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== Rujukan ==
== Rujukan ==
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Revisi per 11 Mei 2023 14.14

Layang-layang
Sebuah layang-layang yang memperlihatkan pasangan sisinya yang sama panjang beserta lingkaran dalamnya.
Sisi dan titik pojok4
Grup simetriD1 (*)

Dalam geometri Euklides, layang-layang adalah sebuah segiempat yang memeiliki simetri cerminan di sekitar sisi diagonalnya. Karena adanya simetri ini, layang-layang memiliki dua sudut yang sama dan memiliki dua pasangan sisi yang sama panjang dan juga berdampingan. Layang-layang juga dikenal dengan sebutan deltoid,[1] tetapi kata deltoid juga dapat mengacu pada kurva deltoid, objek geometri yang tak berkaitan tetapi kajiannya terkadang memiliki kaitan dengan segiempat.[2][3] Layang-layang tidak berupa cembung saja, tetapi layang-layang juga dapat berupa cekung.[4][5]

Setiap layang-layang merupakan segiempat ortodiagonal (yang berarti sisi diagonal berada di sudut siku-siku), dan layang-layang merupakan segiempat garis singgung (sisinya yang bersinggungan dengan sebuah lingkaran dalam) apabila layang-layangnya cembung. Layang-layang cembung setidaknya merupakan segiempat yang sama-sama ortodiagonal dan tangensial (bersinggungan). Terdapat layang-layang siku-siku, layang-lyang yang memiliki dua sudut siku-siku yang berhadapan, yang merupakan kasus istimewa dari layang-layang. Kasus istimewa lainnya adalah belah ketupat yang memiliki dua sumbu simetri yang merupakan diagonalnya, dan persegi yang sama-sama merupakan kasus istimewa dari layang-layang dan juga belah ketupat.

Segiempat dengan perbandingan terbesar dari keliling dan diameter adalah layang-layang dengan sudut 60°, 75°, dan 150°. Layang-layang dari dua bangun datar (yang cembung maupun tak cembung) membentuk prototile dari salah satu bentuk pengubinan Penrose. Layang-layang juga membentuk muka dari polihedron yang memiliki sifat isohedral dan pengubinan. Layang-layang memiliki kaitannya dengan kajian biliar outer (outer billiard), permasalahan dalam matematika lanjutan mengenai sistem dinamika.

Definisi dan klasifikasi

Layang-layang cembung dan cekung

A kite is a quadrilateral with reflection symmetry across one of its diagonals. Equivalently, it is a quadrilateral whose four sides can be grouped into two pairs of adjacent equal-length sides.[1][6] A kite can be constructed from the centers and crossing points of any two intersecting circles.[7] Kites as described here may be either convex or concave, although some sources restrict kite to mean only convex kites. A quadrilateral is a kite if and only if any one of the following conditions is true:

  • The four sides can be split into two pairs of adjacent equal-length sides.[6]
  • One diagonal crosses the midpoint of the other diagonal at a right angle, forming its perpendicular bisector.[8] (In the concave case, the line through one of the diagonals bisects the other.)
  • One diagonal is a line of symmetry. It divides the quadrilateral into two congruent triangles that are mirror images of each other.[6]
  • One diagonal bisects both of the angles at its two ends.[6]

Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape[9][10] and which are in turn named for a hovering bird and the sound it makes.[11][12] According to Olaus Henrici, the name "kite" was given to these shapes by James Joseph Sylvester.[13]

Quadrilaterals can be classified hierarchically, meaning that some classes of quadrilaterals include other classes, or partitionally, meaning that each quadrilateral is in only one class. Classified hierarchically, kites include the rhombi (quadrilaterals with four equal sides) and squares. All equilateral kites are rhombi, and all equiangular kites are squares. When classified partitionally, rhombi and squares would not be kites, because they belong to a different class of quadrilaterals; similarly, the right kites discussed below would not be kites. The remainder of this article follows a hierarchical classification; rhombi, squares, and right kites are all considered kites. By avoiding the need to consider special cases, this classification can simplify some facts about kites.[14]

Like kites, a parallelogram also has two pairs of equal-length sides, but they are opposite to each other rather than adjacent. Any non-self-crossing quadrilateral that has an axis of symmetry must be either a kite, with a diagonal axis of symmetry; or an isosceles trapezoid, with an axis of symmetry through the midpoints of two sides. These include as special cases the rhombus and the rectangle respectively, and the square, which is a special case of both.[1] The self-crossing quadrilaterals include another class of symmetric quadrilaterals, the antiparallelograms.[15]

Kasus istimewa

Sifat

Pengubinan dan polihedron

Biliar outer

Rujukan

  1. ^ a b c Halsted, George Bruce (1896), "Chapter XIV. Symmetrical Quadrilaterals", Elementary Synthetic Geometry, J. Wiley & sons, hlm. 49–53 
  2. ^ Goormaghtigh, R. (1947), "Orthopolar and isopolar lines in the cyclic quadrilateral", The American Mathematical Monthly, 54 (4): 211–214, doi:10.1080/00029890.1947.11991815, JSTOR 2304700, MR 0019934 
  3. ^ See H. S. M. Coxeter's review of (Grünbaum 1960) in MR0125489: "It is unfortunate that the author uses, instead of 'kite', the name 'deltoid', which belongs more properly to a curve, the three-cusped hypocycloid."
  4. ^ Roebyanto, Goenawan (2014), Geometri Pengukuran dan Statistika, hlm. 19, ISBN 978-602-1223-13-0 
  5. ^ Rochajati, Siti; Astutik, Kasni (2020), "Pengetahuan, Sikap, dan Keyakinan Guru Sekolah Dasar Terhadap Pembelajaran Geometri", Jurnal Penelitian Didaktik Matematika, 4 (2), ISSN 2656-5544 
  6. ^ a b c d De Villiers, Michael (2009), Some Adventures in Euclidean Geometry, hlm. 16, 55, ISBN 978-0-557-10295-2 
  7. ^ Szecsei, Denise (2004), The Complete Idiot's Guide to Geometry, Penguin, hlm. 290–291, ISBN 9781592571833 
  8. ^ Usiskin, Zalman; Griffin, Jennifer (2008), The Classification of Quadrilaterals: A Study of Definition, Information Age Publishing, hlm. 49–52, 63–67 
  9. ^ Beamer, James E. (May 1975), "The tale of a kite", The Arithmetic Teacher, 22 (5): 382–386, doi:10.5951/at.22.5.0382, JSTOR 41188788 
  10. ^ Alexander, Daniel C.; Koeberlein, Geralyn M. (2014), Elementary Geometry for College Students (edisi ke-6th), Cengage Learning, hlm. 180–181, ISBN 9781285965901 
  11. ^ Suay, Juan Miguel; Teira, David (2014), "Kites: the rise and fall of a scientific object" (PDF), Nuncius, 29 (2): 439–463, doi:10.1163/18253911-02902004 
  12. ^ Liberman, Anatoly (2009), Word Origins...And How We Know Them: Etymology for Everyone, Oxford University Press, hlm. 17, ISBN 9780195387070 
  13. ^ Henrici, Olaus (1879), Elementary Geometry: Congruent Figures, Longmans, Green, hlm. xiv 
  14. ^ De Villiers, Michael (February 1994), "The role and function of a hierarchical classification of quadrilaterals", For the Learning of Mathematics, 14 (1): 11–18, JSTOR 40248098 
  15. ^ Alsina, Claudi; Nelsen, Roger B. (2020), "Section 3.4: Kites", A Cornucopia of Quadrilaterals, The Dolciani Mathematical Expositions, 55, Providence, Rhode Island: MAA Press and American Mathematical Society, hlm. 73–78, ISBN 978-1-4704-5312-1, MR 4286138 ; see also antiparallelograms, p. 212

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