Lesson 23 — Relative Position of Lines

Criteria and formulas

Parallelism and perpendicularity

Given lines $r: y = m_r x + n_r$ and $s: y = m_s x + n_s$:

Cases with vertical lines: $x = a$ is vertical, $y = b$ is horizontal. Verticals to each other are parallel; vertical with horizontal are perpendicular.

Angle between lines

$\tan\theta = \left|\frac{m_r - m_s}{1 + m_r m_s}\right|$

(Assuming neither line is vertical.)

Point-to-line distance

Point $P_0 = (x_0, y_0)$ and line $Ax + By + C = 0$:

$d(P_0, r) = \frac{|A x_0 + B y_0 + C|}{\sqrt{A^2 + B^2}}$

Distance between parallel lines

For $r: Ax + By + C_1 = 0$ and $s: Ax + By + C_2 = 0$: $d(r, s) = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$

Perpendicular bisector

Set of points equidistant from $A$ and $B$. It is the line perpendicular to $\overline{AB}$ through the midpoint.