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Parallel and Perpendicular Lines

Two lines are parallel when their gradients are equal, and perpendicular when their gradients multiply to −1. These two facts let you build new lines related to a given one.

25 min Video by Zeeshan Zamurred Straight Line Graphs

Edexcel Straight Lines — full chapter playlist (Zeeshan Zamurred)Watch the full walkthrough before the notes below.

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What you'll be able to do

Recognise parallel lines from equal gradients
Find perpendicular gradients (negative reciprocal)
Write equations of parallel and perpendicular lines
Solve geometry problems using these relationships

Parallel lines

Parallel lines never meet — they have the $same gradient$ (but different intercepts).

m_{1} = m_{2} \Rightarrow parallel

Equal gradients ⟶ parallel.

1Same gradient

3

; intercept

5

Answer

y = 3 x + 5

Perpendicular lines

Perpendicular lines meet at a right angle. Their gradients multiply to $- 1$ , so each is the $negative reciprocal$ of the other.

m_{1} \times m_{2} = - 1 \Rightarrow m_{2} = - \frac{1}{m _{1}}

Flip the fraction, change the sign.

1Negative reciprocal of 2.

Answer

- \frac{1}{2}

Tip — Perpendicular gradient = negative reciprocal: gradient 3 → −1/3, gradient −¼ → 4.

Building the equations

Once you have the right gradient (equal for parallel, negative reciprocal for perpendicular), use the point-gradient formula with the given point.

Formula recap

m_{1} = m_{2} \Rightarrow parallel

Parallel: equal gradients.

m_{1} m_{2} = - 1 \Rightarrow perpendicular

Perpendicular: negative reciprocal.

y - y_{1} = m (x - x_{1})

Build the equation.

Common mistakes to avoid

Using the reciprocal but forgetting the minus for perpendicular.

Perpendicular gradient is the NEGATIVE reciprocal.

Thinking parallel lines have the same intercept.

They share a gradient but generally have different intercepts.

Key takeaways

Parallel: equal gradients (m₁ = m₂).
Perpendicular: gradients multiply to −1 (negative reciprocal).
Find the right gradient, then use point-gradient form.

Test yourself

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