Parallel and Perpendicular Lines
Parallel and perpendicular lines are straight lines whose gradients follow fixed rules: parallel lines have equal gradients, while perpendicular lines have gradients that multiply to give -1. This lets you find the equation of a new line, in the form y = mx + c, once you know it must be parallel or perpendicular to a given line through a stated point. It is a core Higher GCSE topic.
Before you start
Make sure you're comfortable with these topics first:
Method
- Find the gradient of the given line, rearranging its equation into the form y = mx + c if needed.
- For a parallel line, use exactly the same gradient (m stays the same).
- For a perpendicular line, find the negative reciprocal of the gradient (flip the fraction and change the sign, so the two gradients multiply to -1).
- Substitute the new gradient and the coordinates of the given point into y - y1 = m(x - x1).
- Rearrange your equation into the form y = mx + c, or ax + by = c if the question asks for integer coefficients.
- Check your answer by substituting the given point back into your final equation.
Worked example
Find the equation of the line that is perpendicular to y = 2x + 3 and passes through the point (4, 1). Give your answer in the form y = mx + c.
- The gradient of y = 2x + 3 is 2.
- The perpendicular gradient is the negative reciprocal of 2, which is -1/2.
- Substitute m = -1/2 and the point (4, 1) into y - y1 = m(x - x1): y - 1 = -1/2(x - 4).
- Expand the brackets: y - 1 = -1/2 x + 2.
- Rearrange to make y the subject: y = -1/2 x + 3.
Practice questions
Try each question, then tap to reveal the answer.
Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
A straight line passes through the points (-2, 5) and (4, -7). Find the equation of the line, giving your answer in the form y = mx + c.
Show that the line y = 3x - 2 is parallel to the line with equation 9x - 3y + 6 = 0.
Triangle DEF has vertices D(-3, 0), E(1, 2) and F(3, -2). Show that triangle DEF has a right angle at E.
Free printable worksheet
Want more practice on paper? Download the parallel and perpendicular lines worksheet pack - 12 pages of exam-style questions with a full mark scheme. No sign-up, no email wall - just the PDF, free for personal and classroom use.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.