Geometric Transformations

One idea organises all of geometry

Geometric transformations are maps of the plane that transform figures. They split into isometries — which preserve distances: translation, rotation, reflection (axial and central) — giving congruent figures (identical in size and shape), and similarities (including dilation), which preserve shape but can change size, giving similar figures.

An isometry preserves distances:

That is why the image is congruent to the original — the same side lengths, the same angles. Only the position or orientation of the figure changes (with a reflection the orientation reverses — the letter $F$ becomes mirrored).

A dilation scales the size:

A dilation with centre $S$ and scale $k$ maps a point $P$ to $P'$ so that $\vec{SP'} = k\,\vec{SP}$. The shape is preserved (a similar figure), but all distances are multiplied by $|k|$ — which is why only a dilation changes the size of a figure.

The Erlangen Program (1872):

Felix Klein, in his famous "Erlangen Program", proposed a revolutionary viewpoint: different geometries can be classified by the groups of transformations that preserve certain properties in them. This unified all of geometry through the theory of groups and symmetry, influencing mathematics and physics (where symmetries underlie conservation laws).

Why you need this for the exam:

Geometric transformations are essential exam material — you need to recognise isometries and similarities and their properties. Trick: ask whether the transformation preserves distances. If yes — the figures are congruent (an isometry). If only the shape, while the size changes — they are similar (a dilation, $k \neq 1$).

Remember: isometries (translation, rotation, reflection) give congruent figures — the same size; a dilation gives similar figures — size × $k$.