12-5 Similar Figures


In previous math courses you have
learned about congruent figures that
have the same size and the same
shape.
In this chapter, we are studying similar
figures that have the same shape but
not necessarily the same size.
12-5: Congruence



When 2 figures are “congruent,” they
share the same size and shape.
The symbol to designate congruence is
an equal sign with a tilde on top of it. 
A congruence transformation is any
transformation of a figure that
preserves size and shape.
Similarity



Similar figures have the same shape but
not necessarily the same size
Similarity is designated by the symbol
“~” or tilde.
A similarity transformation preserves
the shape but not necessarily the size
of a figure.
Size Changes and Isometries


Under size changes, figures and their
images have the same shape.
Under Isometries and their images
have the same size and the same
shape.
Examples of Isometries





Reflections
Translations
Rotations
Let’s look at some examples:
Transformations - Composition NLVM
Definitions


Two figures F and G are similar,
written F~G, if and only if there is a
composite (combination) of size
changes and reflections mapping one
onto the other.
A transformation is a similarity
transformation if and only if it is the
composite of size changes and
reflections.
Hierarchy of Transformations
Transformations
Similarity Transformations
Isometries
Size Changes
Translations
Identities
Rotations
Reflections
Properties of Similar Figures
Preserved Under
Reflections
Angle Measure
Preserved Under
Size
Transformations
Angle Measure
Betweenness
Betweenness
Collinearity
Collinearity
Distance
More Properties of Similarity
Similar Figures Theorem
If two figures are similar, then




1) corresponding angles are congruent
2) corresponding lengths are proportional
Ratio of Similitude

The ratio of a length on an image to the
corresponding length on a similar preimage (what
we referred to as k earlier in the chapter).
Examples


Let’s look at examples of compositions
of size changes and isometries.
compositions.html