Formulas from plane analytic geometry

Distance $d$ between two points $P_1(x_1 \textrm{ , } y_1)$ and $P_2(x_2 \textrm{ , } y_2)$

fig 1
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Slope $m$ of line joining two points $P_1(x_1 \textrm{ , } y_1)$ and $P_2(x_2 \textrm{ , } y_2)$

$m = \frac{y_2 - y_1}{x_2 - y_1} = \textrm { tan } \theta$

Equation of line joining two points $P_1(x_1 \textrm{ , } y_1)$ and $P_2(x_2 \textrm{ , } y_2)$

$\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - y_1} = m \qquad \qquad \qquad \qquad \qquad \qquad \textrm{ or } \qquad \qquad \qquad \qquad \qquad \qquad y - y_1 = m(x - x_1) \\ y = mx + b$
where $b = y_1 - mx_1 = \frac{x_2y_1 - x_1y_2}{x_2 - x_1}$ is the intercept on the $y$ axis, i.e. $y$ intercept.

Equation of line in terms of $x$ intercept $a \ne 0$ and $y$ intercept $b \ne 0$

fig 2
$\frac{x}{a} + \frac{y}{b} = 1$

Normal form for equation of line

$x \textrm{ cos } \alpha + y \textrm{ sin } \alpha = p$
where $p$ = perpendicular distance from origin $O$ to line
and α = angle of inclination of perpendicular with
with positive $x$ axis.
fig 3

General equation of line

$Ax + By + C = 0$

Distance from point $(x_1 \textrm{ , } y_1)$ to line $Ax + By + C = 0$

$\frac{Ax_1 + By_1 + C}{\pm \sqrt{A^2 + B^2}}$
where the sign is chosen so that the distance is nonnegative.

Angle $\psi$ between two lines having slopes $m_1$ and $m_2$

$\textrm{ tan } \psi = \frac{m_2 - m_1}{1 + m_1m_2}$
Lines are parallel or coincident if and only if $m_1 = m_2.$
Lines are perpendicular of and only if $m_2 = -\frac{1}{m_1}.$
fig 4

Area of triangle with vertices at $(x_1 \textrm{ , } y_1) \textrm{ , } (x_2 \textrm{ , } y_2) \textrm{ , } (x_3 \textrm{ , } y_3)$

Area
$= \pm \frac{1}{2} \left| \begin{array}{ccc} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{array} \right| \\ = \pm y_2 ( x_1y_2 + y_1x_3 + y_3x_2 - y_2x_3 - y_1x_2 - x_1y_3 )$
where the sign is chosen so that the area is nonnegative. If the area is zero the points all lie on a line.
fig 5

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Formulas from plane analytic geometry

Distance between two points and

Slope of line joining two points and

Equation of line joining two points and

Equation of line in terms of intercept and intercept