PROPORTION AND RATIOS

PROPORTION AND RATIOS

1. Thales' theorem Thales’ theorem is important because it is the foundation for the study of the similarity of triangles. From these triangles we can check the similarity of any other triangles

   • Parallel lines crossing two other lines Lines a, b and c are parallel and intersect lines r and s. If segments AB and BC are equal, then segments A'B' and B'C' are equal. Check this by measuring them. Lines a, b and c are now parallel too and cross lines r and s. Segment AB is double the segment BC: AB = 2 · BC. So, the second A'B' is also double segment B'C'

2. Using Thales’ theorem: similarity of triangles Triangles, in addition to their simplicity, are basics in the study of other shapes (triangulation is sometimes used to analyse a shape). Therefore, the similarity of triangles deserves special attention.

   • ■ Triangles in Thales’ position The triangles ABC and AB'C' have an angle in common: A ^ . In other words, the small triangle is ‘inserted’ into the large one. Also, the sides opposite A ^ are parallel. This is why we say that both these triangles are Thales’ position
   • ■ A rule on the similarity of triangles So if A ^ = A ^ ', the small triangle can fit into the bigger one. And if, B ^ = B ^ ', the sides a and a' stay parallel. So, the two triangles have been put in Thales’ position and are therefore similar.

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