Perpendicular lines meet at a right angle. If one line has gradient $m$, any line perpendicular to it has gradient $-\dfrac{1}{m}$ — the negative reciprocal. Multiplying the two gradients together always gives $-1$:

Rearrange the given line to find its gradient $m_1$, then flip it and change the sign to get $m_2 = -\dfrac{1}{m_1}$.

For example, $2y = 3x+1 \implies y = \tfrac{3}{2}x + \tfrac{1}{2}$, so $m_1 = \tfrac32$, and a perpendicular gradient is $m_2 = -\tfrac{2}{3}$.

Use the gradient formula from E3.3 first to find $m_1$ from the two coordinates, then negative-reciprocal it exactly as above. This is the most common way perpendicular-line questions are actually set — the "given line" is rarely handed to you already in $y=mx+c$ form.

A perpendicular bisector of segment $AB$ is the line that (a) is perpendicular to $AB$, and (b) passes through the midpoint of $AB$. This combines three separate skills you already have:

1. Find the gradient of $AB$ (E3.3), then negative-reciprocal it.
2. Find the midpoint of $AB$ (E3.4).
3. Substitute the midpoint and the new gradient into $y=mx+c$ to find $c$.

Rotating a line by $90°$ swaps the roles of its rise and run and reverses one of their signs — a line that used to rise $3$ for every run of $2$ (gradient $\tfrac32$) becomes, once rotated a quarter-turn, a line that "runs" $3$ for every "rise" of $-2$ (gradient $-\tfrac23$). Seeing perpendicularity as a $90°$ rotation of the rise/run triangle is a useful way to remember why the gradient flips and negates, rather than just memorising the rule.

Find the equation of the perpendicular bisector of the line joining $A(-3, 8)$ and $B(9, -2)$.

1. Gradient of $AB$: $m_{AB} = \dfrac{-2-8}{9-(-3)} = \dfrac{-10}{12} = -\dfrac{5}{6}$.
2. Perpendicular gradient: $m = \dfrac{6}{5}$.
3. Midpoint of $AB$: $\left( \dfrac{-3+9}{2}, \dfrac{8+(-2)}{2} \right) = (3, 3)$.
4. Substitute $(3,3)$ and $m = \tfrac65$ into $y = mx+c$:

1. Equation: $y = \dfrac{6}{5}x - 0.6$.

A line has gradient $-\dfrac{4}{7}$. State the gradient of a line perpendicular to it.

1. Flip the fraction: $\dfrac{7}{4}$.
2. Change the sign: since the original gradient was negative, the perpendicular gradient is positive.
3. Perpendicular gradient $= \dfrac{7}{4}$. Check: $-\dfrac47 \times \dfrac74 = -1$ ✓.