🔺 Similar Triangle Stacks

Similar triangles often appear in “stacked” configurations with parallel lines or homothety. Master these patterns for efficient problem solving.

🎯 Recognition Cues

Key Triggers

  • Parallel lines - Look for multiple parallel lines creating similar triangles
  • Homothety - Look for scaling transformations or similar figures
  • Ladder problems - Look for figures that look like ladders or steps
  • Gap problems - Look for missing segments between similar triangles

Common Setups

  • Trapezoids with parallel bases
  • Triangles with parallel sides
  • Figures with multiple similar triangles
  • Scaling transformations

🧩 Solution Template

Step 1: Identify Similarity

  • Look for parallel lines or equal angles
  • Check if triangles are similar by AA, SAS, or SSS
  • Mark the similarity relationship

Step 2: Find the Ratio

  • Identify corresponding sides
  • Calculate the similarity ratio
  • Use the ratio to find unknown lengths

Step 3: Apply the Ratio

  • Use the similarity ratio to find missing lengths
  • Apply to all corresponding sides
  • Check that ratios are consistent

Step 4: Verify

  • Ensure all triangles are actually similar
  • Check that ratios are consistent
  • Verify the final answer

🔍 Worked Example

Problem: In trapezoid $ABCD$ with $AB \parallel CD$, $AB = 8$, $CD = 4$, and $AD = 6$. If $E$ is the midpoint of $AD$ and $F$ is the midpoint of $BC$, find $EF$.

Solution: Step 1: Identify similarity

  • $AB \parallel CD$ creates similar triangles
  • $\triangle AEB \sim \triangle CED$ by AA (corresponding angles equal)

Step 2: Find the ratio

  • Similarity ratio: $\frac{AB}{CD} = \frac{8}{4} = 2$
  • This means $AE:CE = 2:1$

Step 3: Apply the ratio

  • Since $E$ is midpoint of $AD$, $AE = 3$
  • From similarity: $CE = \frac{AE}{2} = \frac{3}{2} = 1.5$
  • Therefore, $AC = AE + CE = 3 + 1.5 = 4.5$

Step 4: Find $EF$

  • $EF$ is the midsegment of trapezoid $ABCD$
  • Midsegment length = $\frac{AB + CD}{2} = \frac{8 + 4}{2} = 6$

Answer: $EF = 6$

⚠️ Common Pitfalls

Pitfall: Assuming similarity without proof

  • Fix: Use AA, SAS, or SSS criteria to establish similarity

Pitfall: Wrong similarity ratio

  • Fix: Make sure you’re using corresponding sides in the same order

Pitfall: Forgetting about midsegments

  • Fix: In trapezoids, midsegments have special length formulas

Pitfall: Wrong order in similarity statements

  • Fix: Always match corresponding vertices: $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$

🔗 Related Patterns

💡 Quick Reference

Similarity Criteria

  • AA: Two angles equal (third automatically equal)
  • SAS: Two sides proportional + included angle equal
  • SSS: All three sides proportional

Common Ratios

  • Parallel lines: Create similar triangles with constant ratio
  • Homothety: Scale factor determines similarity ratio
  • Midsegments: Special length formulas in trapezoids

Solution Strategy

  • Identify: Look for parallel lines or equal angles
  • Prove: Use similarity criteria
  • Calculate: Find similarity ratio
  • Apply: Use ratio to find unknown lengths

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