Notes
similarity

Similarity

Two shapes are similar if one can be obtained from the other by a series of rotations, reflections, translations, and scaling.

If two shapes are similar then the following hold:

• Corresponding angles on the two shapes are equal.
• There is a fixed scale factor such that corresponding lengths on the two shapes are related by multiplying by this scale factor.
• The ratio of two lengths in one shape is the same as the ratio of the corresponding lengths in the other.

Similar Triangles

When the two shapes are triangles then the following statements are equivalent:

• The two triangles are similar
• The angles in one triangle are the same as the angles in the other
• The ratios of the lengths in one triangle are the same as in the other

Scale Factors

When two shapes are similar, corresponding lengths are related by a fixed scale factor. Corresponding areas are also related by a scale factor, and if the shape is three dimensional so are volumes. These three scale factors are related as follows:

If the length scale factor is $k$, then the area scale factor is $k^2$, and the volume scale factor is $k^3$.