The French Mathematician Rene Descartes invented the cartesian system. In a Cartesian coordinates form we have a point, which shows the particular place in a segment for representation. Cartesian coordinates system consist of many coordinates which are polar, rectangular etc.

In a Cartesian coordinates we have collinear point coordinates, mid point coordinates and the equidistant point coordinates:

**1) Collinear points:** Collinear points is a point where the three or more points lies on same line.**2) Midpoint: **Mid point is a halfway point where the line segment divided into two equal parts.**3) Equidistant point: **In a line segment a point is equal length from other points which are in congruent then the point are equidistant point.

Related Calculators | |

Cartesian Coordinates Calculator | Cartesian to Polar Calculator |

Convert Polar to Cartesian Calculator | Linear System of Equations Solver |

A system of points on a plane or in space by their coordinates is called a Cartesian system. The co ordinate axis are the two perpendicular lines which intersect at the origin.

These two lines of the Cartesian system are called the X and Y axis:

The value of *x* is positive in the half-plane to the right of *origin*, and is negative to the left of the origin.

The value of *y* is positive in the upper half and negative in the lower half of the XYplane. The plane is therefore divided into four quadrants

The four quadrants are the roman numerals I, II, III, and IV.

The quadrants are labeled counter-clockwise starting from the top right quadrant.

**The values of X and Y in the four quadrants are listed below:**

Quadrant | x values | y values |
---|---|---|

I | > 0 | > 0 |

II | < 0 | > 0 |

III | < 0 | < 0 |

IV | > 0 | < 0 |

The three dimensional cartesian system provides the physical dimensions of height, width, and length of something.

This system is represented using the right-hand rule, and is also called the right-handed coordinate system.

When the thumb, index and the middle finger of the right hand are at perpendicular to each other, we have X, Y, and Z axes respectively, and the points of the cartesian system are seperated by commas, as in (2, 4, 7)

To make a figure larger or smaller we need to multiply the Cartesian coordinates of every point by the same positive number. The reflection of a point is got by changing their signs in the cartesian system.

In coordinate geometry, the tools of algebra are used in studying geometry by establishing 1-1 correspondence between the points in a plane and the ordered pairs of real numbers:

Centroid = $\frac{x_1+x_2+x_3}{3}$, $\frac{y_1 + y_2 + y_3}{3}$

Given below there are few example problems on Cartesian system

Then d (A, B) = $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

= $\sqrt{(4-1)^2 +(3-6)^2}$

= $\sqrt{3^2+(-3)^2}$

= $\sqrt{9+9}$

=$\sqrt {18}$

=3$\sqrt{2}$

Formula = $\frac{(x_1+x_2)}{(2)}$,$\frac{(y_1+ y_2)}{(2)}$ here, $(x_1, y_1)$ = (5,8),($x_2, y_2$) = (-1,-4)

= $\frac{(5-1)}{(2)}$ $\frac{(8-(-5))}{(2)}$

=($\frac{4}{2}$) ($\frac{13}{2}$)

= (2, 6.5)

Formula for centroid= $\frac{(x_1+ x_2+ x_3)}{(3)}$ ,$\frac{( y_1+ y_2+ y_3)}{(3)}$

The centroid of the triangle = $\frac{(0+4+2)}{3}$,$\frac{(-3+6+3)}{3}$

=$\frac{6}{3}$,$\frac{6}{3}$

=(2,2)

The diagonals in a parallelogram bisect each other. Let the point of bisection be O.

Then by mid point formula,

O is the midpoint of BD = ($\frac{(6-3)}{2}$, $\frac{(3-4)}{2}$)

= ($\frac{3}{2}$, $\frac{-1}{2}$)

O is also the midpoint of diagonal CA =($\frac{(x-8)}{2}$, $\frac{(y-4)}{2}$)

But Coordinates of O are ($\frac{3}{2}$, $\frac{1}{2}$)

= ($\frac{3}{2}$, $\frac{-1}{2}$) = ($\frac{(x-8)}{2}$,$\frac{(y-4)}{2}$)

$\rightarrow$ $\frac{(x-8)}{2}$ = $\frac{3}{2}$ , $\frac{(y-4)}{2}$ = $\frac{-1}{2}$

$\rightarrow$ x-8 = 3, y-4 =-1

$\rightarrow$ x=11 , y= 3

Coordinates of C(11, 3).

**Example 5:** Points A and B have vertices (7, -2) and (a, b) respectively. The Coordinates of the mid point are (4, -3). Find the values of a and b. [2 Mark]

**Solution:**

By mid point formula, Coordinates of mid point can be calculated by,

(x,y) = ($\frac{x_1 + x_2}{2}$, $\frac{y_1 + y_2}{2}$)

$\rightarrow$ (4, -3) = ($\frac{(7+a)}{2}$, $\frac{(-2+b)}{2}$ )

$\rightarrow$ 4= $\frac{(7+a)}{2}$, -3 = $\frac{(-2+b)}{2}$

$\rightarrow$ 8= 7+a, -6 = -2+b

$\rightarrow$ 1=a, -4=b

Hence a = 1, b = -4

Coordinates of B(1, -4).

More topics in Cartesian System | |

Interpolation | Ordinate |

Cartesian Coordinates | Coordinate System |

Midpoint Formula | Distance Formula |

Locus | Quadrant |

Section Formula | Polar Coordinates |

Coordinate Geometry | Polar Graphs |

Related Topics | |

Math Help Online | Online Math Tutor |