Hypotenuse
In geometry, a hypotenuse is the side of a right triangle opposite to the right angle.<ref>Template:Cite EB1911</ref> It is the longest side of any such triangle; the two other shorter sides of such a triangle are called catheti or legs. Every rectangle can be divided into a pair of right triangles by cutting it along either diagonal; the diagonals are the hypotenuses of these triangles.
The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two legs. As an algebraic formula, this can be written as <math>a^2 + b^2 = c^2</math>, where Template:Tmath is the length of one leg, Template:Tmath is the length of the other leg, and Template:Tmath is the length of the hypotenuse.<ref>Template:Cite book</ref> For example, if the two legs of a right triangle have lengths 3 and 4, respectively, then the hypotenuse has length Template:Tmath, because Template:Tmath.
Etymology
Template:Wiktionary The word hypotenuse is derived from Greek Template:Lang (sc. Template:Lang or Template:Lang), meaning "[side] subtending the right angle" (Apollodorus),<ref>Template:LSJ, Template:LSJ, Template:LSJ</ref> Template:Lang hupoteinousa being the feminine present active participle of the verb Template:Lang hupo-teinō "to stretch below, to subtend", from Template:Lang teinō "to stretch, extend". The nominalised participle, Template:Lang, was used for the hypotenuse of a triangle in the 4th century BCE (attested in Plato, Timaeus 54d). The Greek term was loaned into Late Latin, as hypotēnūsa.<ref>Template:Cite web</ref><ref>Template:Cite web</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 TLFi).</ref>
Properties and calculations
In a right triangle, the hypotenuse is the 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. Template:Doi. Template:ISBN.</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 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 Template:Nowrap 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>Template:Cite journal</ref> and is available in many software libraries as the hypot function.<ref>Template:Cite book</ref><ref>Template:Cite book</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>Template:Cite web</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>\textstyle \mbox{std::hypot}(x, y, z) = \sqrt{x^2 +y^2 + z^2}</math>;<ref>Template:Cite web</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>Template:Cite web</ref>
Some scientific calculatorsTemplate:Which 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 (c1 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 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:
- <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
- Cathetus
- Triangle
- Space diagonal
- Nonhypotenuse number
- Taxicab geometry
- Trigonometry
- Special right triangles
- Pythagoras
- Norm_(mathematics)#Euclidean_norm