🎯 Extremal Geometry Paths

Extremal geometry problems involve finding maximum or minimum values of geometric quantities. Master these techniques for AMC success.

🎯 Recognition Cues

Key Triggers

  • Minimum distance - Look for shortest path problems
  • Maximum area - Look for largest area problems
  • Reflection method - Look for problems involving reflections
  • Optimization - Look for problems asking for extreme values

Common Setups

  • Shortest path between points
  • Largest area with given constraints
  • Minimum perimeter problems
  • Reflection across lines

🧩 Solution Template

Step 1: Identify the Extremal Problem

  • Determine if you need maximum or minimum
  • Identify the constraint
  • Look for reflection opportunities

Step 2: Apply Reflection Method

  • Reflect one point across the constraint line
  • Draw line from other point to reflected point
  • Find intersection with constraint line

Step 3: Calculate the Extremal Value

  • Use the reflection method result
  • Apply distance or area formulas
  • Solve for the extremal value

Step 4: Verify

  • Check that the solution satisfies constraints
  • Ensure the extremal value is correct
  • Verify the final answer

🔍 Worked Example

Problem: Point $A$ is at $(0,0)$ and point $B$ is at $(4,2)$. Find the shortest path from $A$ to $B$ that touches the line $y = 1$.

Solution: Step 1: Identify extremal problem

  • Need shortest path from $A$ to $B$ touching line $y = 1$
  • This is a minimum distance problem

Step 2: Apply reflection method

  • Reflect point $A$ across line $y = 1$
  • Reflected point $A’$ is at $(0,2)$
  • Draw line from $A’$ to $B$

Step 3: Calculate extremal value

  • Line from $A’(0,2)$ to $B(4,2)$ has equation $y = 2$
  • Intersection with $y = 1$ is at $(2,1)$
  • Shortest path: $A$ to $(2,1)$ to $B$

Step 4: Find distance

  • Distance from $A$ to $(2,1)$: $\sqrt{(2-0)^2 + (1-0)^2} = \sqrt{5}$
  • Distance from $(2,1)$ to $B$: $\sqrt{(4-2)^2 + (2-1)^2} = \sqrt{5}$
  • Total distance: $2\sqrt{5}$

Answer: $2\sqrt{5}$

⚠️ Common Pitfalls

Pitfall: Wrong reflection setup

  • Fix: Reflect the correct point across the constraint line

Pitfall: Forgetting about constraints

  • Fix: Make sure the solution satisfies all given constraints

Pitfall: Wrong distance calculation

  • Fix: Use the distance formula correctly

Pitfall: Forgetting about the reflection method

  • Fix: Use reflection method for shortest path problems

🔗 Related Patterns

💡 Quick Reference

Reflection Method

  • Purpose: Find shortest path touching a line
  • Steps: Reflect one point, draw line to other point, find intersection
  • Result: Shortest path through the intersection point

Common Extremal Problems

  • Shortest path: Use reflection method
  • Largest area: Use calculus or geometric properties
  • Minimum perimeter: Use geometric properties
  • Maximum distance: Use geometric properties

Solution Strategy

  • Identify: Look for extremal problems
  • Apply: Use reflection method or other techniques
  • Calculate: Find the extremal value
  • Verify: Check that solution satisfies constraints

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