The word hypotenuse is derived from Greek ἡ τὴν ὀρθὴν γωνίαν ὑποτείνουσα, meaning " subtending the right angle", 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 BC. The Greek term was loaned into Late Latin, as hypotēnūsa. The spelling in -e, as hypotenuse, is French in origin.
Calculating the hypotenuse
The length of the hypotenuse is calculated using the square root function implied by the Pythagorean theorem. Using the common notation that the length of the two legs of the triangle are a and b and that of the hypotenuse is c, we have 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, which returns the value above. 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 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 at the same time when given x and y. The angle returned is normally given by atan2.
Properties
s:
The length of the hypotenuse equals the sum of the lengths of the orthographic projections of both catheti.
The square of the length of a cathetus equals the product of the lengths of its orthographic projection on the hypotenuse times the length of this.
Also, the length of a cathetus b is the proportional mean between the lengths of its projection m and the hypotenuse a.
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.
