Hypotenuse
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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 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 other two sides. Mathematically, this can be written as , 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 other sides has a length of 3 (when squared, 9) and the other has a length of 4 (when squared, 16), then their squares add up to 25. The length of the hypotenuse is the square root of 25, that is, 5. In other words, if and , then .
Etymology
The word hypotenuse is derived from
Calculating the hypotenuse
The length of the hypotenuse can be calculated using the
The Pythagorean theorem, and hence this length, can also be derived from the law of cosines by observing that the angle opposite the hypotenuse is 90° and noting that its cosine is 0:
Many computer languages support the ISO C standard function hypot(x,y), which returns the value above.[7] 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 scientific calculators[(y,x).
Trigonometric ratios
By means of trigonometric ratios, one can obtain the value of two acute angles, and , of the right triangle.
Given the length of the hypotenuse and of a cathetus , the ratio is:
The trigonometric inverse function is:
in which is the angle opposite the cathetus .
The adjacent angle of the catheti is = 90° –
One may also obtain the value of the angle by the equation:
in which is the other cathetus.
See also
- Cathetus
- Triangle
- Space diagonal
- Nonhypotenuse number
- Taxicab geometry
- Trigonometry
- Special right triangles
- Pythagoras
- Norm_(mathematics)#Euclidean_norm
Notes
- ^ Chisholm, Hugh, ed. (1911). Encyclopædia Britannica (Eleventh ed.). Cambridge University Press – via Wikisource. .
- ISBN 978-1-4486-4707-1.
- 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).
- ^ "hypot(3)". Linux Programmer's Manual. Retrieved 4 December 2021.