Similarity
Criteria
| Property | Statement |
|---|---|
| AA | Two pairs of equal angles |
| SAS | Two sides proportional, included angle equal |
| SSS | All three sides proportional |
Examples
Example 1. Triangles with sides 3,4,5 and 6,8,10. Similar?
Solution. Yes, SSS ratio 1:2.
In Depth
Two figures are similar if one can be obtained from the other by a combination of rotation, reflection, translation, and uniform scaling. Similar figures have the same shape but possibly different sizes. The ratio of corresponding lengths is the scale factor \(k\); areas scale by \(k^2\); volumes scale by \(k^3\).
AA (Angle-Angle) similarity: two triangles are similar if two pairs of corresponding angles are equal. Since angle sums are 180°, two equal angles force the third to be equal too. SAS and SSS similarity criteria provide alternatives.
The basic proportionality theorem (Thales): a line parallel to one side of a triangle divides the other two sides proportionally. This is the foundation of similar triangle proofs and is used in constructions and proofs throughout geometry.
Similar triangles appear in indirect measurement: the height of a tall object can be found by comparing the shadow lengths of the object and a known reference. This method was used by Thales to measure the height of the Great Pyramid.
In trigonometry, the trig ratios (sin, cos, tan) are defined for right triangles and depend only on the angle, not the size — because all right triangles with the same acute angle are similar. This is why trig functions are well-defined.
Key Properties & Applications
The side-splitter theorem: a line parallel to one side of a triangle divides the other two sides proportionally. This is the key tool for proving similar triangles and for constructions involving proportional division of segments.
Scale models and maps use similarity. A 1:50000 map means distances on the map are 1/50000 of real distances; areas are scaled by \((1/50000)^2\). Engineering drawings specify scale factors; architectural models use similarity to represent buildings.
Fractal geometry studies self-similar objects — shapes that are similar to parts of themselves at all scales. The Sierpiński triangle, Koch snowflake, and Mandelbrot set are self-similar. Their fractal dimension (between 1 and 2 for planar fractals) measures their complexity.
Further Reading & Context
The study of similarity connects to many areas of mathematics and its applications. Understanding the foundational definitions and theorems provides the basis for advanced work in analysis, algebra, and applied mathematics.
Historical development: most mathematical concepts evolved over centuries, with contributions from mathematicians across many cultures. The modern axiomatic treatment provides rigor, while computational tools enable practical application.
In modern mathematics, this topic appears in graduate courses and research across pure and applied mathematics. Connections to computer science, physics, and engineering make it a versatile and important area of study. Mastery of the core results and techniques opens doors to research in number theory, analysis, geometry, and beyond.
Recommended next steps: work through the standard theorems with full proofs, explore the connections to related topics listed above, and practice with a variety of problems ranging from computational exercises to theoretical proofs. The interplay between different areas of mathematics is one of the subject's greatest rewards.