# similarity in geometry

Two figures $K$ and $K^{\prime}$ in a Euclidean plane or space (http://planetmath.org/EuclideanVectorSpace) are    iff there exists a bijection $f$ from the set of points of $K$ onto the set of points of $K^{\prime}$ such that, for any $P,Q\in K$, the ratio

 $\frac{P^{\prime}Q^{\prime}}{PQ}$

of the lengths of the line segments $P^{\prime}Q^{\prime}$ and $PQ$ is always the same number $k$, where $P^{\prime}=f(P)$ and $Q^{\prime}=f(Q)$.

The number $k$ is called the ratio of similarity or the line ratio of the figure $K^{\prime}$ with respect to the figure $K$ (N.B. the in which the figures are mentioned!).  The similarity of $K$ and $K^{\prime}$ is often denoted by

 $K^{\prime}\sim K\;\;\;(\mbox{or}\;\;K\sim K^{\prime}).$

Examples

Nonexamples

• Not all rectangles are similar.

• Not all rhombi are similar.

• Not all rectangular prisms are similar.

• Not all ellipses are similar.

• Not all ellipsoids are similar.

• Not all triangles are similar.

Properties

• The corresponding angles (consisting of corresponding points) of two similar figures are equal.

• The lengths of any corresponding arcs of two similar figures are proportional in the ratio $k$.

• The areas of two similar regions are proportional in the ratio $k^{2}$ when $k$ is the line ratio of the regions.

• The volumes of two similar solids are proportional in the ratio $k^{3}$ when $k$ is the line ratio of the solids.

Remarks

• In any Euclidean space $E$, the relation (http://planetmath.org/Relation) of similarity (denoted $\sim$) on the set of figures in $E$ is an equivalence relation.

• If one pair of corresponding line segments in the similar figures $K$ and $K^{\prime}$ are equal, then all pairs of corresponding line segments are equal, i.e. the figures have also equal : They are congruent (http://planetmath.org/Congruence) ($K^{\prime}\cong K$).

 Title similarity in geometry Canonical name SimilarityInGeometry Date of creation 2013-03-22 17:08:48 Last modified on 2013-03-22 17:08:48 Owner pahio (2872) Last modified by pahio (2872) Numerical id 19 Author pahio (2872) Entry type Definition Classification msc 51F99 Classification msc 51M05 Classification msc 51-00 Synonym similarity Synonym similitude Related topic Homothety Related topic ProportionEquation Related topic HarmonicMeanInTrapezoid Defines similar Defines ratio of similarity Defines similitude ratio Defines line ratio