Dilations and Similarity
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Similar triangles - scale factor 2

Similar figures are figures that are the same shape, but not necessarily the same size.

If these figures should also be the same size, the figures are called congruent.

In our study of transformations we have seen many figures that remain congruent after translations (slides), reflections (flips) and rotations (turns).  We have also seen figures (under dilations) which retain their shape but do not remain the same size.  These figures and their images under dilation are similar figures.

The term similar, or similarity, can be defined using the language of transformations:

Definition:   Two figures are similar if one is the image of the other under a transformation from the plane into itself that multiplies all distances by the same positive scale factor, k.  That is to say, one figure is a dilation of the other.

An isometry is a transformation which preserves length.  The only similarity which is an isometry occurs when the scale factor is equal to 1.  Remember that similar figures with a scale factor of 1 are actually congruent (the same size and shape). 

Our discussions of similar triangles with scalar factors not equal to one are, of course, occurring  in Euclidean geometry.  In hyperbolic and elliptical geometry, similar triangles are congruent triangles.