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Triangle Similarity Test - Three sides in proportion (SSS)

 

Definition: Triangles are similar if all three sides in one triangle are in the same proportion to the corresponding sides in the other.

This (SSS) is one of the three ways to test that two triangles are similar . For a list see Similar Triangles.

Try this Drag any orange dot at P,Q,R. The triangle LMN will change to remain similar to the left triangle PQR. (If there is no image below, see support page.)

If all three sides in one triangle are in the same proportion to the corresponding sides in the other, then the triangles are similar. So, for example in the triangle above, the side PQ is exactly twice as long as the corresponding side LM in the other triangle. PR is twice LN and QR is twice MN. All three sides are in the same proportion, in this case 2:1 (two to one), and so the triangles are similar.

It doesn't matter what ratio it is (it could have been, say, 5.3 : 1). But so long as it is the same ratio for all three side pairs, the triangles a similar.

Notice that the the three sides are drawn in thick blue lines to indicate they are the parts being used to test for similarity.

What does this mean?

  1. Since all three corresponding sides are the same length, we can be sure the triangles are similar.
  2. Because the triangles are similar, this means that the three angles at P,Q and R are equal to the angles L,M and N respectively.

But don't forget

Two triangles, similar because all three sides are the same length, but one flipped and rotated Similar triangles can be rotated and/or mirror images of each other (reflected). (See Similar triangles.) In the figure on the right, the two triangles have all three corresponding sides equal in length and so are still similar, even though one is the mirror image of the other and rotated.

Related topics

Similar Polygons

(C) 2007 Copyright John Page