The Three Meanings of Similarity

We can say that two objects are similar if they look alike. In layman’s words, objects with the same shape, whether they have the same size or not are usually called similar. In mathematics, it is quite different. In this post, we are going to learn the three mathematical meanings of similarity.

In mathematics, two objects are similar when either one of the following three conditions is true.

1. When one figure is reduced or enlarged, it will become congruent with the other

The first meaning is based on the definition of congruence. That is, when two figures are similar, if one figure is enlarged or reduced, then they will become congruent with the other. This definition is better illustrated graphically, using a drawing or an applet just as the one shown below. 

In the applet, drag slider k until and make the two similar figures overlap.

2. The ratio of the corresponding segments is constant and the corresponding angles are congruent

The second meaning involves the ratio of segments and angle measures. Two figures are similar if the ratio of the lengths of their corresponding sides is constant or their corresponding angles are congruent. Most authors choose this meaning as the definition because it can easily be used in proofs. The conditions for triangle similarity namely SSS, SAS, AA, etc. use this meaning in proving that two triangles are similar.

3. When they can be placed in a position of similarity

The third meaning involves the positional relationship between figures. If two figures can be placed in a position of similarity by transformation (translation, rotation, reflection), then the two figures are similar. Note that when two figures are in a position of similarity, if their vertices are connected with lines, then these lines intersect at a single point called the point of similarity. This point is also sometimes called the point of similitude or the center of similarity. In the applet above, point O is the point of similarity and ABC and A’B’C’ are in a position of similarity.

Note that each of the three meanings of similarity above can stand alone and each of them follows from the others. Any of these three meanings can be chosen as the definition of similarity. Again, most books just chose the second one because they can easily be used in proving that two figures are similar.

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