Distance on the Coordinate Plane

If two points line up horizontally (same $y$) or vertically (same $x$), the distance is just the difference of the coordinates that change — count the units.

For a diagonal, drop a right triangle: the horizontal change $\Delta x$ and the vertical change $\Delta y$ are the legs, and the distance is the hypotenuse. That's exactly the Pythagorean theorem:

$d = \sqrt{(\Delta x)^2 + (\Delta y)^2}$

So "distance between two points" is never a new idea — it's $a^2 + b^2 = c^2$ applied to the grid.

Take $(0, 0)$ and $(3, 4)$. The horizontal change is $3$, the vertical change is $4$ — the legs of a right triangle whose hypotenuse joins the points:

$d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Axis-aligned points are the easy case: $(1, 2)$ to $(6, 2)$ share a $y$, so the distance is just $|6 - 1| = 5$ — no triangle needed.

Find $\Delta x$ and $\Delta y$, then apply the Pythagorean theorem. Predict before you check.