Plane Geometry: Criteria of Similar Triangles (AA, SSS, SAS) and Area/Perimeter Ratios
Geometric similarity balances shape conservation with proportional scaling by a constant factor k. Two triangles are similar (∼ ) if their corresponding angles are congruent and their corresponding sides are proportional. Unlike general polygons, triangles possess remarkable efficiency: fulfilling angle criteria alone or side proportions alone guarantees similarity. We examine the three standard criteria: 1) Angle-Angle (AA). 2) Side-Side-Side (SSS). 3) Side-Angle-Side (SAS). Furthermore, we prove that the ratio of perimeters equals the scale factor k, whereas the ratio of areas equals k².
1. The Three Similarity Criteria
Conditions for triangle similarity:
- AA Criterion (Angle-Angle): If two angles of one triangle are congruent to two angles of another, the triangles are similar.
- SSS Criterion (Side-Side-Side): If all three pairs of corresponding sides are proportional: AB/DE = BC/EF = AC/DF = k.
- SAS Criterion (Side-Angle-Side): If two pairs of corresponding sides are proportional and their included angles are congruent.
2. Perimeter and Area Relationships
Scale factor k relationships:
- Perimeter Ratio: Perimeter(Δ1) / Perimeter(Δ2) = k.
- Area Ratio: Area(Δ1) / Area(Δ2) = k².
- Scale Factor (k):
• k > 1: Enlargement.
• 0 < k < 1: Reduction.
• k = 1: Congruence.
Geometric Problems:
Solution: Area ratio = k² = 36 / 81 = 4 / 9 ⇒ k = 2 / 3.
Perimeter ratio = 24 / P2 = 2 / 3 ⇒ P2 = (24 × 3) / 2 = 36 cm.
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