Rectangular co-ordinate system is also called as a cartesian coordinate system. It is a coordinate system that defines every point differently in a plane by the pair of numerical coordinates. Rectangular coordinate system was invented by Rene Descartes in 17 |

**''origin'**'. The two axes divide the plane into 4 regions called as

**''quadrants''**and they are numbered from I to IV as shown in the figure below:

Each point on the plane are defined by the pair of numbers called as

**ordered pair.**The ordered pair which is measured on horizontal distances is called as the

**''Abscissa"**and the ordered pair which is measured on vertical distances is called as the

**''Ordinate''**. The number in the ordered pair which are connected with the point are called as the coordinate of the point.

Abscissa is called as the first coordinate or x- coordinate of the ordered pair, and the ordinate is called as second coordinate or y-coordinate.

The two dimensional rectangular coordinates are the one which are defined by 2 axes, forming a **xy** plane having right angles to each other. The horizontal axis is defined as **x**, that is the abscissa and the vertical axis as **y** that is the ordinate. The point of intersection is called as origin.

To define a specific point on a two dimensional coordinate system, we denote the **x** unit first and then the **y** unit in the form of **(x, y)**. The axes intersecting at the origin forming **4 **quadrants which are numerated in a clockwise direction by the roman numbers as shown in the figure below:

The distance between a point **P** and the origin **O** is calculated using the Pythagorean theorem

$OP_1$ = $\sqrt{x_1^2+y_1^2}$

$\therefore$ The distance between P$_1$ and P$_2$ is

$d$ = $P_1P_2$ = $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

**3**axes, located at the right angles to each other. And, they form a three dimensional space. Here, the three axes are defined by

**x**(abscissa),

**y**(ordinate) and

**z**, which is called as applicate. Even here, the point where the axes intersect is called as origin, denoted by

**O**. The particular values on the axes are denoted in the form of

**(x, y, z)**.

Using the Pythagorean theorem, the distances between a point

**P**and the origin

**O**is calculated and we get,

$OP_1$ = $\sqrt{x_1^2+y_1^2+z_1^2}$

The shortest distance left P$_1$ and P$_2$ is also calculated in the same way

$d$ = $P_1P_2$ = $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$

Given below are the few example problems based on rectangular coordinate which will help you to get an idea on how to find the rectangular coordinates.

### Solved Examples

**Question 1:**Find the ordered pair solution of $y$ = $\frac{x}{x-4}$, that corresponds to x = 6.

**Solution:**

**Given:**

$y$ = $\frac{x}{x-4}$

$y$ = $\frac{6}{6-4}$

y = 3

$\therefore$ the ordered pair solution is (6, 3).

**Question 2:**Find the ordered pair solution of $y$ = $\frac{3x}{x+3}$, that corresponds to x = -2.

**Solution:**

**Given:**

$y$ = $\frac{3x}{x+3}$

$y$= $\frac{3(-2)}{-2+3}$

y = -6

$\therefore$ the ordered pair solution is (-2, -6).

**Question 3:**Construct a graph for the ordered pairs solution of y = x

^{2}- 2x, when x = -1, 0, 1 and 2.

**Solution:**

**Given:**

y = x

^{2}- 2x

When x = -1

y = 1 + 2

y = 3.

When x = 0

y = 0

When x = 1

y = -1

When x = 2

y = 0

$\therefore$ the ordered pairs are (-1, 3), (0, 0), (1, -1), and (2, 0).