Parallel and perpendicular lines

Parallel lines have equal gradients, perpendicular gradients multiply to −1, and finding equations of parallel and perpendicular lines through a point.

The rules

Perpendicular gradients

Whatever m you choose (except 0), the two lines meet at right angles and the gradients multiply to −1.

Example 1●●

Find the line parallel to $y = 3x - 2$ through $(1, 5)$.

$m = 3$: $y - 5 = 3(x - 1)$

$y = 3x + 2$

Example 2●●●

Find the line perpendicular to $y = 2x + 3$ through $(4, 1)$.

$m = -\tfrac{1}{2}$: $y - 1 = -\tfrac{1}{2}(x - 4)$

$y = -\tfrac{1}{2}x + 3$

Flashcards

Parallel gradients

Equal

Perpendicular to $m = 3$

$-\tfrac{1}{3}$

Perpendicular to $m = - frac{2}{5}$

$\tfrac{5}{2}$

Practice

Parallel to $y = 3x - 2$ through $(1, 5)$: $y = $ ?
Perpendicular to $y = 2x + 3$ through $(4, 1)$: $y = $ ?
Gradient perpendicular to $y = 2x + 1$?
Gradient perpendicular to $y = -\tfrac{1}{3}x$?
Are $y = 4x + 1$ and $y = 4x - 7$ parallel?
Are $y = 2x$ and $y = 0.5x + 3$ perpendicular?
Are $y = \tfrac{3}{4}x$ and $y = -\tfrac{4}{3}x + 2$ perpendicular?
Line $y = kx + 1$ is parallel to $2x + y = 5$. Find $k$.