Cartesian Plane

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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.

Cartesian Plane Quadrants
Cartesian Plane Quadrants

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 xis distance on x-axis and x is distance on y-axis.

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

Solution:

\begin{aligned}&if \hspace{2px} p = (3, 5) \hspace{2px} then\\ \\
&x = 3\\ \\
&y = 5
\end{aligned}
Cartesian Plane - Plotting a Point
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.

Pythagorean Theorem and Distance Formula
Pythagorean Theorem and Distance Formula

The above triangle has three sides – a, b and c, then Pythagorean theorem is given by

\begin{aligned}
&a^2 + b^2 = c^2\\ \\
&c = \sqrt{a^2 + b^2}
\end{aligned}

Suppose there are two points on the Cartesian plane.

\begin{aligned}
&p(x_1, y_1) = (2, 4)\\ \\
&q(x_2, y_2) = (2. 2)
\end{aligned}

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,

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Distance formula for two points is

\begin{aligned}
&d = \sqrt{(|x2 - x1|)^2 +(|y2 - y1|)^2}\\ \\\
&d = \sqrt{(|2 - 2|)^2 +(|2 - 4|)^2}\\ \\
&d = \sqrt{(0)^2 + (-2)^2}\\ \\
&d = 2
\end{aligned}
Distance between Points
Distance between Points

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.

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