7.2 Parallel and Perpendicular Lines

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📖 Main Content

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1. What is a Gradient?

Before we look at parallel and perpendicular lines, you need to understand what gradient means.

The gradient of a line tells you how steep the line is. It measures how much the line goes up or down for every step it moves to the right.

The Gradient Formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Here, $(x_1, y_1)$ and $(x_2, y_2)$ are any two points on the line, and $m$ stands for the gradient.

In plain English: Take two points on a line. Divide the vertical distance (difference in $y$ values) by the horizontal distance (difference in $x$ values).

Quick Example: Find the gradient of the line passing through $(1, 3)$ and $(5, 11)$.

$m = \frac{11 - 3}{5 - 1} = \frac{8}{4} = 2$

So the gradient is 2. This means for every 1 step to the right, the line goes 2 steps up.

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2. The Equation of a Straight Line

A straight line can be written in the form:

$y = mx + c$

• $m$ is the gradient (the steepness of the line)
• $c$ is the y-intercept (where the line crosses the y-axis)

This form is very useful because you can read off the gradient immediately just by looking at the equation.

Example: In the equation $y = 3x + 7$, the gradient is $3$ and the y-intercept is $7$.

Sometimes the equation is written in a different form, such as $ax + by = c$. In that case, rearrange it into the form $y = mx + c$ to find the gradient.

Example: Rearrange $2x + 4y = 8$ into $y = mx + c$.

$4y = -2x + 8$ $y = -\frac{1}{2}x + 2$

The gradient is $-\frac{1}{2}$.

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🔑 The Condition for Parallel Lines:

$m_1 = m_2$

Here, $m_1$ is the gradient of the first line and $m_2$ is the gradient of the second line.

Think of it this way: If two lines lean at exactly the same angle, they will never cross — they are parallel.