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A right-angled triangle and its hypotenuse

In geometry, a hypotenuse is the side of a right triangle opposite the right angle.^[1] 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 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 $a^{2}+b^{2}=c^{2}$ , where a is the length of one leg, b is the length of another leg, and c is the length of the hypotenuse.^[2]

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 $a=3$ and $b=4$ , then $c={\sqrt {a^{2}+b^{2}}}=5$ .

Etymology

The word hypotenuse is derived from Greek ἡ τὴν ὀρθὴν γωνίαν ὑποτείνουσα (sc. γραμμή or πλευρά), meaning "[side] subtending the right angle" (Apollodorus),^[3] ὑποτείνουσα hupoteinousa being the feminine present active participle of the verb ὑποτείνω hupo-teinō "to stretch below, to subtend", from τείνω teinō "to stretch, extend". The nominalised participle, ἡ ὑποτείνουσα, 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.^[4]^[5] The spelling in -e, as hypotenuse, is French in origin (Estienne de La Roche 1520).^[6]

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.^[7] 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 a and b, and the hypotenuse labelled c, it is written as $a^{2}+b^{2}=c^{2}$ . Using the square root function on both sides of the equation, it follows that

c={\sqrt {a^{2}+b^{2}}}.

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 ${\sqrt {169}}$ , 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 $c^{2}=a^{2}+b^{2}-2ab\cos \theta$ 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°:

c^{2}=a^{2}+b^{2}-2ab\cos \theta =a^{2}+b^{2}\implies c={\sqrt {a^{2}+b^{2}}}.

Many computer languages support the ISO C standard function hypot(x,y), which returns the value above.^[8] 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 ${\mbox{std::hypot}}(x,y,z)={\sqrt {x^{2}+y^{2}+z^{2}}}$ ;^[9] this gives the length of the diagonal of a rectangular cuboid with edges x, y, and z. Python 3.8 extended ${\mbox{math.hypot}}$ to handle an arbitrary number of arguments. ^[10]

Some scientific calculators^[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, $\alpha \,$ and $\beta \,$ , of the right triangle.

Given the length of the hypotenuse $c\,$ and of a cathetus $b\,$ , the ratio is:

{\frac {b}{c}}=\sin(\beta )\,

The trigonometric inverse function is:

\beta \ =\arcsin \left({\frac {b}{c}}\right)\,

in which $\beta \,$ is the angle opposite the cathetus $b\,$ .

The adjacent angle of the catheti $b\,$ is $\alpha \,$ = 90° – $\beta \,$

One may also obtain the value of the angle $\beta \,$ by the equation:

\beta \ =\arccos \left({\frac {a}{c}}\right)\,

in which $a\,$ is the other cathetus.

Notes

^ "Triangle (geometry)" . Encyclopædia Britannica. Vol. 27 (11th ed.). 1911. p. 258. ...Also a right-angled triangle has one angle a right angle, the side opposite this angle being called the hypotenuse;...
^ Jr, Jesse Moland (August 2009). I Hate Trig!: A Practical Guide to Understanding Trigonometry. Jesse Moland. p. 1. ISBN 978-1-4486-4707-1.
^ u(po/, tei/nw, pleura/. Liddell, Henry George; Scott, Robert; A Greek–English Lexicon at the Perseus Project
^ "hypotenuse | Origin and meaning of hypotenuse by Online Etymology Dictionary". www.etymonline.com. Retrieved 2019-05-14.
^ "hypotenuse definition and word origin". Collins Dictionary. Collins. Retrieved 2022-04-12.
^ Estienne de La Roche, l'Arismetique (1520), fol. 221r (cited after TLFi).
^ 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.
^ "hypot(3)". Linux Programmer's Manual. Retrieved 4 December 2021.
^ "C++ std::hypot". C++ Language Manual. Retrieved 6 June 2024.
^ "Python math.hypot". Python Language Manual. Retrieved 6 June 2024.

References

[1] "Triangle (geometry)" . Encyclopædia Britannica. Vol. 27 (11th ed.). 1911. p. 258. ...Also a right-angled triangle has one angle a right angle, the side opposite this angle being called the hypotenuse;...

[2] Jr, Jesse Moland (August 2009). I Hate Trig!: A Practical Guide to Understanding Trigonometry. Jesse Moland. p. 1. ISBN 978-1-4486-4707-1.

[3] u(po/, tei/nw, pleura/. Liddell, Henry George; Scott, Robert; A Greek–English Lexicon at the Perseus Project

[4] "hypotenuse | Origin and meaning of hypotenuse by Online Etymology Dictionary". www.etymonline.com. Retrieved 2019-05-14.

[5] "hypotenuse definition and word origin". Collins Dictionary. Collins. Retrieved 2022-04-12.

[6] Estienne de La Roche, l'Arismetique (1520), fol. 221r (cited after TLFi).

[7] 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.

[8] "hypot(3)". Linux Programmer's Manual. Retrieved 4 December 2021.

[9] "C++ std::hypot". C++ Language Manual. Retrieved 6 June 2024.

[10] "Python math.hypot". Python Language Manual. Retrieved 6 June 2024.

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