The first five axioms for plane geometry have enabled us to prove a number of theorems about congruent figures. However, there are many important properties of the plane that depend on scaling or similarity -- the ability in the plane to make a scaled copy of a figure.
A figure S' is similar to a figure S if there is a correspondence between the points of the figures and there is a constant k > 0 so that all corresponding angles are congruent and for all points A and B and corresponding points A' and B' the distance |A'B'| = k|AB|.
The constant k, which = |A'B'|/|AB|, is called the scaling ratio or the ratio of similitude from S to S'.
Note: Given S' similar to S with ratio k, then S is also similar to S' with scaling ratio 1/k.
Note: S' is similar to S with scaling ratio k = 1 if and only if S' is congruent to S.
Note: If S' is similar to S with scaling ratio k and S'' is similar to S' with scaling ratio h, then S'' is similar to S with scaling ratio hk.
From the definition, triangle DEF is similar to triangle ABC with scaling ratio k if and only if
None of the axioms so far account of similarity, so we will add one more axiom.
DILATION AXIOM: Given two rays OA and OB, with point A' is on ray OA and B' on ray OB, if |OA'|/|OA| = |OB'|/|OB|, then triangle OA'B' is similar to triangle OAB.
This figure shows an example with k = 2/5:
Discussion: For this figure angle AOB = angle A'OB' (same angles). Also, if we denote by k the ratio |OA'|/|OA|, then we are assuming |OA'|/|OA| = k and |OB'|/|OB| = k. The axiom says that we can then conclude that
In other words, we are assuming one angle and its corresponding angle are congruent and the two pairs of corresponding sides adjacent to the angle have the same ratio k, so this a special case of Side-Angle-Side (SAS) for similarity.
The first figure has k = 7/3. The second figure shows isosceles triangles with k = 2.
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For these consequenes, see B&B