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{{short description|Triangle containing a 90-degree angle}}
[[File:Rtriangle.svg|right|thumb|upright=1.25|A right triangle {{math|△''ABC''}} with its right angle at {{mvar|C}}, hypotenuse {{mvar|c}}, and legs {{mvar|a}} and {{mvar|b}},]]
A '''right triangle''' or '''right-angled triangle''', sometimes called an '''orthogonal triangle''' or '''rectangular triangle''', is a [[triangle]] in which two [[Edge (geometry)|sides]] are [[perpendicular]], forming a [[right angle]] ({{frac|1|4}} [[turn (unit)|turn]] or 90 [[degree (angle)|degrees]]).

The side opposite to the right angle is called the ''[[hypotenuse]]'' (side <math>c</math> in the figure). The sides adjacent to the right angle are called ''legs'' (or ''catheti'', singular: ''[[cathetus]]''). Side <math>a</math> may be identified as the side ''adjacent'' to angle <math>B</math> and ''opposite'' (or ''opposed to'') angle <math>A,</math> while side <math>b</math> is the side adjacent to angle <math>A</math> and opposite angle <math>B.</math>

Every right triangle is half of a [[rectangle]] which has been divided along its [[diagonal]]. When the rectangle is a [[square]], its right-triangular half is [[isosceles triangle|isosceles]], with two congruent sides and two congruent angles. When the rectangle is not a square, its right-triangular half is [[scalene triangle|scalene]].

Every triangle whose [[base (geometry)|base]] is the [[diameter]] of a [[circle]] and whose [[apex (geometry)|apex]] lies on the circle is a right triangle, with the right angle at the apex and the hypotenuse as the base; conversely, the [[circumcircle]] of any right triangle has the hypotenuse as its diameter. This is [[Thales' theorem]].

The legs and hypotenuse of a right triangle satisfy the [[Pythagorean theorem]]: the sum of the areas of the squares on two legs is the area of the square on the hypotenuse, <math>a^2 + b^2 = c^2.</math> If the lengths of all three sides of a right triangle are integers, the triangle is called a '''Pythagorean triangle''' and its side lengths are collectively known as a ''[[Pythagorean triple]]''.

The relations between the sides and angles of a right triangle provides one way of defining and understanding [[trigonometry]], the study of the metrical relationships between lengths and angles.

==Principal properties==

===Sides===
[[File:Illustration to Euclid's proof of the Pythagorean theorem2.svg|thumb|The diagram for Euclid's proof of the Pythagorean theorem: each smaller square has area equal to the rectangle of corresponding color.]]
{{main|Pythagorean theorem}}

The three sides of a right triangle are related by the [[Pythagorean theorem]], which in modern algebraic notation can be written

:<math>a^2 + b^2 = c^2,</math>

where <math>c</math> is the length of the ''hypotenuse'' (side opposite the right angle), and <math>a</math> and <math>b</math> are the lengths of the ''legs'' (remaining two sides). [[Pythagorean triple]]s are integer values of <math>a, b, c</math> satisfying this equation. This theorem was proven in antiquity, and is proposition I.47 in [[Euclid's Elements|Euclid's ''Elements'']]: "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle."

===Area===
As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the base then the other is height, so the area of a right triangle is one half the product of the two legs. As a formula the area <math>T</math> is
:<math>T=\tfrac{1}{2}ab</math>

where <math>a</math> and <math>b</math> are the legs of the triangle.

If the [[Incircle and excircles of a triangle|incircle]] is tangent to the hypotenuse <math>AB</math> at point <math>P,</math> then letting the [[semi-perimeter]] be <math>s = \tfrac12(a+b+c),</math> we have  <math>|PA| = s-a</math> and <math>|PB| = s-b,</math> and the area is given by
:<math>T= |PA| \cdot |PB| = (s-a)(s-b).</math>

This formula only applies to right triangles.<ref>Di Domenico, Angelo S., "A property of triangles involving area", ''[[Mathematical Gazette]]'' 87, July 2003, pp. 323–324.</ref>

===Altitudes===
[[Image:Teorema.png|thumb|right|Altitude {{mvar|f}} of a right triangle]]<!--Better diagram needed here.-->
If an [[Altitude (triangle)|altitude]] is drawn from the vertex, with the right angle to the hypotenuse, then the triangle is divided into two smaller triangles; these are both [[Similarity (geometry)|similar]] to the original, and therefore similar to each other. From this:
* The altitude to the hypotenuse is the [[geometric mean]] ([[Ratio#Euclid's definitions|mean proportional]]) of the two segments of the hypotenuse.<ref name=Posamentier/>{{rp|243}}
* Each leg of the triangle is the mean proportional of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.
In equations,
:<math>f^2=de,</math> (this is sometimes known as the [[right triangle altitude theorem]])
:<math>b^2=ce,</math>
:<math>a^2=cd</math>

where <math>a,b,c,d,e,f</math> are as shown in the diagram.<ref>Wentworth p. 156</ref> Thus
:<math>f=\frac{ab}{c}.</math>

Moreover, the altitude to the hypotenuse is related to the legs of the right triangle by<ref>Voles, Roger, "Integer solutions of <math>a^{-2} + b^{-2} = d^{-2}</math>," ''Mathematical Gazette'' 83, July 1999, 269–271.</ref><ref>Richinick, Jennifer, "The upside-down Pythagorean Theorem," ''Mathematical Gazette'' 92, July 2008, 313–317.</ref>
:<math>\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{f^2}.</math>

For solutions of this equation in integer values of <math>a,b,c,f,</math> see [[Integer triangle#Pythagorean triangles with integer altitude from the hypotenuse|here]].

The altitude from either leg coincides with the other leg. Since these intersect at the right-angled vertex, the right triangle's [[orthocenter]]—the intersection of its three altitudes—coincides with the right-angled vertex.

===Inradius and circumradius===
The radius of the [[incircle]] of a right triangle with legs <math>a</math> and <math>b</math> and hypotenuse <math>c</math> is
:<math>r = \frac{a+b-c}{2} = \frac{ab}{a+b+c}.</math>

The radius of the [[circumcircle]] is half the length of the hypotenuse,
:<math>R = \frac{c}{2}.</math>

Thus the sum of the circumradius and the inradius is half the sum of the legs:<ref name="Crux">''Inequalities proposed in "[[Crux Mathematicorum]]"'', [http://www.imomath.com/othercomp/Journ/ineq.pdf].</ref>

:<math>R+r = \frac{a+b}{2}.</math>

One of the legs can be expressed in terms of the inradius and the other leg as
:<math>a=\frac{2r(b-r)}{b-2r}.</math>

==Characterizations==
A triangle <math>\triangle ABC</math> with sides <math>a \le b < c</math>, [[semiperimeter]] <math display=inline>s = \tfrac12(a+b+c)</math>, [[area]] <math>T,</math> [[altitude (triangle)|altitude]] <math>h_c</math> opposite the longest side, [[Circumscribed circle|circumradius]] <math>R,</math> [[Incircle and excircles of a triangle#Relation to area of the triangle|inradius]] <math>r,</math> [[Incircle and excircles of a triangle#Relation to area of the triangle|exradii]] <math>r_a, r_b, r_c</math> tangent to <math>a,b,c</math> respectively, and [[median (geometry)|medians]] <math>m_a, m_b, m_c</math> is a right triangle [[if and only if]] any one of the statements in the following six categories is true. Each of them is thus also a property of any right triangle.

===Sides and semiperimeter===
* <math>a^2+b^2=c^2\quad (\text{Pythagorean theorem})</math>
* <math>(s-a)(s-b) = s(s-c)</math>
* <math>s=2R+r.</math><ref>{{Cite web |url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=411120 |title=Triangle right iff s = 2R + r, ''Art of problem solving'', 2011 |access-date=2012-01-02 |archive-date=2014-04-28 |archive-url=https://web.archive.org/web/20140428221212/http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=411120 |url-status=dead }}</ref>
* <math>a^2+b^2+c^2=8R^2.</math><ref name=Andreescu/>

===Angles===
* <math>A</math> and <math>B</math> are [[complementary angles|complementary]].<ref>{{Cite web |url=http://www.ricksmath.com/right-triangles.html |title=Properties of Right Triangles |access-date=2012-02-15 |archive-date=2011-12-31 |archive-url=https://web.archive.org/web/20111231222001/http://www.ricksmath.com/right-triangles.html |url-status=dead }}</ref>
* <math>\cos{A}\cos{B}\cos{C}=0.</math><ref name=Andreescu/><ref name="CTK">CTK Wiki Math, ''A Variant of the Pythagorean Theorem'', 2011, [http://www.cut-the-knot.org/wiki-math/index.php?n=Trigonometry.AVariantOfPythagoreanTheorem] {{Webarchive|url=https://web.archive.org/web/20130805051705/http://www.cut-the-knot.org/wiki-math/index.php?n=Trigonometry.AVariantOfPythagoreanTheorem|date=2013-08-05}}.</ref>
* <math>\sin^2{A}+\sin^2{B}+\sin^2{C}=2.</math><ref name=Andreescu/><ref name=CTK/>
* <math>\cos^2{A}+\cos^2{B}+\cos^2{C}=1.</math><ref name=CTK/>
* <math>\sin{2A}=\sin{2B}=2\sin{A}\sin{B}.</math>

===Area===
* <math>T=\frac{ab}{2}</math>
* <math>T=r_ar_b=rr_c</math>
* <math>T=r(2R+r)</math>
* <math>T=\frac{(2s-c)^2-c^2}{4}=s(s-c)</math>
* <math>T=|PA| \cdot |PB|,</math> where <math>P</math> is the tangency point of the [[Incircle and excircles of a triangle|incircle]] at the longest side <math>AB.</math><ref>{{citation
|last=Darvasi |first=Gyula
|journal=The Mathematical Gazette
|pages=72–76
|title=Converse of a Property of Right Triangles
|volume=89
|number=514
|date=March 2005|doi=10.1017/S0025557200176806
|s2cid=125992270
|doi-access=free
}}.</ref>

===Inradius and exradii===
* <math>r=s-c=(a+b-c)/2</math>
* <math>r_a=s-b=(a-b+c)/2</math>
* <math>r_b=s-a=(-a+b+c)/2</math>
* <math>r_c=s=(a+b+c)/2</math>
* <math>r_a+r_b+r_c+r=a+b+c</math>
* <math>r_a^2+r_b^2+r_c^2+r^2=a^2+b^2+c^2</math>
* <math>r=\frac{r_ar_b}{r_c}.</math><ref name=Bell>{{citation|last=Bell |first=Amy|journal=Forum Geometricorum|pages=335–342|title=Hansen's Right Triangle Theorem, Its Converse and a Generalization|url=http://forumgeom.fau.edu/FG2006volume6/FG200639.pdf |archive-url=https://web.archive.org/web/20080725014729/http://forumgeom.fau.edu/FG2006volume6/FG200639.pdf |archive-date=2008-07-25 |url-status=live|volume=6|year=2006}}</ref>

===Altitude and medians===
{{right_angle_altitude.svg}}
* <math>h_c=\frac{ab}{c}</math>
* <math>m_a^2+m_b^2+m_c^2=6R^2.</math><ref name=Crux/>{{rp|Prob. 954, p. 26}}
* The length of one [[Median (geometry)|median]] is equal to the [[Circumscribed circle|circumradius]].
* The shortest [[Altitude (triangle)|altitude]] (the one from the vertex with the biggest angle) is the [[geometric mean]] of the [[line segment]]s it divides the opposite (longest) side into. This is the [[right triangle altitude theorem]].

===Circumcircle and incircle===
* The triangle can be inscribed in a [[semicircle]], with one side coinciding with the entirety of the diameter ([[Thales' theorem]]).
* The [[Circumscribed circle|circumcenter]] is the [[midpoint]] of the longest side.
* The longest side is a [[diameter]] of the [[Circumscribed circle#Circumscribed circles of triangles|circumcircle]] <math>(c=2R).</math>
* The circumcircle is [[tangent]] to the [[nine-point circle]].<ref name=Andreescu>Andreescu, Titu and Andrica, Dorian, "Complex Numbers from A to...Z", Birkhäuser, 2006, pp. 109–110.</ref>
* The [[Altitude (triangle)#Orthocenter|orthocenter]] lies on the circumcircle.<ref name=Crux/>
* The distance between the [[Incircle and excircles of a triangle|incenter]] and the orthocenter is equal to <math>\sqrt{2}r</math>.<ref name=Crux/>

==Trigonometric ratios==
{{main|Trigonometric functions#Right-angled triangle definitions|l1=Trigonometric functions – Right-angled triangle definitions}}

The [[trigonometric functions]] for acute angles can be defined as ratios of the sides of a right triangle. For a given angle, a right triangle may be constructed with this angle, and the sides labeled opposite, adjacent and hypotenuse with reference to this angle according to the definitions above. These ratios of the sides do not depend on the particular right triangle chosen, but only on the given angle, since all triangles constructed this way are [[similar triangles|similar]]. If, for a given angle α, the opposite side, adjacent side and hypotenuse are labeled <math>O,</math> <math>A,</math> and <math>H,</math> respectively, then the trigonometric functions are
:<math>\sin\alpha =\frac {O}{H},\,\cos\alpha =\frac {A}{H},\,\tan\alpha =\frac {O}{A},\,\sec\alpha =\frac {H}{A},\,\cot\alpha =\frac {A}{O},\,\csc\alpha =\frac {H}{O}.</math>

For the expression of [[hyperbolic function]]s as ratio of the sides of a right triangle, see the [[hyperbolic sector#Hyperbolic triangle|hyperbolic triangle]] of a [[hyperbolic sector]].

==Special right triangles==
{{main|Special right triangles}}

The values of the trigonometric functions can be evaluated exactly for certain angles using right triangles with special angles. These include the 30-60-90 triangle which can be used to evaluate the trigonometric functions for any multiple of <math>\tfrac16\pi,</math> and the isosceles right triangle or 45-45-90 triangle which can be used to evaluate the trigonometric functions for any multiple of <math>\tfrac14\pi.</math>

===Kepler triangle===
Let  <math>H,</math> <math>G,</math> and <math>A</math> be the [[harmonic mean]], the [[geometric mean]], and the [[arithmetic mean]] of two positive numbers <math>a</math> and <math>b</math> with <math>a > b.</math> If a right triangle has legs <math>H</math> and <math>G</math> and hypotenuse <math>A,</math> then<ref>Di Domenico, A., "The golden ratio — the right triangle — and the arithmetic, geometric, and harmonic means," ''Mathematical Gazette'' 89, July 2005, 261. Also Mitchell, Douglas W., "Feedback on 89.41", vol 90, March 2006, 153–154.</ref>
:<math>
\frac{A}{H} = \frac{A^{2}}{G^{2}} = \frac{G^{2}}{H^{2}} = \phi, \qquad
\frac{a}{b} = \phi^{3},
</math>

where <math>\phi = \tfrac12\bigl(1 + \sqrt{5}\bigr)</math> is the [[golden ratio]]. Since the sides of this right triangle are in [[geometric progression]], this is the [[Kepler triangle]].

==Thales' theorem==
{{main|Thales' theorem}}
[[Image:thm mediane.svg|thumb|300px|right|Median of a right angle of a triangle]]
'''Thales' theorem''' states that if <math>BC</math> is the diameter of a circle and <math>A</math> is any other point on the circle, then <math>\triangle ABC</math> is a right triangle with a right angle at <math>A.</math> The converse states that the hypotenuse of a right triangle is the diameter of its [[circumcircle]]. As a corollary, the circumcircle has its center at the midpoint of the diameter, so the [[median (geometry)|median]] through the right-angled vertex is a radius, and the circumradius is half the length of the hypotenuse.

==Medians==
The following formulas hold for the [[median (geometry)|medians]] of a right triangle:
:<math>m_a^2 + m_b^2 = 5m_c^2 = \frac{5}{4}c^2.</math>

The median on the hypotenuse of a right triangle divides the triangle into two isosceles triangles, because the median equals one-half the hypotenuse.

The medians <math>m_a</math> and <math>m_b</math> from the legs satisfy<ref name=Crux/>{{rp|p.136,#3110}}

:<math>4c^4+9a^2b^2=16m_a^2m_b^2.</math>

==Euler line==
In a right triangle, the [[Euler line]] contains the median on the hypotenuse—that is, it goes through both the right-angled vertex and the midpoint of the side opposite that vertex. This is because the right triangle's orthocenter, the intersection of its altitudes, falls on the right-angled vertex while its circumcenter, the intersection of its [[Bisection#Perpendicular bisectors|perpendicular bisectors of sides]], falls on the midpoint of the hypotenuse.

==Inequalities==

In any right triangle the diameter of the incircle is less than half the hypotenuse, and more strongly it is less than or equal to the hypotenuse times <math>(\sqrt{2}-1).</math><ref name=PL>Posamentier, Alfred S., and Lehmann, Ingmar. ''[[The Secrets of Triangles]]''. Prometheus Books, 2012.</ref>{{rp|p.281}}

In a right triangle with legs <math>a,b</math> and hypotenuse <math>c,</math>

:<math>c \geq \frac{\sqrt{2}}{2}(a+b)</math>

with equality only in the isosceles case.<ref name=PL/>{{rp|p.282, p.358}}

If the altitude from the hypotenuse is denoted <math>h_c,</math> then

:<math>h_c \leq \frac{\sqrt {2}}{4}(a+b)</math>

with equality only in the isosceles case.<ref name=PL/>{{rp|p.282}}

==Other properties==
If segments of lengths <math>p</math> and <math>q</math> emanating from vertex <math>C</math> trisect the hypotenuse into segments of length <math>\tfrac13 c,</math> then<ref name=Posamentier>Posamentier, Alfred S., and Salkind, Charles T. ''Challenging Problems in Geometry'', Dover, 1996.</ref>{{rp|pp. 216–217}}
:<math>p^2 + q^2 = 5\left(\frac{c}{3}\right)^2.</math>

The right triangle is the only triangle having two, rather than one or three, distinct inscribed squares.<ref>Bailey, Herbert, and DeTemple, Duane, "Squares inscribed in angles and triangles", ''[[Mathematics Magazine]]'' 71(4), 1998, 278–284.</ref>

Given any two positive numbers <math>h</math> and <math>k</math> with <math>h > k.</math> Let <math>h</math> and <math>k</math>  be the sides of the two inscribed squares in a right triangle with hypotenuse <math>c.</math> Then
:<math>\frac{1}{c^2} + \frac{1}{h^2} = \frac{1}{k^2}.</math>

These sides and the incircle radius <math>r</math> are related by a similar formula:

:<math>\frac{1}{r}=-{\frac{1}{c}}+\frac{1}{h}+\frac{1}{k}.</math>

The perimeter of a right triangle equals the sum of the radii of [[Incircle|the incircle and the three excircles]]:

:<math>a+b+c=r+r_a+r_b+r_c.</math>

==See also==
* [[Acute and obtuse triangles]] (oblique triangles)
* [[Spiral of Theodorus]]
* [[Trirectangular spherical triangle]]

==References==
{{reflist}}
* {{MathWorld |title=Right Triangle |urlname=RightTriangle}}
* {{cite book |title=A Text-Book of Geometry |first=G.A. |last=Wentworth |publisher=Ginn & Co. |year=1895 
|url=https://archive.org/details/atextbookgeomet10wentgoog}}

==External links==
{{Commons category|Right triangles}}
* {{usurped|1=[https://web.archive.org/web/20170930064506/http://www.kurztutorial.info/mathematik/trigonometrie/en/dreieck.html Calculator for right triangles]}}
* [http://www.triangle-calculator.com/?what=rt Advanced right triangle calculator]

{{Polygons}}

[[Category:Types of triangles]]
[[Category:Orthogonality]]
[[Category:Trigonometry]]