What is Similarity?

What is Similarity?

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Before defining similarity, let us recap on the concept of congruence.

Two geometrical figures are said to be congruent when they can be superimposed on each other exactly. In other words, two congruent figures are alike in every respect or they have the same shape and same size as well.

Definition of Similarity

The simplest definition of similarity is: "Two figures are similar if they have the same shape (they may not have the same size)".

For example, two circles (of any radii) will always be similar:

Similar circles

If the two circles happen to have equal radii, they will be congruent as well.

Two squares (of any side lengths) will always be similar:

Similar squares

They will be congruent only when their side lengths are equal to each other, but they will always be similar.

Suppose that you have a picture or figure of an arbitrary quadrilateral. You shrink that picture, and compare it with the original quadrilateral:

Similar quadrilaterals

What do you observe? The two figures have the same shape, but not the same size. These two quadrilaterals are similar, but not congruent. Note how all the angles are preserved.

You can intuitively think of similarity as follows: Two figures \(A\) and \(B\) will be similar if \(B\) can be enlarged or shrunk (by a certain factor) to make it congruent with \(A\). If \(A\) and \(B\) are congruent, to begin with, then they are similar as well. Thus, the similarity is more general than congruent, that is, congruence is a special case of similarity.

✍Note: All congruent figures are similar but all similar figures need not be congruent.

Let’s consider some more examples of similar figures.

Solved Examples:

Example 1: Two similar triangles

Similar triangles

Note how the (corresponding) angles of the two triangles are the same. Can you figure out what the relationship is between their corresponding sides?

✍Note: If two polygons are similar then their corresponding angles are equal and their corresponding sides are in the same ratio (or proportion).

Example 2: Two similar pentagons

Similar pentagons

Once again, observe how the corresponding angles are equal.

Example 3: Two similar arcs

Similar arcs

The two arcs under consideration have been highlighted. The two arcs are similar because they subtend the same angle at the center \(O\) of their parent circles.

✍Note: Two figures are similar if one can be obtained from the other by uniform scaling, that is, by uniform enlarging or shrinking.

yesChallenge: Are all equilateral triangles always similar?

⚡Tip: All the three angles of an equilateral triangle are equal.

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