One way of thinking about a transformation is to consider what happens to every point on the plane under that transformation. For example, under translation, every (pre-image) point on the plane moves to another (image) point; and no two distinct pre-image points move to the same image point. Under reflection, every point goes to a distinct image-point except for points on the mirror line of symmetry (which stay fixed). In addition, a transformation defines how points move on the plane, and conversely, any statement of how points move on the plane defines some sort of transformation.
Example
In the dynamic sketch below, consider AB and its midpoint C. What happens when you drag (only) point A?
(Points A and C are traced so you can compare A to C, the position of which depends on A.)
Question
How could you describe C as a transformation of A?
Click on the 'Show Answer' button in the dynamic sketch below to check your answer.
Mystery Transformation
Explore More
Click on the 'Link to More Mystery Transform' button above to navigate to another activity on this 'mystery' transformation.
Application
Here is a nice example of applying the above 'mystery transformation': Interesting Locus Result
Related Links
Miscellaneous Dynamic Transformations (of Geometric Figures & Graphs)
All parabola are similar
Some Transformations of Graphs
Interesting Locus Result
Reasoning Backwards: Triangle Midpoints (Rethinking Proof activity)
Four Collinear Points
Some Trapezoid (Trapezium) Explorations (see Investigations 3 & 4)
Midpoint trapezium (trapezoid) theorem generalized
Rugby Place-kicking Problem
Path of Balancing Point
A Fundamental Theorem of Similarity
External Links
Investigate Geometric Transformations as Functions
Dilation (geometry) (Wikipedia)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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By Michael de Villiers. Created, 24 July 2013; updated 24 June 2022; 28 Jan 2026.