($\frac{4}{5}$) – 1 = $\frac{7}{5}$.
Therefore the lines intersect at the point
    ($\frac{4}{5}$, $\frac{7}{5}$).

Example 2: Calculate the difference between the two points (15, 20) and (35, 5).
Solution: For A is at (15, 20) its x-coordinate is the first number or 15. The x-coordinate of B is 35. So the difference is 20. Now, the difference in the point's y-coordinates would be 20 - 5 = 15.
Now using the distance formula for finding distance between two points which is $\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$, we will get, distance = $\sqrt{(20)^{2}+(15)^{2}}$ = 25.

Example 3: Find the coordinates (x, y) of the midpoint of the segment that connects the points (-4, 6) and (3, -8).
Solution: Applying the midpoint formula, we get, (x, y) = $[(\frac{x_{1}+x_{2}}{2}),(\frac{y_{1}+y_{2}}{2})]$.

(x, y) = $[(\frac{-4+3}{2}),(\frac{6-8}{2})]$
The answer is (-1.5, -1).

More topics in  Coordinate System
Cartesian Coordinate System Polar Coordinate System
Cylindrical Coordinate System Spherical Coordinate System
Curvilinear Coordinates Distance Formula for Polar Coordinates
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