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Understanding Similar Polygons and Their Properties

Apr 16, 2025

Lecture on Similar Polygons

Definition and Properties

  • Similar Polygons: Two polygons are similar if and only if:
    • Their corresponding angles are congruent.
    • Their corresponding sides are proportional.
  • Shape vs. Size: Similar polygons have the same shape but not the same size.
    • Angles remain the same.
    • Sides are scaled proportionately.

Example of Quadrilaterals

  • Consider quadrilateral ABCD and quadrilateral EFGH.
  • Use the "∼" symbol to denote similarity.
  • Importance of letter order:
    • A and E should correspond, B and F, C and G, D and H.
  • Congruent Angles:
    • (\angle A \cong \angle E )
    • (\angle B \cong \angle F )
    • (\angle C \cong \angle G )
    • (\angle D \cong \angle H )
  • Proportional Sides:
    • (\frac{AB}{EF} = \frac{BC}{FG} = \frac{CD}{GH} = \frac{AD}{EH})

Example with Triangles

  • Triangle ABC is similar to triangle DEF.
  • Congruent Angles:
    • (\angle A \cong \angle D )
    • (\angle B \cong \angle E )
    • (\angle C \cong \angle F )
  • Proportional Sides:
    • (\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD})
  • Orientation may differ, but similarity can be confirmed by statements.

Scale Factor

  • Defined as the ratio of corresponding side lengths.
  • Example: If (AB = 6) and (DE = 3), then scale factor (ABC \to DEF) is (\frac{6}{3} = 2:1).
  • Reverse scale factor (DEF \to ABC) is (1:2).

Determining Similarity and Scale Factor

  • Example: Determine if two triangles are similar.
  • Check for congruent angles and proportional sides.
  • If similar, write a similarity statement and find the scale factor.
  • Example triangles: ABC and HFJ.
    • Congruent angles and proportional sides confirmed.
    • Similarity statement: (\triangle ABC \sim \triangle HFJ).
    • Scale factor: (2:1).

Solving for Variables

  • Use proportions to solve for unknowns in similar polygons.
  • Example with pentagons: Find x and y given similarity.
    • Set up proportions using known and unknown side lengths.
    • Solve for unknowns.

Calculating Perimeter and Scale Factor

  • Find perimeter of similar polygons using known side lengths.
  • Compare perimeters to confirm similarity and calculate scale factor.
  • Example results:
    • Pentagon ABCDE perimeter: 26
    • Pentagon RSTUV perimeter: 45.5
    • Scale factor: (\frac{7}{4}) or verified by perimeter ratio.

These notes summarize the key points about similar polygons, including definitions, properties, examples, and calculations involved in determining similarity and scale factors.

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