# Cartesian Plane

Cartesian plane is a two dimensional co-ordinate system. This system has two axis – the x-axis and the y-axis. The center of the Cartesian plane is called the origin.

The x-axis and the y-axis divide the plane into 4 different quadrants as shown in the figure above. The axis are real number line with 0 at the origin.

Any point on the plane is plotted in terms of horizontal distance on x-axis and vertical distance on y-axis. $p = (x, y)$

where $x$ is distance on x-axis and $x$ is distance on y-axis.

Example: Plot a point for $p = (3, 5)$

Solution:

If $p = (3, 5)$  then $x = 3$ $y = 5$ Cartesian Plane – Plotting a Point

## Pythagorean Theorem and Distance Formula

Pythagorean theorem is used for finding the distance of hypotenuse of a right triangle. The formula is modified to find the distance of two point on the Cartesian plane.

The above triangle has three sides – a, b and c, then Pythagorean theorem is given by $a^2 + b^2 = c^2$ $c = \sqrt{a^2 + b^2}$

Suppose there are two points on the Cartesian plane.

p  (x1, y1) = (2, 4)

q (x2, y2) = (2, 2)


and we have to find the distance between them.

Using Pythagorean theorem, we get

a = | y2 – y1 | = length of a

b = | x2 – x1 | = length of b

Therefore,

Distance formula for two points is $d = \sqrt{(|x2 - x1|)^2 +(|y2 - y1|)^2}$ $d = \sqrt{(|2 - 2|)^2 +(|2 - 4|)^2}$ $d = \sqrt{(0)^2 + (-2)^2}$ $d = 2$