GCSE MATHS • HIGHER TIER
Parallel and Perpendicular Lines Made Simple
Use gradients to identify parallel and perpendicular lines and find equations of related straight lines.
Clear method
Exam practice
Worked proofs
Parallel and Perpendicular Lines in Four Moves
1
Identify
Write each line in the form y = mx + c and identify its gradient.
2
Compare
For parallel lines, check whether the gradients are the same.
3
Transform
For perpendicular lines, flip the gradient and change its sign.
4
Solve
Use y = mx + c and any given coordinate to find the required equation.
What You’ll Learn
Understand parallel and perpendicular lines.
Use gradients to identify parallel lines.
Find equations of parallel lines.
Find perpendicular gradients using negative reciprocals.
Apply gradient rules to coordinate geometry problems.
Find equations of perpendicular lines.
Key Rules
Parallel lines
Parallel lines have the same gradient.
Perpendicular lines
Flip the gradient and change its sign to find the perpendicular gradient.
Perpendicular check
m 1 ×m 2 =−1 for perpendicular lines.
Finding the equation
Use \(y = mx + c\) with the required gradient, then substitute the given point to find \(c\).
Worked Proof: Parallel and Perpendicular Lines
Step-by-Step Method
- 1. Write the original line as y = mx + c.
- 2. Identify gradient m.
- 3. Decide whether the required line is parallel or perpendicular.
- 4. For parallel lines, keep the same gradient.
- 5. For perpendicular lines, flip the gradient and change its sign.
- 6. Substitute the gradient into y = mx + c.
- 7. Use the given coordinate to find c and write the final equation.
More Worked Examples
1
Determine whether y = 5x + 2 and y = 3x + 6 are parallel.
Step 1: Identify gradients.
5 and 3
Step 2: Compare. They are different.
Step 2: Compare. They are different.
2
Find the gradient of a line perpendicular to y = 2x + 5.
Step 1: Identify the gradient.
2
Step 2: Flip and change sign. -1/2
Step 2: Flip and change sign. -1/2
3
Find the gradient of a line perpendicular to y = -4x + 1.
Step 1: Identify gradient.
-4
Step 2: Flip and change sign. 1/4
Step 2: Flip and change sign. 1/4
4
Find the equation of a line parallel to y = 3x + 2 that passes through (0, 7).
Step 1: Keep the same gradient.
m = 3
Step 2: Use y = mx + c. y = 3x + c
Step 3: Substitute (0, 7).
7 = 3(0) + c
c = 7
Step 2: Use y = mx + c. y = 3x + c
Step 3: Substitute (0, 7).
7 = 3(0) + c
c = 7
- Common GCSE Mistakes
- Thinking parallel lines have the same intercept.
- Forgetting that only gradients matter for parallel lines.
- Forgetting to flip and change the sign for perpendicular gradients.
- Mixing up reciprocal values.
- Arithmetic errors when finding equations.
- Exam Tips
- Parallel lines need the same gradient, not the same intercept.
- For perpendicular lines, use the negative reciprocal.
- Check perpendicular gradients by multiplying them; the product should be -1.
- Use the given coordinate to calculate c.
- Keep fractions exact when working with perpendicular gradients.
- Always finish by writing the complete equation.
Skills Used in Parallel and Perpendicular Lines
Linear Graphs
Parallel Lines
Perpendicular Lines
Gradient
Negative Reciprocals
Equation of a Line
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Frequently Asked Questions
Parallel lines have the same gradient.
No. It is the gradient that must be the same.
Flip the gradient and change its sign.
Multiply the gradients. For non-vertical lines, the product should be -1.
Use the same gradient, then substitute the given point into y = mx + c to find c.
Find the negative reciprocal gradient, then use the given point to find c.