Lines and Points Formulas

Parameter	Formula
zero point	$\frac{-b}{a}$
y-intercept	$b$
value x for given y	$\frac{y-b}{a}$
angle between line and X-axis	$\arctan(a)$
angle between line and Y-axis	$\frac{1}{2}\pi - \arctan(a)$ or $\arctan(\frac{1}{a})$
angle between two lines	$\arctan\left(\left\vert \frac{a_2-a_1}{1 + a_1a_2} \right\vert\right)$
two lines intersection point	$$\left( \frac{b_2-b_1}{a_1-a_2}, \frac{a_1\cdot[b_2-b_1]}{a_1-a_2} + b_1 \right)$$
line segment center	$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$
distance between two parallel lines	$$\frac{\vert b_2-b_1 \vert}{\sqrt{1+a^2}}$$
distance between line and point	$$\frac{\vert ax_p-y_p+b \vert}{\sqrt{1+a^2}}$$
distance between two points	$\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
derivative	$$f'(x) = a$$
indefinite integral	$$\int f\ dx = \frac{1}{2}ax^2 + bx + C$$
definite integral	$$\overset{\quad u}{\underset{l}{\int}} f\ dx = \frac{a(u^2-l^2)}{2} + (u-l)b$$
normal vector length	$\|\|v\|\| = \sqrt{1+a^2}$
area of triangle $\Delta_{z0b}$ ($a \cdot b \neq 0$)	$$A_{\Delta z0b} = \frac{b^2}{\|2a\|}$$

Lengths

for line $f(x) = mx + i$

Length	Formula
line length in range $\langle x; x \pm r \rangle$ in X-axis	$$\sqrt{r^2 + (rm)^2}$$
line length in range $\langle y; y \pm r \rangle$ in Y-axis	$$\sqrt{\left(\frac{r}{m}\right)^2 + r^2}$$
length of X-axis range in which line is $\ell$ in length	$$\frac{\ell}{\sqrt{1+m^2}}$$
length of Y-axis range in which line is $\ell$ in length	$$\frac{\ell}{\sqrt{1+\frac{1}{m^2}}}$$

or:

$||v|| = \sqrt{1+m^2}$

Length	Formula
line length in range $⟨ x; x \pm r ⟩$ in X-axis	$\sqrt{r^{2} + (r m)^{2}}$
line length in range $⟨ y; y \pm r ⟩$ in Y-axis	$\sqrt{{(\frac{r}{m})}^{2} + r^{2}}$
length of X-axis range in which line is $ℓ$ in length	$\frac{ℓ}{\sqrt{1 + m^{2}}}$
length of Y-axis range in which line is $ℓ$ in length	$\frac{ℓ}{\sqrt{1 + \frac{1}{m^{2}}}}$

Parameter	Formula
zero point	$\frac{- b}{a}$
y-intercept	$b$
value x for given y	$\frac{y - b}{a}$
angle between line and X-axis	$\arctan (a)$
angle between line and Y-axis	$\frac{1}{2} π - \arctan (a)$ or $\arctan (\frac{1}{a})$
angle between two lines	$\arctan (\| \frac{a_{2} - a_{1}}{1 + a_{1} a_{2}} \|)$
two lines intersection point	$(\frac{b_{2} - b_{1}}{a_{1} - a_{2}}, \frac{a_{1} \cdot [b_{2} - b_{1}]}{a_{1} - a_{2}} + b_{1})$
line segment center	$(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$
distance between two parallel lines	$\frac{\| b_{2} - b_{1} \|}{\sqrt{1 + a^{2}}}$
distance between line and point	$\frac{\| a x_{p} - y_{p} + b \|}{\sqrt{1 + a^{2}}}$
distance between two points	$\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$
derivative	$f^{'} (x) = a$
indefinite integral	$\int f d x = \frac{1}{2} a x^{2} + b x + C$
definite integral	$\overset{u}{\int_{l}} f d x = \frac{a (u^{2} - l^{2})}{2} + (u - l) b$
normal vector length	$\| \| v \| \| = \sqrt{1 + a^{2}}$
area of triangle $Δ_{z 0 b}$ ( $a \cdot b \neq 0$ )	$A_{Δ z 0 b} = \frac{b^{2}}{\| 2 a \|}$