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Transformations & Advanced Geometry

Reflections, rotations, inversions, projective techniques, and trigonometric Ceva/Menelaus

Key Concepts

  • Reflections and rotations are isometries (preserve distance)
  • Homothety scales a figure about a centre
  • Spiral similarity combines rotation and scaling
  • Transformations map circles and lines to circles and lines

Important Formulae

Homothety Ratio k scales all lengths by |k|
Rotation Preserves distances and angles about its centre

Quick Tips

  • Use a rotation or reflection to move parts of a figure into a usable position.
  • Homothety is ideal for problems involving tangent or nested circles.

Sample Practice Questions

  1. A homothety (enlargement) with ratio 2 scales every length by a factor of:

    • 1/2
    • 2
    • 4
    • 0
    Show answer

    Answer: 2

  2. A rotation about a point preserves:

    • Distances
    • Only area
    • Only angles, not lengths
    • Nothing
    Show answer

    Answer: Distances

  3. The reflection of the point (3, 2) in the x-axis is:

    • (−3, 2)
    • (3, −2)
    • (−3, −2)
    • (2, 3)
    Show answer

    Answer: (3, −2)

  4. A rotation of 360° maps a figure to:

    • Its mirror image
    • Itself
    • A larger copy
    • A point
    Show answer

    Answer: Itself

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Practice Questions

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Frequently Asked Questions

What are the key concepts in Transformations & Advanced Geometry for RMO?

Reflections and rotations are isometries (preserve distance); Homothety scales a figure about a centre; Spiral similarity combines rotation and scaling; Transformations map circles and lines to circles and lines.

Which formulae are important for Transformations & Advanced Geometry in RMO?

Homothety: Ratio k scales all lengths by |k|; Rotation: Preserves distances and angles about its centre.

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Geometry