Parallel and Perpendicular Lines Revision Notes for Edexcel GCSE Maths

Parallel and perpendicular lines

Understanding the relationship between parallel and perpendicular lines is essential for working with coordinate geometry. These concepts help you determine how lines relate to each other and find equations of new lines based on existing ones.

Parallel lines and their gradients

Parallel lines are lines that never meet - they run alongside each other at exactly the same angle. The key mathematical property that defines parallel lines is that they have identical gradients.

When you write any straight line in the form $y = mx + c$ , the value of $m$ (the gradient) tells you the steepness and direction of the line. For lines to be parallel, their $m$ values must be exactly the same, even if their $c$ values (y-intercepts) are different.

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The gradient $m$ determines the direction and steepness of a line, while the y-intercept $c$ determines where the line crosses the y-axis. Parallel lines can have different y-intercepts but must have the same gradient.

For example, the lines $y = 2x + 3$ , $y = 2x$ , and $y = 2x - 4$ are all parallel because they each have a gradient of $2$ . They're positioned at different heights on the coordinate plane, but they all slope upwards at the same rate.

Finding equations of parallel lines

When you need to find the equation of a line parallel to a given line, the key principle is that both lines must have the same gradient. Here's the systematic approach:

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Worked Example: Finding a Parallel Line

Find the equation of a line parallel to $y = -0.25x + 1$ that passes through the point $(2, 3)$ .

Step 1: Identify the gradient of the original line From $y = -0.25x + 1$ , the gradient is $-0.25$

Step 2: Use the same gradient for the parallel line The parallel line will also have gradient $-0.25$

Step 3: Substitute the coordinates into $y = mx + c$ $3 = -0.25(2) + c$

Step 4: Solve for $c$ $3 = -0.5 + c$ $c = 3.5$

Step 5: Write the complete equation $y = -0.25x + 3.5$

Perpendicular lines and their gradients

Perpendicular lines meet at right angles (90°). The mathematical relationship between their gradients is more complex than parallel lines, but follows a clear pattern.

If one line has gradient $m$ , then any line perpendicular to it will have gradient $-\frac{1}{m}$ . This means the gradients are negative reciprocals of each other.

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The product of perpendicular gradients always equals $-1$ . This gives you a quick way to check if two lines are perpendicular: if $m_1 \times m_2 = -1$ , then the lines are perpendicular.

An important property to remember is that when you multiply the gradients of two perpendicular lines together, you always get $-1$ . This gives you a quick way to check if two lines are perpendicular.

Finding equations of perpendicular lines

To find the equation of a line perpendicular to a given line, you need to calculate the negative reciprocal of the original gradient:

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Worked Example: Finding a Perpendicular Line

Find the equation of line B perpendicular to line A with equation $3y - x = 6$ , passing through $(3, 3)$ .

Step 1: Find the gradient of line A Rearrange to standard form: $3y = x + 6$ , so $y = \frac{1}{3}x + 2$ Gradient of line A = $\frac{1}{3}$

Step 2: Calculate the perpendicular gradient Perpendicular gradient = $-\frac{1}{\frac{1}{3}} = -3$

Step 3: Use the perpendicular gradient in $y = mx + c$ $3 = -3(3) + c$

Step 4: Solve for $c$ $3 = -9 + c$ $c = 12$

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

Key relationships to remember

The gradient relationships form the foundation for solving parallel and perpendicular line problems. Understanding these relationships allows you to quickly identify the type of relationship between lines and solve complex geometry problems.

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When working with line equations that aren't in $y = mx + c$ form, always rearrange them first to easily identify the gradient. This is a essential first step in most parallel and perpendicular line problems.

The fundamental relationships are:

Parallel lines: gradients are equal ( $m_1 = m_2$ )
Perpendicular lines: gradients are negative reciprocals ( $m_1 = -\frac{1}{m_2}$ )
Perpendicular check: $m_1 \times m_2 = -1$

These relationships work regardless of how the original equation is presented. You might need to rearrange equations into $y = mx + c$ form first, but the gradient relationships remain consistent.

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Key Points to Remember:

Parallel lines have exactly the same gradient - they never meet because they're angled identically
Perpendicular lines have gradients that are negative reciprocals of each other
To find a perpendicular gradient, take the negative reciprocal: if gradient is $m$ , perpendicular gradient is $-\frac{1}{m}$
The product of perpendicular gradients always equals $-1$
Always rearrange equations into $y = mx + c$ form to easily identify the gradient

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