Similarity (geometry)
In geometry, similarity refers to a transformation that preserves angles but may change size. Two geometric figures are said to be similar if they have the same shape but possibly different sizes. This contrasts with congruence, where figures must have the same shape and size.
More formally, two figures are similar if one can be obtained from the other by a sequence of transformations including:
- Translation: Sliding the figure without changing its orientation.
- Rotation: Turning the figure around a point.
- Reflection: Flipping the figure over a line.
- Dilation (Scaling): Enlarging or reducing the figure by a scale factor.
The first three transformations (translation, rotation, and reflection) are rigid transformations that preserve both size and shape and are part of congruence transformations. Dilation is the key transformation distinguishing similarity from congruence.
The ratio of corresponding side lengths in similar figures is constant and is called the scale factor or similarity ratio. If two figures are similar with a scale factor of k, then the ratio of their perimeters is also k, and the ratio of their areas is k2.
For example, two triangles are similar if their corresponding angles are congruent (equal) and their corresponding sides are proportional. This leads to similarity postulates such as AA (Angle-Angle), SSS (Side-Side-Side), and SAS (Side-Angle-Side) similarity.
The concept of similarity is fundamental in many areas of geometry, including trigonometry, fractal geometry, and geometric proofs. It is also crucial in real-world applications such as mapmaking, architectural design, and computer graphics, where scaling and proportion are essential.