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Hypotenuse
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{{short description|Longest side of a right-angled triangle, the side opposite of the right angle}} {{technical|date=May 2019}} [[File:Hypotenuse.svg|thumb|right|A right-angled triangle and its hypotenuse]] In [[geometry]], a '''hypotenuse''' is the side of a [[right triangle]] opposite the [[right angle]].<ref>{{Cite EB1911 |wstitle= Triangle (geometry) |volume= 27 |page= 258|quote=...Also a right-angled triangle has one angle a right angle, the side opposite this angle being called the hypotenuse;...|short=1}}</ref> It is the longest side of any such triangle; the two other shorter sides of such a triangle are called ''[[catheti]]'' or ''legs''. The [[length]] of the hypotenuse can be found using the [[Pythagorean theorem]], which states that the [[Square (algebra)|square]] of the length of the hypotenuse equals the sum of the squares of the lengths of the two legs. Mathematically, this can be written as <math>a^2 + b^2 = c^2</math>, where ''a'' is the length of one leg, ''b'' is the length of another leg, and ''c'' is the length of the hypotenuse.<ref>{{Cite book |last=Jr |first=Jesse Moland |url=https://books.google.com/books?id=dWAXwWpQa5EC&q=hypotenuse |title=I Hate Trig!: A Practical Guide to Understanding Trigonometry |date=August 2009 |publisher=Jesse Moland |isbn=978-1-4486-4707-1 |pages=1 |language=en}}</ref> For example, if one of the legs of a right angle has a length of 3 and the other has a length of 4, then their squares add up to 25 = 9 + 16 = 3 × 3 + 4 × 4. Since 25 is the square of the hypotenuse, the length of the hypotenuse is the [[square root]] of 25, that is, 5. In other words, if <math>a = 3</math> and <math>b = 4</math>, then <math>c = \sqrt{a^2 + b^2} = 5</math>. ==Etymology== {{wiktionary|ὑποτείνουσα}} The word ''hypotenuse'' is derived from [[Ancient Greek|Greek]] {{lang|grc|ἡ τὴν ὀρθὴν γωνίαν <u>ὑποτείνουσα</u>}} (sc. {{lang|grc|γραμμή}} or {{lang|grc|πλευρά}}), meaning "[side] <u>subtending</u> the right angle" ([[Apollodorus of Athens|Apollodorus]]),<ref>{{LSJ|u(po/}}, {{LSJ|tei/nw}}, {{LSJ|pleura/|ref}}</ref> {{lang|grc|ὑποτείνουσα}} ''hupoteinousa'' being the feminine present active participle of the verb {{lang|grc|ὑποτείνω}} ''hupo-teinō'' "to stretch below, to subtend", from {{lang|grc|τείνω}} ''teinō'' "to stretch, extend". The nominalised participle, {{lang|grc|ἡ ὑποτείνουσα}}, was used for the hypotenuse of a triangle in the 4th century BCE (attested in [[Plato]], ''[[Timaeus (dialogue)|Timaeus]]'' 54d). The Greek term was [[Romanization of Greek|loaned]] into [[Late Latin]],<!--"Late Latin" is from etymonline, without specification; we have evidence for the term being used in New Latin, but so far we have no reference to actual Late Latin--> as ''hypotēnūsa''.<ref>{{Cite web|url=https://www.etymonline.com/word/hypotenuse|title=hypotenuse {{!}} Origin and meaning of hypotenuse by Online Etymology Dictionary|website=www.etymonline.com|language=en|access-date=2019-05-14}}</ref><ref>{{cite web|url=https://www.collinsdictionary.com/dictionary/english/hypotenuse|title=hypotenuse definition and word origin|website=Collins Dictionary|publisher=Collins|access-date=2022-04-12}}</ref> The spelling in ''-e'', as ''hypotenuse'', is French in origin ([[Estienne de La Roche]] 1520).<ref>Estienne de La Roche, ''l'Arismetique'' (1520), fol. 221r (cited after [http://www.cnrtl.fr/etymologie/hypotenuse TLFi]).</ref> <!--WP:DUE A [[folk etymology]] mentioned in the 1940s incorrectly claims that ''tenuse'' means "side" and ''hypotenuse'' means a support like a prop or [[buttress]].<ref>{{cite book |title=Romping Through Mathematics |last=Anderson |first=Raymond |coauthors= |year=1947 |publisher=Faber |location= |isbn= |pages=52}}</ref> [[Merriam-Webster's Collegiate Dictionary]]{{year needed|date=October 2018}} offers the alternative unetymological spelling ''hypothenuse'', but this is very rarely seen. --> ==Properties and calculations== {{anchor|Calculating the hypotenuse}} [[File:Triangle Sides ABC.svg|alt=A right triangle with the legs a and b, and the hypotenuse c|frame|A right triangle with the hypotenuse ''c'']] In a [[right triangle]], the hypotenuse is the [[Edge (geometry)|side]] that is opposite the [[right angle]], while the other two sides are called the ''[[catheti]]'' or ''legs''.<ref>Millian, Richard S.; Parker, George D. (1981). ''Geometry: A Metric Approach with Models''. Undergraduate Texts in Mathematics. New York: Springer. p. 133. {{doi|10.1007/978-1-4684-0130-1}}. {{ISBN|978-1-4684-0130-1}}.</ref> The length of the hypotenuse can be calculated using the [[square root]] function implied by the [[Pythagorean theorem]]. It states that the sum of the two legs [[Square (algebra)|squared]] equals the hypotenuse squared. In mathematical notation, with the respective legs labelled <math>a</math> and <math>b</math>, and the hypotenuse labelled <math>c</math>, it is written as {{nowrap|<math>a^2 + b^2 = c^2</math>.}} Using the square root function on both sides of the equation, it follows that :<math>c = \sqrt { a^2 + b^2 } .</math> This calculation of <math>c</math> from <math>a</math> and <math>b</math> is called [[Pythagorean addition]],<ref>{{cite journal | last1 = Moler | first1 = Cleve | author1-link = Cleve Moler | last2 = Morrison | first2 = Donald | citeseerx = 10.1.1.90.5651 | doi = 10.1147/rd.276.0577 | issue = 6 | journal = IBM Journal of Research and Development | pages = 577–581 | title = Replacing square roots by Pythagorean sums | url = https://scholar.archive.org/work/vn5a4l3fhfbkzomjj4dypacw7q | volume = 27 | year = 1983}}</ref> and is available in many [[software library|software libraries]] as the <code>hypot</code> function.<ref>{{cite book|title=Ivor Horton's Beginning Java 2|first=Ivor|last=Horton|publisher=John Wiley & Sons|year=2005|isbn=9780764568749|page=57|url=https://books.google.com/books?id=h0Uhz4lBoF8C&pg=PA57}}</ref><ref>{{cite book|title=Learning Scientific Programming with Python|first=Christian|last=Hill|edition=2nd|publisher=Cambridge University Press|year=2020|isbn=9781108787468|page=14|url=https://books.google.com/books?id=XAIHEAAAQBAJ&pg=PA14}}</ref> As a consequence of the Pythagorean theorem, the hypotenuse is the longest side of any right triangle; that is, the hypotenuse is longer than either of the triangle's legs. For example, given the length of the legs ''a'' = 5 and ''b'' = 12, then the sum of the legs squared is (5 × 5) + (12 × 12) = 169, the square of the hypotenuse. The length of the hypotenuse is thus the square root of 169, denoted <math>\sqrt{169}</math>, which equals 13. The Pythagorean theorem, and hence this length, can also be derived from the [[law of cosines]] in [[trigonometry]]. In a right triangle, the [[cosine]] of an angle is the [[ratio]] of the leg adjacent of the angle and the hypotenuse. For a right angle ''γ'' (gamma), where the adjacent leg equals 0, the cosine of ''γ'' also equals 0. The law of cosines formulates that <math>c^2 = a^2 + b^2 - 2ab\cos\theta</math> holds for some angle ''θ'' (theta). By observing that the angle opposite the hypotenuse is right and noting that its cosine is 0, so in this case ''θ'' = ''γ'' = 90°: :<math>c^2 = a^2 + b^2 - 2ab\cos\theta = a^2 + b^2 \implies c = \sqrt{a^2 + b^2}.</math> Many computer languages support the ISO C standard function hypot(''x'',''y''), which returns the value above.<ref>{{cite web |title=hypot(3) |url=https://manpages.debian.org/bullseye/manpages-dev/hypot.3.en.html |work=Linux Programmer's Manual |access-date=4 December 2021}}</ref> The function is designed not to fail where the straightforward calculation might overflow or underflow and can be slightly more accurate and sometimes significantly slower. Some languages have extended the definition to higher dimensions. For example, C++17 supports <math>\mbox{std::hypot}(x, y, z) = \sqrt{x^2 +y^2 + z^2}</math>;<ref>{{cite web |title=C++ std::hypot |url=https://en.cppreference.com/w/cpp/numeric/math/hypot |work=C++ Language Manual |access-date=6 June 2024}}</ref> this gives the length of the diagonal of a [[rectangular cuboid]] with edges ''x'', ''y'', and ''z''. Python 3.8 extended <math>\mbox{math.hypot}</math> to handle an arbitrary number of arguments. <ref>{{cite web |title=Python math.hypot |url=https://docs.python.org/3/library/math.html |work=Python Language Manual |access-date=6 June 2024}}</ref> Some scientific calculators{{which|date=December 2021}} provide a function to convert from [[rectangular coordinates]] to [[polar coordinates]]. This gives both the length of the hypotenuse and the [[angle]] the hypotenuse makes with the base line (''c<sub>1</sub>'' above) at the same time when given ''x'' and ''y''. The angle returned is normally given by [[atan2]](''y'',''x''). == Trigonometric ratios == By means of [[Trigonometry#Trigonometric ratios|trigonometric ratios]], one can obtain the value of two acute angles, <math>\alpha\,</math>and <math> \beta\,</math>, of the right triangle. Given the length of the hypotenuse <math> c\,</math>and of a cathetus <math> b\,</math>, the ratio is: [[File:Euklidova veta.svg|upright=1.35|right]] :::<math> \frac{b}{c} = \sin (\beta)\,</math> The trigonometric inverse function is: :::<math> \beta\ = \arcsin\left(\frac {b}{c} \right)\,</math> in which <math>\beta\,</math> is the angle opposite the cathetus <math> b\,</math>. The adjacent angle of the catheti <math> b\,</math> is <math>\alpha\,</math> = 90° – <math>\beta\,</math> One may also obtain the value of the angle <math>\beta\,</math>by the equation: :::<math> \beta\ = \arccos\left(\frac {a}{c} \right)\,</math> in which <math> a\,</math> is the other cathetus. ==See also== {{Portal|Mathematics}} *[[Cathetus]] *[[Triangle]] *[[Space diagonal]] *[[Nonhypotenuse number]] *[[Taxicab geometry]] *[[Trigonometry]] *[[Special right triangles]] *[[Pythagoras]] *[[Norm_(mathematics)#Euclidean_norm]] == Notes == {{Reflist}} == References == * [http://www.encyclopediaofmath.org/index.php/Hypotenuse ''Hypotenuse'' at Encyclopaedia of Mathematics] * {{mathworld|urlname=Hypotenuse|title=Hypotenuse}} {{wiktionary|hypotenuse}} [[Category:Parts of a triangle]] [[Category:Trigonometry]] [[Category:Pythagorean theorem]]
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