Parallel and perpendicular lines

#### Tier: #Foundation #Higher

Description

Parallel lines never meet — they have the same gradient. Perpendicular lines meet at right angles — their gradients multiply to give $-1$.

$$m_1 \times m_2 = -1 \implies m_2 = -\frac{1}{m_1}$$

Example: Line $y = 3x + 2$ has gradient 3.

• A parallel line has gradient 3: $y = 3x + 5$ (different $c$)
• A perpendicular line has gradient $-\frac{1}{3}$: $y = -\frac{1}{3}x + 1$

Finding the equation of a perpendicular line through a given point:

1. Find the gradient of the original line
2. Take the negative reciprocal
3. Substitute the point into $y = mx + c$ to find $c$

This is a common Higher-tier question — often finding the perpendicular from a point to a line, or a perpendicular bisector.

Common error: adding 1 to the gradient rather than taking the negative reciprocal, e.g. writing gradient $= 3-1 = 2$ instead of $-\frac{1}{3}$.

Links

Equation of a straight line Gradients and intercepts Coordinates