Lemniscate of Bernoulli: Difference between revisions

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[[File:Hypotenuse.svg|thumb|150px|Aright|A right-angled triangle and its hypotenuse.]]
In [[geometry]], a '''hypotenuse''' is the longest side of a [[Right triangle|right-angled triangle]], the side opposite of the [[right angle]]. The length of the hypotenuse of a [[right triangle]] 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 other two sides.
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.
==Etymology==
The word ''hypotenuse'' means essentially "length under", and derives from [[Latin]] ''hypotēnūsa'', a [[Romanization of Greek|transliteration]] of [[Ancient Greek]] {{Unicode|''hypoteínousa'' (''pleurā&#769;'' or ''grammē&#769;'')}}, the [[grammatical gender|feminine]] [[present tense|present]] [[participle]] of ''hypoteínō'', a combination of ''hypó'' ("under") and ''teínō'' ("I stretch" or "length").<ref>{{OEtymD|hypotenuse}}</ref><ref>{{LSJ|u(potei/nw}}, {{LSJ|u(po/}}, {{LSJ|tei/nw}}, {{LSJ|pleura/|ref}}</ref> The word ὑποτείνουσα was used for the hypotenuse of a triangle by [[Plato]] in the [[Timaeus (dialogue)]] 54d and by many other ancient authors.
 
A [[folk etymology]] says that ''tenuse'' means "side", so ''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> but this is inaccurate.
 
==Calculating the hypotenuse==
[[File:Triangle Sides.svg|200px|frame|right|A right-angled triangle and its hypotenuse, ''h'', along with [[Cathetus|catheti]], ''c<sub>1</sub>'' and ''c<sub>2</sub>''.]]
 
Usually the length of the hypotenuse is calculated using the [[square root]] function derived from the [[Pythagorean theorem]]. Setting x&nbsp;=&nbsp;c<sub>1</sub> and y&nbsp;=&nbsp;c<sub>2</sub> to avoid subscripts:
 
In mathematical notation;
 
:<math>h = \sqrt { x^2 + y^2 } </math>
 
The length can also be derived from the [[law of cosines]] by setting ''&#611;'' to 90&deg;:
 
:<math>c^2 = a^2 + b^2 - 2ab\cos90^\circ = a^2 + b^2 \therefore c = \sqrt{a^2 + b^2}</math>
 
Many computer languages support the ISO C standard function hypot(''x'',''y''), 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.
 
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 (''c<sub>1</sub>'' above) at the same time when given ''x'' and ''y''. The angle returned will normally be that given by [[atan2]](''y'',''x'').
 
== Properties ==
[[File:Triângulo retângulo.svg|thumb|225px|right|In the figure, '''a''' is the hypotenuse and '''b''' and '''c''' are the catheti. The orthographic projection of '''b''' is '''m''', and of '''c''' is '''n'''.]]
 
[[Orthographic projection]]s:
 
* The length of the hypotenuse equals the sum  of the lengths  of the orthographic projections of both catheti. And
 
* The square of the length of a cathetus equals the [[Product (mathematics)|product]] of the lengths of its orthographic projection on the hypotenuse times the length of this.
 
::'''b² = a · m'''
::'''c² = a · n'''
 
* Also, the length of a cathetus '''b''' is the proportional mean between the lengths of its projection '''m''' and the hypotenuse '''a'''.
 
::'''a/b = b/m'''
::'''a/c = c/n'''
 
== Trigonometric ratios ==
 
By means of [[Trigonometry|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|330px|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>, will be <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]]
 
== Notes ==
{{Reflist}}
 
== References ==
* [http://www.encyclopediaofmath.org/index.php/Hypotenuse ''Hypotenuse'' at Encyclopaedia of Mathematics]
* {{mathworld|urlname=Hypotenuse|title=Hypotenuse}}
 
[[Category:Elementary geometry]]
[[Category:Triangles]]
[[Category:Trigonometry]]
 
[[de:Rechtwinkliges Dreieck#Hypotenuse]]
[[vi:Tam giác#Phân loại tam giác]]

Latest revision as of 15:16, 15 October 2014

If you were young, perhaps you were one of those fortunate persons who didnt need fat reduction help. Maybe we were chasing kids, working at an outside job, cooking, cleaning and living an active lifestyle. Its not that guys over 50 plus post-menopausal females are not active. However frequently they are less active. This could be considering of physical limitations, including arthritis, or because they have simply gotten chosen to a more sedentary life-style inside retirement or when they are still functioning.

Kiley's e-book Personal Path to Pregnancy can be worth buying. The cost is far within the expense is to go see a doctor. The e-book costs less than thirty five dollars plus in case you are not content with it she usually give you a refund, a thing healthcare professionals do not provide! Kiley has 9 free bonuses together with a baby names list, bmi chart, the greatest way to workout and ways to avoid stretch marks. She delivers we with several secrets regarding how to conceive and considering it's natural there isn't anything to get rid of. Doctors do not provide this info because their practice is medicine hence they naturally lean to treatments as an answer. Beth helped a lot of couples and those people have given their own testimonials about their ordeal. This is well-worth the chance.

Strength Training: There is a great deal of information available on lifting weights plus strength training, nevertheless bmi chart men being thoughtful to commence this inside the "right" technique is significant because you receive older. Running Planet has completed a nice job w/ laying out "The 8 rules of Strength Training". We have some advantageous videos on our Resources page.

Lets state an adult man is 60 and weighs 200 pounds. According to the chart, his BMI would be 27.1, which puts him into the overweight category. If he loses 17 pounds, the same man, today at 183 pounds, would have a BMI of 24.8, which would place him in the general weight category.

29. Reading about Running: There are so many fabulous books out there on running that are fun to read. They can motivate and excite you. We have a limited posted on our Amazon Store.

Massage: Another Dara Torres staple plus 1 of my personal favorites. It does not bmi chart women matter should you have a fabulous husband like I do or receive from a professional, it works to relieve the stress of training plus tired muscles. You can even do it yourself w/ several videos by Rich Poley whom wrote "Self Massage for Athletes".

Let's break it down a bit. "Ideal body weight" is defined as the fat persons are expected to weigh for good health based on age, sex plus height. Body composition is the makeup of the body inside terms of fat-free mass (the part of the body composition that represents everything nevertheless fat--blood, bones, connective cells, organs and muscles) plus fat mass (actual amount of essential plus non-essential fat inside the body.) Essential fat is the amount of fat necessary for general working of the body.

If you think you are regular weight obese, the best thing you are able to do is consult the doctor, and start to create healthier options inside your life. For more information feel free to go and visit a few of the hyperlinks I've provided.

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