Parallel and Perpendicular Lines Revision Notes for AQA GCSE Maths

Parallel and perpendicular lines

Understanding how to work with parallel and perpendicular lines is essential for solving coordinate geometry problems. These special types of lines have unique properties that make them easier to identify and work with once you know the key relationships.

Parallel lines have the same gradient

When two lines run alongside each other and never meet, they are called parallel lines. The most important thing to remember is that parallel lines always have exactly the same gradient (slope).

This means that if you have the equation of one line, you can easily find the equation of any line parallel to it. All parallel lines follow the pattern $y = mx + c$ , where the 'm' value (gradient) stays the same, but the 'c' value (y-intercept) can be different.

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For example, the lines $y = 2x + 3$ , $y = 2x$ , and $y = 2x - 4$ are all parallel to each other because they all have the same gradient of 2.

Finding the equation of a parallel line

Let's work through a systematic approach using this example: Line J has a gradient of -0.25, and we need to find the equation of Line K, which is parallel to Line J and passes through the point (2, 3).

Since Lines J and K are parallel, they must have the same gradient. This means Line K also has a gradient of -0.25.

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Worked Example: Finding a Parallel Line Equation

Given: Line J has gradient -0.25, Line K is parallel to Line J and passes through (2, 3)

Step 1: Use the same gradient Since the lines are parallel: gradient of Line K = -0.25

Step 2: Start building the equation $y = -0.25x + c$

Step 3: Find the y-intercept using the given point Substitute (2, 3) into the equation: $3 = -0.25(2) + c$ $3 = -0.5 + c$ $c = 3.5$

Step 4: Write the complete equation $y = -0.25x + 3.5$

Perpendicular line gradients

Perpendicular lines meet at right angles (90°), and they have a special mathematical relationship. When you multiply the gradients of two perpendicular lines together, you always get -1.

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Key Rule for Perpendicular Lines: If the gradient of the first line is $m$ , then the gradient of the perpendicular line will be $-\frac{1}{m}$ .

The key insight is that perpendicular gradients are negative reciprocals of each other. This means you flip the fraction and change the sign.

Finding the equation of a perpendicular line

Here's how to tackle this type of problem step by step: Lines A and B are perpendicular and intersect at (3, 3). If Line A has the equation $3y - x = 6$ , what is the equation of Line B?

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Worked Example: Finding a Perpendicular Line Equation

Given: Lines A and B are perpendicular, intersect at (3, 3), Line A: $3y - x = 6$

Step 1: Find the gradient of Line A Rearrange into standard form: $3y - x = 6$ $3y = x + 6$ $y = \frac{1}{3}x + 2$

So the gradient of Line A is $\frac{1}{3}$

Step 2: Find the gradient of Line B Since Line B is perpendicular to Line A: Gradient of Line B = $-\frac{1}{\frac{1}{3}} = -1 \times 3 = -3$

Step 3: Build the equation $y = -3x + c$

Step 4: Find the y-intercept Using the intersection point (3, 3): $3 = -3(3) + c$ $3 = -9 + c$ $c = 12$

Step 5: Write the complete equation $y = -3x + 12$

Key problem-solving strategy

A reliable approach for these problems is to use the gradient relationship first, then use the known coordinates to work out the y-intercept. This method works whether you're dealing with parallel or perpendicular lines.

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Strategy Summary:

For parallel lines: use the same gradient
For perpendicular lines: use the negative reciprocal of the gradient
Always substitute the given point coordinates into your equation to find the y-intercept value

Remember!

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Key Points to Remember:

Parallel lines have identical gradients - they never change direction relative to each other
Perpendicular lines have gradients that multiply together to equal -1
To find a perpendicular gradient, take the negative reciprocal (flip the fraction and change the sign)
Always substitute known point coordinates to find the y-intercept
The standard form $y = mx + c$ makes it easy to identify gradients and apply these relationships

Parallel and Perpendicular Lines (AQA GCSE Maths): Revision Notes