Parallel and perpendicular lines

Revise Parallel and perpendicular lines for Mathematics D 4024 (O Level) — revision notes, 36 practice questions and instant AI marking.

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Cambridge O Level Mathematics (Syllabus D) 4024

Parallel and perpendicular lines

The big idea: Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to −1, so one is the negative reciprocal of the other. With a gradient and one point you can write the equation of either.

Summary — What This Chapter Covers

Find the gradient and equation of a line parallel to a given line.
Find the gradient and equation of a line perpendicular to a given line.
Find the equation of the perpendicular bisector of a line segment.

1. Parallel Lines

Syllabus 3.6 · Paper 1 & 2

Same gradient, different intercept

Parallel lines never meet because they have the same gradient. To find a parallel line through a given point, keep m and find the new c.

Worked Example

Find the equation of the line parallel to y = 4x − 1 that passes through (1, −3).

Practice Question 1

Find the equation of the line parallel to 3x + 2y = 7 that passes through (4, 1).

2. Perpendicular Lines

Syllabus 3.7 · Paper 1 & 2

Negative reciprocal gradient

Two lines are perpendicular (meet at 90°) if the product of their gradients is −1. To find the perpendicular gradient, turn the gradient upside down and change its sign.

Perpendicular gradients (not given) m₁ × m₂ = −1 · m₂ = −1/m₁ e.g. 2 → −1/2, −3/4 → 4/3

Worked Example

Find the gradient of a line perpendicular to 2y = 3x + 1.

Practice Question 2

(a) Show that the lines 3x + 2y = 6 and 2x − 3y = 12 are perpendicular.
(b) Find the equation of the line perpendicular to y = 2x − 3 that passes through (4, 1).

3. Perpendicular Bisectors

Syllabus 3.7 · Paper 1 & 2

Through the midpoint, at right angles

The perpendicular bisector of AB cuts AB in half at right angles. It passes through the midpoint of AB and has the negative reciprocal of the gradient of AB.

Worked Example

Find the equation of the perpendicular bisector of the line joining (−3, 8) and (9, −2).

Practice Question 3

Memorise

Parallel

Same gradient: m₁ = m₂.

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Parallel and perpendicular lines