Gradient, length and midpoint

The three formulae

For two points $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$:

Gradient is change in $y$ over change in $x$ — the $y$ values on top. Length comes straight from Pythagoras: the two differences are the legs of a right-angled triangle and $AB$ is its hypotenuse. Midpoint is the average of each coordinate separately.

Reading a gradient

• positive gradient: the line rises left to right;
• negative gradient: it falls;
• gradient $0$: horizontal, equation $y=c$;
• gradient undefined: vertical, equation $x=k$ — the denominator $x_{2}-x_{1}$ is zero, and dividing by zero has no value at all. "Gradient zero" and "gradient undefined" are opposite cases.

Order does not matter — as long as it is consistent

Taking $A$ first or $B$ first changes the sign of both the numerator and the denominator, so the gradient is unchanged. Taking $A$ first on top and $B$ first underneath, however, flips the sign — and that is the single most common slip in this topic.