📐 Coordinate Geometry Topics

🎯 Core Subtopics

Distance and Midpoint

  • Distance Formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
  • Midpoint Formula: $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
  • Micro-example: Distance from $(3,4)$ to $(0,0)$ is $\sqrt{3^2 + 4^2} = 5$
  • Trap: Forgetting to square the differences before adding

Slope and Linear Equations

  • Slope Formula: $m = \frac{y_2-y_1}{x_2-x_1}$
  • Point-Slope Form: $y - y_1 = m(x - x_1)$
  • Slope-Intercept Form: $y = mx + b$
  • Micro-example: Line through $(2,3)$ with slope $-\frac{1}{2}$: $y - 3 = -\frac{1}{2}(x - 2)$
  • Trap: Mixing up rise and run in slope calculation

Parallel and Perpendicular Lines

  • Parallel Lines: Same slope ($m_1 = m_2$)
  • Perpendicular Lines: Negative reciprocal slopes ($m_1 \cdot m_2 = -1$)
  • Micro-example: Line perpendicular to $y = 2x + 1$ has slope $-\frac{1}{2}$
  • Trap: Forgetting to flip the sign when finding perpendicular slope

Circles

  • Standard Form: $(x-h)^2 + (y-k)^2 = r^2$ where $(h,k)$ is center
  • General Form: $x^2 + y^2 + Dx + Ey + F = 0$
  • Micro-example: Circle with center $(2,-1)$ and radius $3$: $(x-2)^2 + (y+1)^2 = 9$
  • Trap: Forgetting to change signs when completing the square

Parabolas

  • Vertex Form: $y = a(x-h)^2 + k$ where $(h,k)$ is vertex
  • Standard Form: $y = ax^2 + bx + c$
  • Micro-example: Parabola with vertex $(1,2)$ opening up: $y = (x-1)^2 + 2$
  • Trap: Confusing vertex coordinates with axis of symmetry

Transformations

  • Translations: $(x,y) \to (x+h, y+k)$
  • Reflections: Over x-axis: $(x,y) \to (x,-y)$; Over y-axis: $(x,y) \to (-x,y)$
  • Micro-example: Reflecting $(3,4)$ over x-axis gives $(3,-4)$
  • Trap: Forgetting to change the sign of the coordinate being reflected

🚨 Common Traps

  1. Sign Errors: Mixing up positive/negative in distance formula
  2. Slope Confusion: Rise over run vs. run over rise
  3. Circle Completing Square: Forgetting to add to both sides
  4. Transformation Direction: Confusing which coordinate changes
  5. Midpoint Formula: Adding instead of averaging coordinates

💡 Quick Tips

  • Distance: Always use absolute value for differences
  • Slope: Vertical lines have undefined slope, horizontal lines have slope 0
  • Circles: Complete the square to find center and radius
  • Transformations: Draw a quick sketch to verify
  • Midpoint: Average the x-coordinates and y-coordinates separately