Parallel and perpendicular lines

6 minAS Level1.3.11.3.2

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can use the gradient rules for parallel and perpendicular lines to find equations, including the perpendicular bisector of a line segment.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$m_1$	gradient of the first line
$m_2$	gradient of the second line
$m_{AB}$	gradient of the line through points $A$ and $B$
$\perp$	"is perpendicular to" (meets at a right angle)

Reminder: the gradient is the steepness, $\dfrac{\text{rise}}{\text{run}}$ (how far up for each 1 across). A line through $(x_1, y_1)$ with gradient $m$ is $y - y_1 = m(x - x_1)$ .

Two rules, one of them sneaky

Parallel lines have equal gradients: $m_1 = m_2$ . Easy.
Perpendicular lines have gradients that multiply to $-1$ : $m_1 m_2 = -1$ .

For the second one, "flip it and change the sign": the perpendicular gradient to $m$ is $-\dfrac{1}{m}$ .

Gradient	Perpendicular gradient
$2$	$-\tfrac12$
$-\tfrac34$	$\tfrac43$
$\tfrac25$	$-\tfrac52$

The exception: horizontal lines ( $m = 0$ ) are perpendicular to vertical lines (no gradient at all). The product rule needs two actual gradients.

Why it's −1

Take a line with gradient $\dfrac{\text{rise}}{\text{run}}$ . Turn it through 90°:

the run and the rise swap places, and
one of them reverses direction (gets a minus sign).

So the turned line has gradient $-\dfrac{\text{run}}{\text{rise}}$ .

Multiply the two gradients:

$\frac{\text{rise}}{\text{run}} \times \left(-\frac{\text{run}}{\text{rise}}\right) = -1$

Everything cancels except the minus sign.

Quick check

Fill in the blanksQuestion 1 · Recall

Answer the 2 questions above to continue.