11.4 Parallel and perpendicular lines — Learn

Prerequisites: identify the gradient and vertical intercept in $y=mx+c$.

Objectives

By the end of this section, you can use gradients to identify parallel and perpendicular lines and find their equations.

Parallel and perpendicular gradients

Parallel lines have equal gradients. Perpendicular lines meet at a right angle. Their gradients multiply to $-1$, so a line with non-zero gradient $m$ has perpendicular gradient $-1/m$.

Worked example. Find the line perpendicular to $y=2x+1$ through $(0,4)$.

The given gradient is 2. The perpendicular gradient is $-1/2$, because $2\\times(-1/2)=-1$. The point $(0,4)$ gives the vertical intercept 4. Therefore, the equation is $y=-\\frac12x+4$.

Watch out. Do not change only the sign of the gradient. For gradient 2, the perpendicular gradient is $-1/2$, and their product is $-1$.

Progress check

A line with gradient $1/15$ is perpendicular to a line with gradient $-15$, since $(1/15)(-15)=-1$.

You should now be able to

• use gradients to identify parallel and perpendicular lines and find their equations.;
• check your result by substitution or by interpreting the graph;
• explain the method in your own words.