8A - Distances between points

Distance from coordinate differences

From $A(-1,2)$ to $B(5,6)$, you move $6$ units across and $4$ units up. Is the straight-line distance $10$ units?

That sum measures a journey along two sides of a right-angled triangle. The direct route is its hypotenuse, so Pythagoras gives its square as $6^2+4^2=52$.

[DIAGRAM: asset_name: Coordinate differences form a right triangle; asset_slug: coordinate-distance; caption: The direct distance is the hypotenuse, with horizontal change 6 and vertical change 4; description: Equal-scale coordinate axes show A at minus 1 comma 2 and B at 5 comma 6, a right-angle corner at 5 comma 2, horizontal side 6, vertical side 4 and hypotenuse 2 square root 13.]

A coordinate difference may be negative; its square supplies the squared side length. Reversing both differences leaves the distance unchanged.

Now let the horizontal change be negative. The sign changes the direction of travel, not the length.

The same formula includes horizontal and vertical segments. If the $y$-coordinates agree, the distance is $|x_2-x_1|$; if the $x$-coordinates agree, it is $|y_2-y_1|$. For example, $(-2,3)$ to $(5,3)$ has length $7$, and $(4,-1)$ to $(4,5)$ has length $6$.

Use an exact root in the next calculation.

Equal-length conditions

When two lengths are equal, their squares are equal too, so you can often solve the condition without carrying square roots through the algebra.

For nonnegative lengths $a$ and $b$, $a=b$ is equivalent to $a^2=b^2$. This equivalence is why comparing squared distances loses no geometric information.