🔄 Transformations

Transformations are powerful tools in AMC 12 geometry that can simplify complex problems by moving figures to more convenient positions.

🎯 Key Concepts

Reflections

Definition: Flip figure across a line (mirror) Properties:

  • Preserves distances and angles
  • Creates congruent figures
  • Useful for minimizing distances (reflection method)

Common Applications:

  • Finding shortest paths
  • Proving equal distances
  • Creating symmetric figures

Rotations

Definition: Turn figure around a point by given angle Properties:

  • Preserves distances and angles
  • Creates congruent figures
  • Useful for proving equal angles

Common Applications:

  • Proving equal angles
  • Creating similar figures
  • Simplifying coordinate problems

Translations

Definition: Slide figure in given direction by given distance Properties:

  • Preserves distances and angles
  • Creates congruent figures
  • Useful for parallel line problems

Common Applications:

  • Proving parallel lines
  • Creating similar figures
  • Simplifying coordinate problems

Homothety (Scaling)

Definition: Scale figure by given factor from given center Properties:

  • Preserves angles, scales distances
  • Creates similar figures
  • Useful for ratio problems

Common Applications:

  • Proving similarity
  • Finding ratios
  • Creating tangent circles

🔍 Micro-Examples

Reflection Example

Reflect point $(3,4)$ across line $y = x$:

  • Swap coordinates: $(4,3)$
  • This is the reflection

Rotation Example

Rotate point $(1,0)$ by $90°$ counterclockwise around origin:

  • New coordinates: $(0,1)$
  • This is the rotation

Translation Example

Translate point $(2,3)$ by vector $(1,2)$:

  • New coordinates: $(2+1, 3+2) = (3,5)$
  • This is the translation

Homothety Example

Scale triangle with vertices $(0,0)$, $(1,0)$, $(0,1)$ by factor 2 from origin:

  • New vertices: $(0,0)$, $(2,0)$, $(0,2)$
  • This is the homothety

⚠️ Common Traps

Pitfall: Wrong reflection line

  • Fix: Remember that reflection across $y = x$ swaps coordinates

Pitfall: Wrong rotation direction

  • Fix: Counterclockwise is positive, clockwise is negative

Pitfall: Wrong homothety center

  • Fix: All points scale from the same center

Pitfall: Forgetting transformation properties

  • Fix: Reflections and rotations preserve distances, homothety scales them

🎯 AMC-Style Worked Example

Problem: Point $A$ is at $(2,3)$ and point $B$ is at $(4,1)$. Find the coordinates of the reflection of $B$ across the line through $A$ and the origin.

Solution: First, find the equation of the line through $(0,0)$ and $(2,3)$: Slope = $\frac{3-0}{2-0} = \frac{3}{2}$ Equation: $y = \frac{3}{2}x$

Now, find the reflection of $(4,1)$ across this line:

  • Distance from $(4,1)$ to line: $d = \frac{|3(4) - 2(1)|}{\sqrt{3^2 + 2^2}} = \frac{|12 - 2|}{\sqrt{13}} = \frac{10}{\sqrt{13}}$
  • Reflection point: $(4,1) - 2d \cdot \frac{(3,2)}{\sqrt{13}} = (4,1) - \frac{20}{\sqrt{13}} \cdot \frac{(3,2)}{\sqrt{13}} = (4,1) - \frac{20}{13}(3,2) = (4,1) - (\frac{60}{13}, \frac{40}{13}) = (\frac{52}{13}, \frac{13}{13}) - (\frac{60}{13}, \frac{40}{13}) = (-\frac{8}{13}, -\frac{27}{13})$

Answer: $(-\frac{8}{13}, -\frac{27}{13})$

🔗 Related Topics

💡 Quick Reference

Transformation Properties

  • Reflections: Preserve distances and angles, create congruent figures
  • Rotations: Preserve distances and angles, create congruent figures
  • Translations: Preserve distances and angles, create congruent figures
  • Homothety: Preserve angles, scale distances, create similar figures

Common Applications

  • Reflections: Shortest paths, equal distances
  • Rotations: Equal angles, similar figures
  • Translations: Parallel lines, similar figures
  • Homothety: Similar figures, ratio problems

Coordinate Formulas

  • Reflection across $y = x$: $(x,y) \rightarrow (y,x)$
  • Rotation by $90°$: $(x,y) \rightarrow (-y,x)$
  • Translation by $(a,b)$: $(x,y) \rightarrow (x+a,y+b)$
  • Homothety by factor $k$: $(x,y) \rightarrow (kx,ky)$

Next: Coordinate Geometry → | Prev: Length & Area Classics → | Back to: Topics Overview →