📐 Coordinate Geometry Formulas

🎯 Essential Formulas

Distance Formula

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

Usage: Find distance between any two points Micro-example: Distance from $(0,0)$ to $(3,4)$ is $\sqrt{3^2 + 4^2} = 5$

Midpoint Formula

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

Usage: Find midpoint of a line segment Micro-example: Midpoint of $(2,6)$ and $(8,4)$ is $\left(\frac{2+8}{2}, \frac{6+4}{2}\right) = (5,5)$

Slope Formula

$$m = \frac{y_2-y_1}{x_2-x_1}$$

Usage: Find slope of line through two points Micro-example: Slope through $(1,3)$ and $(4,9)$ is $\frac{9-3}{4-1} = 2$

Point-Slope Form

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

Usage: Write equation of line given point and slope Micro-example: Line through $(2,5)$ with slope $3$: $y - 5 = 3(x - 2)$

Slope-Intercept Form

$$y = mx + b$$

Usage: Write equation of line given slope and y-intercept Micro-example: Line with slope $2$ and y-intercept $-1$: $y = 2x - 1$

Circle Standard Form

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

Usage: Write equation of circle given center and radius Micro-example: Circle with center $(2,-3)$ and radius $5$: $(x-2)^2 + (y+3)^2 = 25$

Circle General Form

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

Usage: Convert to standard form by completing the square Micro-example: $x^2 + y^2 - 4x + 6y - 3 = 0$ becomes $(x-2)^2 + (y+3)^2 = 16$

🔗 Relationship Formulas

Parallel Lines

$$m_1 = m_2$$

Usage: Two lines are parallel if they have the same slope Micro-example: $y = 2x + 1$ and $y = 2x - 5$ are parallel

Perpendicular Lines

$$m_1 \cdot m_2 = -1$$

Usage: Two lines are perpendicular if their slopes are negative reciprocals Micro-example: $y = 2x + 1$ and $y = -\frac{1}{2}x + 3$ are perpendicular

🎨 Transformation Formulas

Translation

$$(x,y) \to (x+h, y+k)$$

Usage: Move point $h$ units right, $k$ units up Micro-example: $(3,4)$ translated by $(2,-1)$ becomes $(5,3)$

Reflection Over X-Axis

$$(x,y) \to (x,-y)$$

Usage: Flip point over x-axis Micro-example: $(3,4)$ reflected over x-axis becomes $(3,-4)$

Reflection Over Y-Axis

$$(x,y) \to (-x,y)$$

Usage: Flip point over y-axis Micro-example: $(3,4)$ reflected over y-axis becomes $(-3,4)$

💡 Quick Reference

⚠️ Common Pitfalls

• Distance formula: Don’t forget to square the differences
• Midpoint formula: Average the coordinates, don’t add them
• Slope formula: Rise over run, not run over rise
• Circle completing square: Add to both sides of equation
• Transformations: Draw a sketch to verify direction