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similarity in geometry (Definition)

Two figures $ K$ and $ K'$ in a Euclidean plane or space are similar iff there exists a bijection $ f$ from the set of points of $ K$ onto the set of points of $ K'$ such that, for any $ P,Q \in K$, the ratio

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

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

$\displaystyle K' \sim K\;\;\;($or$\displaystyle \;\;K \sim K').$

Examples

Nonexamples

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 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'$ are equal, then all pairs of corresponding line segments are equal, i.e. the figures have also equal sizes: They are congruent ( $ K' \cong K$).



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"similarity in geometry" is owned by pahio. [ full author list (4) ]
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See Also: homothety, proportion equation, harmonic mean in trapezoid

Other names:  similarity
Also defines:  similar, ratio of similarity, line ratio

Attachments:
compass and straightedge construction of similar triangles (Algorithm) by Wkbj79
$n$-section of line segment with compass and straightedge (Algorithm) by pahio
similarity of triangles (Theorem) by pahio
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Cross-references: equivalence relation, Euclidean space, solids, volumes, regions, areas, arcs, angles, properties, triangles, ellipsoids, ellipses, prisms, rhombi, rectangles, equilateral triangles, spheres, central angle, sectors, parabolas, circles, cubes, squares, number, line segments, lengths, ratio, onto, points, bijection, iff, Euclidean plane
There are 268 references to this entry.

This is version 15 of similarity in geometry, born on 2007-05-24, modified 2008-02-13.
Object id is 9455, canonical name is SimilarityInGeometry.
Accessed 5467 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
 51M05 (Geometry :: Real and complex geometry :: Euclidean geometries and generalizations)
 51F99 (Geometry :: Metric geometry :: Miscellaneous)
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