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Text of the page (random words):
er figures shown in the same color are similar if two angles of a triangle have measures equal to the measures of two angles of another triangle then the triangles are similar corresponding sides of similar polygons are in proportion and corresponding angles of similar polygons have the same measure two congruent shapes are similar with a scale factor of 1 however some school textbooks specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar citation needed contents 1 similar triangles 2 other similar polygons 3 similar curves 4 in euclidean space 5 area ratio and volume ratio 6 similarity with a center 7 in general metric spaces 8 topology 9 self similarity 10 psychology 11 see also 12 notes 13 references 14 further reading 15 external links similar triangles edit two triangles abc and a b c are similar if and only if corresponding angles have the same measure this implies that they are similar if and only if the lengths of corresponding sides are proportional 1 it can be shown that two triangles having congruent angles equiangular triangles are similar that is the corresponding sides can be proved to be proportional this is known as the aaa similarity theorem 2 note that the aaa is a mnemonic each one of the three a s refers to an angle due to this theorem several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent 3 there are several criteria each of which is necessary and sufficient for two triangles to be similar any two pairs of congruent angles 4 which in euclidean geometry implies that their all three angles are congruent 5 if bac is equal in measure to b a c and abc is equal in measure to a b c then this implies that acb is equal in measure to a c b and the triangles are similar all the corresponding sides are proportional 6 ab a b bc b c ac a c this is equivalent to saying that one triangle or its mirror image is an enlargement of the other any two pairs of sides are proportional and the angles included between these sides are congruent 7 ab a b bc b c and abc is equal in measure to a b c this is known as the sas similarity criterion 8 the sas is a mnemonic each one of the two s s refers to a side the a refers to an angle between the two sides symbolically we write the similarity and dissimilarity of two triangles abc and a b c as follows 9 a b c a b c displaystyle abc sim a b c a b c a b c displaystyle abc nsim a b c there are several elementary results concerning similar triangles in euclidean geometry 10 any two equilateral triangles are similar two triangles both similar to a third triangle are similar to each other transitivity of similarity of triangles corresponding altitudes of similar triangles have the same ratio as the corresponding sides two right triangles are similar if the hypotenuse and one other side have lengths in the same ratio 11 there are several equivalent conditions in this case such as the right triangles having an acute angle of the same measure or having the lengths of the legs sides being in the same proportion given a triangle abc and a line segment de one can with ruler and compass find a point f such that abc def the statement that the point f satisfying this condition exists is wallis s postulate 12 and is logically equivalent to euclid s parallel postulate 13 in hyperbolic geometry where wallis s postulate is false similar triangles are congruent in the axiomatic treatment of euclidean geometry given by george david birkhoff see birkhoff s axioms the sas similarity criterion given above was used to replace both euclid s parallel postulate and the sas axiom which enabled the dramatic shortening of hilbert s axioms 8 similar triangles provide the basis for many synthetic without the use of coordinates proofs in euclidean geometry among the elementary results that can be proved this way are the angle bisector theorem the geometric mean theorem ceva s theorem menelaus s theorem and the pythagorean theorem similar triangles also provide the foundations for right triangle trigonometry 14 other similar polygons edit similar rectangles the concept of similarity extends to polygons with more than three sides given any two similar polygons corresponding sides taken in the same sequence even if clockwise for one polygon and counterclockwise for the other are proportional and corresponding angles taken in the same sequence are equal in measure however proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles otherwise for example all rhombi would be similar likewise equality of all angles in sequence is not sufficient to guarantee similarity otherwise all rectangles would be similar a sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional for given n all regular n gons are similar similar curves edit several types of curves have the property that all examples of that type are similar to each other these include lines any two lines are even congruent line segments circles parabolas 15 hyperbolas of a specific eccentricity 16 ellipses of a specific eccentricity 16 catenaries graphs of the logarithm function for different bases graphs of the exponential function for different bases logarithmic spirals are self similar in euclidean space edit a similarity also called a similarity transformation or similitude of a euclidean space is a bijection f from the space onto itself that multiplies all distances by the same positive real number r so that for any two points x and y we have d f x f y r d x y displaystyle d f x f y r d x y where d x y is the euclidean distance from x to y 17 the scalar r has many names in the literature including the ratio of similarity the stretching factor and the similarity coefficient when r 1 a similarity is called an isometry rigid transformation two sets are called similar if one is the image of the other under a similarity as a map f ℝ n ℝ n a similarity of ratio r takes the form f x r a x t displaystyle f x rax t where a o n ℝ is an n n orthogonal matrix and t ℝ n is a translation vector similarities preserve planes lines perpendicularity parallelism midpoints inequalities between distances and line segments 18 similarities preserve angles but do not necessarily preserve orientation direct similitudes preserve orientation and opposite similitudes change it 19 the similarities of euclidean space form a group under the operation of composition called the similarities group s 20 the direct similitudes form a normal subgroup of s and the euclidean group e n of isometries also forms a normal subgroup 21 the similarities group s is itself a subgroup of the affine group so every similarity is an affine transformation one can view the euclidean plane as the complex plane 22 that is as a 2 dimensional space over the reals the 2d similarity transformations can then be expressed in terms of complex arithmetic and are given by f z az b direct similitudes and f z a z b opposite similitudes where a and b are complex numbers a 0 when a 1 these similarities are isometries area ratio and volume ratio edit the tessellation of the large triangle shows that it is similar to the small triangle with an area ratio of 5 the similarity ratio is 5 h h 1 5 this can be used to construct an non periodic infinite tiling main article square cube law the ratio between the areas of similar figures is equal to the square of the ratio of corresponding lengths of those figures for example when the side of a square or the radius of a circle is multiplied by three its area is multiplied by nine i e by three squared the altitudes of similar triangles are in the same ratio as corresponding sides if a triangle has a side of length b and an altitude drawn to that side of length h then a similar triangle with corresponding side of length kb will have an altitude drawn to that side of length kh the area of the first triangle is a 1 2 bh while the area of the similar triangle will be a 1 2 kb kh k 2 a similar figures which can be decomposed into similar triangles will have areas related in the same way the relationship holds for figures that are not rectifiable as well the ratio between the volumes of similar figures is equal to the cube of the ratio of corresponding lengths of those figures for example when the edge of a cube or the radius of a sphere is multiplied by three its volume is multiplied by 27 i e by three cubed galileo s square cube law concerns similar solids if the ratio of similitude ratio of corresponding sides between the solids is k then the ratio of surface areas of the solids will be k 2 while the ratio of volumes will be k 3 similarity with a center edit example where each similarity composed with itself several times successively has a center at the center of a regular polygon that it shrinks example of direct similarity of center s decomposed into a rotation of 135 angle and a homothety that halves areas examples of direct similarities that have each a center if a similarity has exactly one invariant point a point that the similarity keeps unchanged then this only point is called center of the similarity on the first image below the title on the left one or another similarity shrinks a regular polygon into a concentric one the vertices of which are each on a side of the previous polygon this rotational reduction is repeated so the initial polygon is extended into an abyss of regular polygons the center of the similarity is the common center of the successive polygons a red segment joins a vertex of the initial polygon to its image under the similarity followed by a red segment going to the following image of vertex and so on to form a spiral actually we can see more than three direct similarities on this first image because every regular polygon is invariant under certain direct similarities more precisely certain rotations the center of which is the center of the polygon and a composition of direct similarities is also a direct similarity for example we see the image of the initial regular pentagon under a homothety of negative ratio k displaystyle text ratio k which is a similarity of 180 angle and a positive ratio equal to k displaystyle text equal to k below the title on the right the second image shows a similarity decomposed into a rotation and a homothety similarity and rotation have the same angle of 135 degrees modulo 360 degrees similarity and homothety have the same ratio of 2 2 displaystyle text of frac sqrt 2 2 multiplicative inverse of the ratio 2 displaystyle text ratio sqrt 2 square root of 2 of the inverse similarity point s is the common center of the three transformations rotation homothety and similarity for example point w is the image of f under the rotation and point t is the image of w under the homothety more briefly t h w h r f h r f d f by naming r h and d displaystyle r h text and d the previous rotation homothety and similarity with d like direct displaystyle text d text like direct this direct similarity that transforms triangle efa into triangle atb can be decomposed into a rotation and a homothety of same center s in several manners for example d r h h r displaystyle d r circ h h circ r the last decomposition being only represented on the image to get d displaystyle d we can also compose in any order a rotation of 45 displaystyle text of 45 circ angle and a homothety of ratio 2 2 displaystyle text of ratio frac sqrt 2 2 with m like mirror displaystyle text m text like mirror and i like indirect displaystyle text i text like indirect if m displaystyle m is the reflection with respect to line cw then m d i displaystyle m circ d i is the indirect similarity that transforms segment bf like d displaystyle text like d into segment ct but transforms point e into b and point a into a itself square acbt is the image of abef under similarity i of ratio 1 2 displaystyle i text of ratio 1 sqrt 2 point a is the center of this similarity because any point k being invariant under it fulfills a k a k 2 displaystyle ak ak sqrt 2 only possible if a k 0 displaystyle text if ak 0 otherwise written a k displaystyle a k how to construct the center s of direct similarity d displaystyle d from square a b e f displaystyle text from square abef how to find point s center of a rotation of 135 angle that transforms ray se into ray sa this is an inscribed angle problem plus a question of orientation the set of points p such that p e p a 135 displaystyle p text such that angle overrightarrow pe overrightarrow pa 135 circ is an arc of circle e a displaystyle overset frown ea that joins e and a of which the two radius leading to e and a form a central angle of 2 180 135 2 45 90 displaystyle text of 2 180 circ 135 circ 2 times 45 circ 90 circ this set of points is the blue quarter of circle of center f inside square abef in the same manner point s is a member of the blue quarter of circle of center t inside square bcat so point s is the intersection point of these two quarters of circles in general metric spaces edit sierpiński triangle a space having self similarity dimension log 3 log 2 log 2 3 which is approximately 1 58 from hausdorff dimension in a general metric space x d an exact similitude is a function f from the metric space x into itself that multiplies all distances by the same positive scalar r called f s contraction factor so that for any two points x and y we have d f x f y r d x y displaystyle d f x f y rd x y weaker versions of similarity would for instance have f be a bi lipschitz function and the scalar r a limit lim d f x f y d x y r displaystyle lim frac d f x f y d x y r this weaker version applies when the metric is an effective resistance on a topologically self similar set a self similar subset of a metric space x d is a set k for which there exists a finite set of similitudes f s s s with contraction factors 0 r s 1 such that k is the unique compact subset of x for which a self similar set constructed with two similitudes z 0 1 4 i z 4 and z 0 1 4 7i z 5 2i s s f s k k displaystyle bigcup _ s in s f_ s k k these self similar sets have a self similar measure μ d with dimension d given by the formula s s r s d 1 displaystyle sum _ s in s r_ s d 1 which is often but not always equal to the set s hausdorff dimension and packing dimension if the overlaps between the f s k are small we have the following simple formula for the measure μ d f s 1 f s 2 f s n k r s 1 r s 2 r s n d displaystyle mu d f_ s_ 1 circ f_ s_ 2 circ cdots circ f_ s_ n k r_ s_ 1 cdot r_ s_ 2 cdots r_ s_ n d topology edit this section needs additional citations for verification please help improve this article by adding citations to reliable sources unsourced material may be challenged and removed august 2018 learn how and when to remove this template message in topology a metric space can be constru...