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Parallel and Perpendicular Lines

FoundationHigherAQAEdexcelOCR

Practise parallel and perpendicular lines with this free Foundation and Higher GCSE Maths worksheet from Virtus Academy. You'll work through equations of parallel and perpendicular lines, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Perpendicular gradients multiply to give −1.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

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Higher

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Topic overview

Perpendicular lines meet at a right angle. On a graph, two lines are perpendicular when the product of their gradients is \(-1\).

The practical rule is that the gradient of a perpendicular line is the negative reciprocal of the original. Flip the fraction and change the sign, so a gradient of \(2\) becomes \(-\frac{1}{2}\), and \(-\frac{3}{4}\) becomes \(\frac{4}{3}\).

Both steps are required. Flipping without changing the sign, or changing the sign without flipping, gives a line that is not perpendicular. Multiplying the two gradients together is the quickest way to check: the answer must be exactly \(-1\).

Revision notes

The negative reciprocal

To find a perpendicular gradient, turn the fraction upside down and reverse the sign.

A gradient of \(3\) is \(\frac{3}{1}\), so the perpendicular gradient is \(-\frac{1}{3}\). A gradient of \(-\frac{2}{5}\) becomes \(\frac{5}{2}\).

Checking with the product

Multiply the two gradients. If the answer is \(-1\), the lines are perpendicular.

For \(4\) and \(-\frac{1}{4}\): the product is \(-1\), so they are perpendicular. This check takes seconds and catches sign errors.

Finding the equation

Use the perpendicular gradient with a given point to find the intercept, exactly as with parallel lines.

For a line perpendicular to \(y = 2x + 1\) through \((4, 3)\): the gradient is \(-\frac{1}{2}\), so \(3 = -\frac{1}{2}(4) + c\), giving \(c = 5\) and \(y = -\frac{1}{2}x + 5\).

Key points

  • Perpendicular lines meet at 90 degrees.
  • The product of their gradients is \(-1\).
  • Take the negative reciprocal: flip and change sign.
  • Both steps are needed, not just one.
  • A gradient of \(m\) pairs with \(-\frac{1}{m}\).
  • Multiply the gradients to check.

Worked examples

Example 1

Find the gradient of a line perpendicular to one with gradient \(5\).

Working

\[5 = \tfrac{5}{1}\]write the gradient as a fraction
\[-\tfrac{1}{5}\]flip the fraction and change the sign

Example 2

Are the lines \(y = 3x + 2\) and \(y = -\tfrac{1}{3}x - 1\) perpendicular?

Working

\[3 \times -\tfrac{1}{3}\]multiply the two gradients
\[= -1\]the product is −1, so the lines are perpendicular

Example 3

Find the equation of the line perpendicular to \(y = 4x - 3\) through \((8, 1)\).

Working

\[m = -\tfrac{1}{4}\]take the negative reciprocal of 4
\[1 = -\tfrac{1}{4}(8) + c \Rightarrow c = 3\]substitute the point to find c
\[y = -\tfrac{1}{4}x + 3\]write the equation

Common mistakes

  • Flipping without changing the sign.

    The perpendicular gradient to 2 is −½, not ½. Both steps are required.

  • Changing the sign without flipping.

    The perpendicular gradient to 2 is not −2. Multiply to check: −4, not −1.

  • Not rearranging first.

    Read the gradient only once the equation is in the form y = mx + c.

  • Using the original gradient to find c.

    Substitute the point into the new equation with the perpendicular gradient.

Exam tips

  • Write the gradient as a fraction before flipping it.
  • Multiply the two gradients as a check — the answer must be −1.
  • Rearrange any equation into y = mx + c first.
  • Substitute the given point back to verify your equation.

Key terms

Perpendicular
Meeting at a right angle.
Negative reciprocal
The result of flipping a fraction and reversing its sign.
Product
The result of multiplying two values.
Gradient
The steepness of a line.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.