# 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

$x = 3$

$y = 5$

## 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

$d = \sqrt{(|x2 - x1|)^2 +(|y2 - y1|)^2}$

$d = \sqrt{(|2 - 2|)^2 +(|2 - 4|)^2}$

$d = \sqrt{(0)^2 + (-2)^2}$

$d = 2$

The above diagram verify the results and it shows that the distance is actually 2 units. Hence, the distance formula is correct and applies to the Cartesian plane.