Distance of Point from Origin in Cartesian Coordinates

Theorem

Let $P = \tuple {x, y}$ be a point in the cartesian plane.

Then $P$ is at a distance of $\sqrt {x^2 + y^2}$ from the origin.

Proof

By definition of the cartesian plane, the point $P$ is $x$ units from the $y$-axis and $y$ units from the $x$-axis.

The $y$-axis and $x$-axis are perpendicular to each other, also by definition.

Thus $x$, $y$ and $OP$ form a right-angled triangle.

By Pythagoras' Theorem:

$OP^2 = x^2 + y^2$

Hence the result.

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions