Analytic Geometry

• The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the formula $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$
• The midpoint $M=(x_m,y_m)$ of a line segment joining the points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the formula $M=(x_m,y_m)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Equation of a line

$m$ is the slope of the line

	Equation of a line
Two-point form	$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$	$(x_1,y_1)$ and $(x_2,y_2)$ are any two points of the line, (if $x_1\ne x_2$ then $m=\frac{y_2-y_1}{x_2-x_1}=\tan (\theta )$ is the slope)
Slope–intercept form (y)	$y=mx+y_0$	assuming non-vertical line. $y_0$ is y-intercept
Slope–intercept form (x)	$y=m(x-x_0)$	assuming non-horizontal line. $x_0$ is x-intercept
Standard Form	$Ax+By=C$
Point–slope form	$y=m(x-x_0)+y_0$	$(x_0,y_0)$ is any point of the line. (assuming non-vertical line)
Intercept form	$\frac{x}{x_0}+\frac{y}{y_0}=1$	assuming the line is not parallel to an axis and does not pass through the origin. The intercept values $x_0$ and $y_0$ are nonzero

• Angle Between Two Lines - If two nonperpendicular lines have slopes $m_1$ and $m_2$, then the tangent of the angle between the two lines is $\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|$

• If two nonvertical lines are parallel, then their slopes are equal.

  • Alternative Form: ${\ell }_1\parallel {\ell }_2⟺m_1=m_2$

• If two lines (neither horizontal nor vertical) are perpendicular, then the product of their slopes is $-1$.

  • Alternative Form: ${\ell }_1\perp {\ell }_2⟺m_1{m}_2=-1$ where neither ${\ell }_1$ nor ${\ell }_2$ is a vertical line or a horizontal line.

• Let ${\ell }_1,{\ell }_2$ be two non-vertical lines in the plane (so their slopes $m_1,m_2\in \mathbb{R}$ exist), and let ${\alpha }_1,{\alpha }_2\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ be the angles they make with the positive $x$-axis, (i.e. $m_i=\tan {\alpha }_i$ for $i=1,2$), and let $\theta \in \left[0,\frac{\pi }{2}\right]$ be the angle between them. Then:

  • ${\ell }_1\parallel {\ell }_2⟺m_1=m_2⟺{\alpha }_1={\alpha }_2$.
  • ${\ell }_1\perp {\ell }_2⟺|{\alpha }_2-{\alpha }_1|=\frac{\pi }{2}⟺m_1{m}_2=-1$.
  • ${\ell }_1/\perp {\ell }_2⟹\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|$.

• If $m=0$ is the slope of a line $\ell$, then

• A line $\ell$ is said to be horizontal if its slope is $m=0$.

• A line $\ell$ is said to be vertical if $\ell =\{(x,y)\in {\mathbb{R}}^2\mid x=a\}$ for some $a\in \mathbb{R}$, i.e. if it is parallel to the $y$-axis. It has no slope.

• A line $\ell$ is said to be oblique if it is neither horizontal nor vertical.

Angle Type	Degrees	Radians
Zero angle	$\theta =0^{\circ }$	$\theta =0$
Acute angle	$0^{\circ }<\theta <90^{\circ }$	$0<\theta <\frac{\pi }{2}$
Right angle	$\theta =90^{\circ }$	$\theta =\frac{\pi }{2}$
Obtuse angle	$90^{\circ }<\theta <180^{\circ }$	$\frac{\pi }{2}<\theta <\pi$
Straight angle	$\theta =180^{\circ }$	$\theta =\pi$
Reflex angle	$180^{\circ }<\theta <360^{\circ }$	$\pi <\theta <2\pi$
Full angle	$\theta =360^{\circ }$	$\theta =2\pi$