📐 Coordinate Geometry

When pure geometry becomes complex, coordinate methods often provide the most direct path to solutions. Master these essential techniques for AMC success.

🎯 Key Concepts

Line Equations

Slope-Intercept Form: $y = mx + b$ Point-Slope Form: $y - y_1 = m(x - x_1)$ Two-Point Form: $\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}$

Key Formulas:

  • Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$
  • Midpoint: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
  • Distance: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Circle Equations

Standard Form: $(x - h)^2 + (y - k)^2 = r^2$ (center at $(h,k)$, radius $r$) General Form: $x^2 + y^2 + Dx + Ey + F = 0$

Key Properties:

  • Center: $(-D/2, -E/2)$
  • Radius: $r = \sqrt{D^2/4 + E^2/4 - F}$

Shoelace Formula

Purpose: Find area of polygon from vertex coordinates Formula: $A = \frac{1}{2}|x_1y_2 + x_2y_3 + \cdots + x_ny_1 - y_1x_2 - y_2x_3 - \cdots - y_nx_1|$

When to Use: Given vertex coordinates, need area AMC Appearance: Common in problems with specific coordinates

🔍 Micro-Examples

Line Example

Line through $(1,2)$ and $(3,4)$:

  • Slope: $m = \frac{4-2}{3-1} = 1$
  • Equation: $y - 2 = 1(x - 1)$ or $y = x + 1$

Circle Example

Circle with center $(2,3)$ and radius 5:

  • Equation: $(x-2)^2 + (y-3)^2 = 25$

Shoelace Example

Triangle with vertices $(0,0)$, $(3,0)$, $(0,4)$:

  • $A = \frac{1}{2}|0 \cdot 0 + 3 \cdot 4 + 0 \cdot 0 - 0 \cdot 3 - 0 \cdot 0 - 4 \cdot 0| = \frac{1}{2}|12| = 6$

⚠️ Common Traps

Pitfall: Wrong slope formula

  • Fix: $m = \frac{y_2 - y_1}{x_2 - x_1}$, not $\frac{x_2 - x_1}{y_2 - y_1}$

Pitfall: Wrong circle center formula

  • Fix: Center is $(-D/2, -E/2)$, not $(D/2, E/2)$

Pitfall: Wrong shoelace formula order

  • Fix: Go around polygon in order, don’t skip vertices

Pitfall: Forgetting absolute value in shoelace

  • Fix: Area is always positive, use absolute value

🎯 AMC-Style Worked Example

Problem: Triangle $ABC$ has vertices $A(0,0)$, $B(4,0)$, and $C(2,3)$. Find the area of the triangle.

Solution: Using the shoelace formula: $A = \frac{1}{2}|x_1y_2 + x_2y_3 + x_3y_1 - y_1x_2 - y_2x_3 - y_3x_1|$

Substituting the coordinates: $A = \frac{1}{2}|0 \cdot 0 + 4 \cdot 3 + 2 \cdot 0 - 0 \cdot 4 - 0 \cdot 2 - 3 \cdot 0|$ $A = \frac{1}{2}|0 + 12 + 0 - 0 - 0 - 0|$ $A = \frac{1}{2}|12|$ $A = 6$

Answer: $A = 6$

🔗 Related Topics

💡 Quick Reference

Essential Formulas

  • Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$
  • Distance: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
  • Midpoint: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
  • Circle: $(x-h)^2 + (y-k)^2 = r^2$

Shoelace Formula

  • Triangle: $A = \frac{1}{2}|x_1y_2 + x_2y_3 + x_3y_1 - y_1x_2 - y_2x_3 - y_3x_1|$
  • General: $A = \frac{1}{2}|\sum_{i=1}^n x_iy_{i+1} - \sum_{i=1}^n y_ix_{i+1}|$ (with $x_{n+1} = x_1$)

Common Applications

  • Lines: Finding equations, intersections, parallel/perpendicular
  • Circles: Finding equations, intersections, tangency
  • Areas: Shoelace formula for any polygon
  • Distances: Between points, from point to line

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