Coordinate Geometry Formulas

Distance Formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$:

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

Midpoint Formula

The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$:

$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Slope Formula

The slope of a line passing through $(x_1, y_1)$ and $(x_2, y_2)$:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}$

Special Cases

Type	Slope
Horizontal line	$m = 0$
Vertical line	$m$ is undefined
Rising left to right	$m > 0$
Falling left to right	$m < 0$

Equations of Lines

Slope-Intercept Form

$y = mx + b$

where $m$ = slope, $b$ = y-intercept

Point-Slope Form

$y - y_1 = m(x - x_1)$

where $m$ = slope, $(x_1, y_1)$ = point on line

Standard Form

$Ax + By = C$

where $A$, $B$, $C$ are integers and $A \geq 0$

Two-Point Form

$\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}$

Intercept Form

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

where $a$ = x-intercept, $b$ = y-intercept

Parallel and Perpendicular Lines

Parallel Lines

Same slope: $m_1 = m_2$

Perpendicular Lines

Slopes are negative reciprocals: $m_1 \cdot m_2 = -1$

$m_2 = -\frac{1}{m_1}$

Standard Equation of a Circle

Center at Origin

$x^2 + y^2 = r^2$

Center at $(h, k)$

$(x - h)^2 + (y - k)^2 = r^2$

where $(h, k)$ = center, $r$ = radius

General Form

$x^2 + y^2 + Dx + Ey + F = 0$

Center: $\left(-\frac{D}{2}, -\frac{E}{2}\right)$

Radius: $r = \sqrt{\frac{D^2}{4} + \frac{E^2}{4} - F}$

Quadratic Formula

For $ax^2 + bx + c = 0$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Discriminant

$\Delta = b^2 - 4ac$

Discriminant	Nature of Roots
$\Delta > 0$	Two distinct real roots
$\Delta = 0$	One repeated real root
$\Delta < 0$	Two complex conjugate roots

Distance from Point to Line

Distance from point $(x_0, y_0)$ to line $Ax + By + C = 0$:

$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$