Parallel & Perpendicular Lines (DP IB Analysis & Approaches (AA): HL): Revision Note

Parallel lines are always equidistant meaning they never intersect

Parallel lines have the same gradient

• If gradient of line $l_1$ is $m_1$ and gradient of line $l_2$ is $m_2$ ,

  • $m_1=m_2\Rightarrow l_1\& l_2 \mathrm{are} \mathrm{parallel}$

  • $l_1 \& l_2 \mathrm{are}\mathrm{ }\mathrm{parallel}\Rightarrow m_1=m_2$

To determine if two lines are parallel:

• Rearrange into the gradient-intercept form, $y=mx+c$

• Compare the coefficients of $x$

• If they are equal then the lines are parallel

Perpendicular lines intersect at right angles

The gradients of two perpendicular lines are negative reciprocals

• If gradient of line $l_1$ is $m_1$ and gradient of line $l_2$ is $m_2$ ,

  • $m_1\times m_2=-1\Rightarrow l_1\&l_2\mathrm{are}\mathrm{perpendicular}$

  •   $l_1\&l_2\mathrm{are}\mathrm{perpendicular}\Rightarrow m_1\times m_2=-1$

• For example, if $m_1=\frac{1}{3}$ and $m_2=-3$ then the lines are perpendicular

  • If $m_1=5$ and $m_2=\frac{1}{5}$ then the lines are not perpendicular

To determine if two lines are perpendicular:

• Rearrange into the gradient-intercept form, $y=mx+c$

• Compare the values of $m$ for each

• If the product of the two values of $m$ is -1, then they are perpendicular

Be careful with horizontal and vertical lines

• $x=p$ and $y=q$ are perpendicular where $p$ and $q$ are constants