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Understanding Similar Polygons and Ratios

Feb 25, 2025

Chapter 8: Similarity

8.1 Similar Polygons

  • Definition: Similar polygons have the same shape but different sizes.
    • All corresponding angles are congruent.
    • Corresponding sides are in proportion, not congruent.

Example with Triangles

  • Given: Two triangles are similar.
    • All corresponding angles are congruent.
    • Sides are proportional.
    • Example ratios from smaller to larger triangle:
      • Smallest side: 3 : 6
      • Medium side: 4 : 8
      • Largest side: 5 : 10
    • Observation: Ratios are equivalent (e.g., 3/6 = 1/2, 4/8 = 1/2, 5/10 = 1/2).

Writing Similarity Statements

  • Notation: Similar symbol is like congruent symbol without the equal sign.
  • Naming Example:
    • Small triangle: ABC
    • Large triangle: DEF
    • Similarity statement: ΔABC ~ ΔDEF
    • Ensure corresponding angles match: A with E, B with F, C with D.

Side Ratios

  • Side ratio is the proportion between corresponding sides.
  • Example: Side ratio is 1/2 (e.g., 3/6, 4/8, 5/10).
  • Also known as similarity ratio or perimeter ratio.
    • Scale factor tells how much larger one figure is compared to another.

Solving for Sides

  • Ensure proper side matching using triangle naming.
  • Proportion setup: X/17 = 10/14
  • Solve using cross-multiplication:
    • 14X = 170
    • X = 170/14 = 85/7 or approximately 12.14

Additional Examples

Squares

  • Side ratio: 2 : 3
  • Area ratio: 2² : 3² = 4 : 9
  • Pattern: Side ratio squared equals area ratio.
    • To find side ratio from area ratio, take the square root.

Application with Figures

  • Given two similar figures with side ratio 3 : 4:
    • Incorrect setup: 3/4 = 20/X (wrong because sides and areas aren't directly proportional)
    • Correct setup via area ratio: 9/16 = 20/X
    • Solve proportion: 9X = 320, X = 320/9 = 35.5 square feet

Practice Example with Socks

  • Given side ratio 4 : 5:
    • Find area ratio: 4² : 5² = 16 : 25
    • Setup proportion with areas: 16/25 = X/30
    • Solve: X = 480/25 = 19.2 square feet
  • Note: Ensure similar shapes are stated for correct ratio applications.
  • Key Concepts: Similarity involves proportional reasoning, and ratios of sides are foundational to understanding similarity and area calculations.

End of Chapter 8.1. Next session will continue with further concepts.

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