Defining Similarity

HSG.SRT.A.2

High School Geometry

Learning Objectives

• Define similarity transformations as sequences of rigid motions followed by dilations.
• Use the definition of similarity to determine if two figures are similar.
• Explain why similar triangles have equal corresponding angles and proportional corresponding sides.
• Distinguish between congruence and similarity.
• Apply similarity transformations to verify that two figures are similar.

Review: Congruence

• Two figures are congruent if one can be mapped onto the other using Rigid Motions.
• Rigid motions (Translation, Rotation, Reflection) preserve:
  • Distance (Size)
  • Angle Measure (Shape)

Introducing: Similarity Transformation

A similarity transformation is a sequence consisting of:

1. Rigid Motion(s) (to align position and orientation)
2. A Dilation (to match the size)

Two figures are similar if a similarity transformation maps one onto the other.

Mapping Similarity

1. Pre-image (Original)
2. Translate & Rotate (Rigid Motions)
3. Dilate (Scale by k)
4. Image (Final match)

Congruence is a Special Case

• Similarity Transformation with scale factor k = 1 is a Congruence Transformation.
• Every congruent pair is also similar.
• Not every similar pair is congruent.

Comparison Table

Similarity in Triangles

If $ riangle ABC \sim riangle DEF$ (with factor k):

1. Corresponding angles are equal:
   
2. Corresponding sides are proportional:
   

Why are Angles Preserved?

• Rigid Motions preserve angles by definition.
• Dilations preserve angles (lines map to parallel lines).
• Conclusion: A similarity transformation (the sequence) MUST preserve angles.

Why are Sides Proportional?

• Rigid Motions keep length ratios at 1:1.
• Dilations multiply ALL lengths by factor k.
• Conclusion: Every side in the image is k times the corresponding side in the original.

Determining Similarity

How to verify if two figures are similar:

1. Identify corresponding points and sides.
2. Determine rigid motions to align orientation.
3. Calculate scale factor .
4. Check: Is k consistent for ALL corresponding sides?

Example 1: Similar Rectangles

• Rect A:
• Rect B:

Ratios:

• Width: 6/4 = 1.5
• Height: 9/6 = 1.5

Result: Similar with k = 1.5

Example 2: Not Similar

• Rect A:
• Rect C:

Ratios:

• Width: 5/4 = 1.25
• Height: 9/6 = 1.5

Result: NOT similar. No single k works.

Example 3: Triangle Mapping

• Step 1: Translate to align vertex A with D.
• Step 2: Rotate to align side AB with DE.
• Step 3: Dilate from D to match size.

Similarity for General Figures

The transformation definition works for EVERYTHING, not just triangles!

• All circles are similar. (Dilate one radius to match another).
• All squares are similar.
• All regular hexagons are similar.

Real-World Similarity

• Maps: Scale models of the terrain.
• Architectural Models: Smaller versions of buildings.
• Photography: Enlarging or reducing a print.
• Microscopes: Dilating small objects to see detail.

Watch Out: Colloquial Use

Mistake: "These triangles are similar because they both have a right angle."

Correction: Similarity is an EXACT relationship.

• All angles must match.
• All sides must be proportional.
• Being "kind of alike" is not enough for geometry.

Watch Out: Orientation

Mistake: "They aren't similar because one is upside down."

Correction: Similarity transformations include Reflections and Rotations.

• Orientation does not affect similarity.
• Check the angles and ratios, not the "tilt."

Summary

1. Similarity Transformation: Sequence of Rigid Motions then Dilation.
2. Definition: Figure A ~ Figure B if a transformation maps A onto B.
3. Triangle Rule: Similar triangles have equal angles and proportional sides.
4. General Rule: Similarity applies to all shapes (circles, polygons, etc.).

Next Steps

AA Criterion for Similarity

Do we really need to check every side and every angle?
(Spoiler: For triangles, we only need two angles!)

Similarity in the Coordinate Plane

Using the distance formula to prove similarity algebraically.