Gradient, distance and midpoint

Take $A(1, 1)$ and $B(4, 5)$ (see the diagram). Walk from $A$ to $B$ the lazy way: straight across, then straight up. You've drawn a right-angled triangle:

• across (the "run"): from $x = 1$ to $x = 4$, that's $4 - 1 = 3$;
• up (the "rise"): from $y = 1$ to $y = 5$, that's $5 - 1 = 4$.

That triangle answers three questions:

• How steep is $AB$? The gradient is rise ÷ run $= \tfrac43$: you go up $\tfrac43$ for every 1 across.
• How long is $AB$? $AB$ is the hypotenuse, so by Pythagoras $AB = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
• Where's the middle? Halfway across from 1 to 4 is $\tfrac{1 + 4}{2} = 2.5$; halfway up from 1 to 5 is $\tfrac{1 + 5}{2} = 3$. The midpoint is $(2.5, 3)$: just average the $x$'s and average the $y$'s.

To write rules that work for every pair of points, call them $A(x_1, y_1)$ and $B(x_2, y_2)$. The little 1 and 2 are labels, not powers: $x_1$ means "the $x$-coordinate of the first point", $y_2$ means "the $y$-coordinate of the second point". In the example, $x_1 = 1$, $y_1 = 1$, $x_2 = 4$, $y_2 = 5$.

Then the run is $x_2 - x_1$ and the rise is $y_2 - y_1$, and the three rules are exactly what we just did with numbers:

• Positive: uphill as you go right. Negative: downhill.
• Zero: a horizontal line ($y = k$).
• Undefined: a vertical line ($x = k$), because the run is 0 and you can't divide by 0.

Subtract in the same order on top and bottom. $\dfrac{y_2 - y_1}{x_2 - x_1}$ and $\dfrac{y_1 - y_2}{x_1 - x_2}$ are both fine. Mixing them flips the sign.