📐 Area Ratio in Triangle

Area ratios in triangles are powerful tools for finding relationships between different parts of a triangle. Master these patterns for efficient problem solving.

🎯 Recognition Cues

Key Triggers

  • Medians - Look for lines from vertices to midpoints of opposite sides
  • Cevians - Look for lines from vertices to opposite sides
  • Shared heights - Look for triangles with the same height
  • Shared bases - Look for triangles with the same base

Common Setups

  • Triangles with medians or cevians
  • Figures with shared heights or bases
  • Area comparisons
  • Ratio problems

🧩 Solution Template

Step 1: Identify the Configuration

  • Determine if you have medians, cevians, or shared heights/bases
  • Mark given lengths and relationships
  • Identify what ratios you need to find

Step 2: Apply Area Ratio Principles

  • Shared height: Area ratio = base ratio
  • Shared base: Area ratio = height ratio
  • Medians: Divide triangle into six equal areas
  • Cevians: Use area ratios based on side splits

Step 3: Set Up the Ratio

  • Write the area ratio equation
  • Substitute known values
  • Solve for the unknown

Step 4: Verify

  • Check that the ratio makes geometric sense
  • Ensure all areas are positive
  • Verify the final answer

🔍 Worked Example

Problem: In triangle $ABC$, $D$ is the midpoint of $BC$ and $E$ is the midpoint of $AC$. If the area of triangle $ABC$ is 24, find the area of triangle $ADE$.

Solution: Step 1: Identify configuration

  • $D$ is midpoint of $BC$ (median)
  • $E$ is midpoint of $AC$ (median)
  • Area of $ABC = 24$

Step 2: Apply area ratio principles

  • Medians divide triangle into six equal areas
  • Each small triangle has area $\frac{24}{6} = 4$
  • Triangle $ADE$ is one of these small triangles

Step 3: Calculate area

  • Area of $ADE = 4$

Step 4: Verify

  • Check that medians create six equal areas ✓
  • Area is positive ✓
  • Answer makes sense ✓

Answer: Area of $ADE = 4$

⚠️ Common Pitfalls

Pitfall: Wrong area ratio formula

  • Fix: Remember that shared height means area ratio = base ratio

Pitfall: Confusing medians and cevians

  • Fix: Medians go to midpoints, cevians go to any point on opposite side

Pitfall: Forgetting about shared heights

  • Fix: Look for triangles with the same height

Pitfall: Wrong median area division

  • Fix: Medians divide triangle into six equal areas

🔗 Related Patterns

💡 Quick Reference

Area Ratio Principles

  • Shared height: Area ratio = base ratio
  • Shared base: Area ratio = height ratio
  • Medians: Divide triangle into six equal areas
  • Cevians: Use area ratios based on side splits

Common Ratios

  • Median areas: Each small triangle = $\frac{1}{6}$ of total area
  • Cevian areas: Based on ratio of side splits
  • Similar triangles: Area ratio = (length ratio)²

Solution Strategy

  • Identify: Look for medians, cevians, or shared heights/bases
  • Apply: Use appropriate area ratio principle
  • Calculate: Solve for unknown areas or ratios
  • Verify: Check that answer makes geometric sense

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