Enlargements

Translations, rotations and reflections are examples of congruence transformations, because the image of a figure under one of these transformations is congruent to the original. We defined two figures to be congruent if one could be mapped to the other by a sequence of these transformations.

Here a fourth type of transformation of the plane called an enlargement is introduced, in which all lengths are increased or decreased in the same ratio.

To specify an enlargement, we need to specify two things:

• the centre O of the enlargement; the centre of enlargement stays fixed in the one place, while the enlargement expands or shrinks everything else around it
• an enlargement factor k (or enlargement ratio 1 : k); the distances of all points from the enlargement centre increase or decrease by this factor.

With enlargement, lengths of sides are increased or decreased by the same enlargement factor.

Here the centre of enlargement is O and the enlargement factor is 2.

Detailed description

Note that \(OA^\prime = 2OA, OB^\prime = 2OB\ \text{and}\ OC^\prime = 2OC.\ \text{Also}, A^\prime B^\prime = 2AB, A^\prime C^\prime = 2AC,\ \text{and}\ B^\prime C^\prime = 2BC\).