By using the Pythagorean theorem, we can work out the length of the hypotenuse of a right-angled triangle if we know the length of the other two sides. To find the distance between two points $\left(x_1\,y_1\right)$ and $\left(x_2\,y_2\right)$, we first locate the points on the Cartesian plane.

Next, we can construct a right triangle by drawing a horizontal line through $\left(x_1\,y_1\right)$ and a vertical line through $\left(x_2\,y_2\right)$.

From the Pythagorean Theorem we know that $c\=\sqrt{a^2\+b^2}$.

Now, we need to determine the lengths a and b.

$a\=\left|x_2-x_1\right|$        $b\=\left|y_2-y_1\right|$

Then square each side to get closer to the form  $\sqrt{a^2\+b^2}$.

$a^2\= {\left|x_2-x_1\right|}^2\={\left(x_2-x_1\right)}^2$        $b^2\={\left|y_2-y_1\right|}^2\={\left(y_2-y_1\right)}^2$

Then

$c\=\sqrt{a^2\+b^2}\=\sqrt{{\left(x_2-x_1\right)}^2\+{\left(y_2-y_1\right)}^2}$