Movatterモバイル変換

Jump to content

Right triangle

From Wikipedia, the free encyclopedia

(Redirected fromRight-angled triangle)

Triangle containing a 90-degree angle

A right triangle△ABC with its right angle atC, hypotenusec, and legsa andb,

Aright triangle orright-angled triangle, sometimes called anorthogonal triangle orrectangular triangle, is atriangle in which twosides areperpendicular, forming aright angle (1⁄4turn or 90degrees).

The side opposite to the right angle is called thehypotenuse (side $c {\displaystyle c}$ in the figure). The sides adjacent to the right angle are calledlegs (orcatheti, singular:cathetus). Side $a {\displaystyle a}$ may be identified as the sideadjacent to angle $B {\displaystyle B}$ andopposite (oropposed to) angle $A, {\displaystyle A,}$ while side $b {\displaystyle b}$ is the side adjacent to angle $A {\displaystyle A}$ and opposite angle $B . {\displaystyle B.}$

Every right triangle is half of arectangle which has been divided along itsdiagonal. When the rectangle is asquare, its right-triangular half isisosceles, with two congruent sides and two congruent angles. When the rectangle is not a square, its right-triangular half isscalene.

Every triangle whosebase is thediameter of acircle and whoseapex lies on the circle is a right triangle, with the right angle at the apex and the hypotenuse as the base; conversely, thecircumcircle of any right triangle has the hypotenuse as its diameter. This isThales' theorem.

The legs and hypotenuse of a right triangle satisfy thePythagorean theorem: the sum of the areas of the squares on two legs is the area of the square on the hypotenuse, $a^{2}+b^{2}=c^{2}.$ If the lengths of all three sides of a right triangle are integers, the triangle is called aPythagorean triangle and its side lengths are collectively known as aPythagorean triple.

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

Principal properties

Sides

Main article:Pythagorean theorem

TheBride's Chair from the proof of thePythagorean theorem, in the colored version used by Byrne's 1847 edition. The proof shows that the black and yellow areas are equal, as are the red and blue areas.

The three sides of a right triangle are related by thePythagorean theorem, which in modern algebraic notation can be written $a^{2}+b^{2}=c^{2},$ where $c {\displaystyle c}$ is the length of thehypotenuse (side opposite the right angle), and $a {\displaystyle a}$ and $b {\displaystyle b}$ are the lengths of thelegs (remaining two sides). This theorem was proven in antiquity, and is proposition I.47 inEuclid'sElements: "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."^[1] Any integer values of $a, b, c {\displaystyle a,b,c}$ satisfying this equation is calledPythagorean triple.

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 $T {\displaystyle T}$ is

T={\tfrac {1}{2}}ab

where $a {\displaystyle a}$ and $b {\displaystyle b}$ are the legs of the triangle.

If theincircle is tangent to the hypotenuse $A B {\displaystyle AB}$ at point $P, {\displaystyle P,}$ then letting thesemi-perimeter be $s={\tfrac {1}{2}}(a+b+c),$ we have $|PA|=s-a$ and $|PB|=s-b,$ and the area is given by

T=|PA|\cdot |PB|=(s-a)(s-b).

This formula only applies to right triangles.^[2]

Altitudes

Altitudef of a right triangle

If analtitude is drawn from the vertex, with the right angle to the hypotenuse, then the triangle is divided into two smaller triangles; these are bothsimilar to the original, and therefore similar to each other. From this:

The altitude to the hypotenuse is thegeometric mean (mean proportional) of the two segments of the hypotenuse.^[3]^: 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,

f^{2}=de,

(this is sometimes known as theright triangle altitude theorem)

b^{2}=ce,

a^{2}=cd

where $a, b, c, d, e, f {\displaystyle a,b,c,d,e,f}$ are as shown in the diagram.^[4] Thus

f={\frac {ab}{c}}.

Moreover, the altitude to the hypotenuse is related to the legs of the right triangle by^[5]^[6]

{\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}={\frac {1}{f^{2}}}.

For solutions of this equation in integer values of $a, b, c, f, {\displaystyle a,b,c,f,}$ seehere.

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

Inradius and circumradius

The radius of theincircle of a right triangle with legs $a {\displaystyle a}$ and $b {\displaystyle b}$ and hypotenuse $c {\displaystyle c}$ is

r={\frac {a+b-c}{2}}={\frac {ab}{a+b+c}}.

ByThales's theorem, the hypotenuse is the diameter of thecircumcircle, and so the radius of the circumcircle is half the length of the hypotenuse:

R={\frac {c}{2}}.

Thus the sum of the circumradius and the inradius is half the sum of the legs:^[7]

R+r={\frac {a+b}{2}}.

One of the legs can be expressed in terms of the inradius and the other leg as

a={\frac {2r(b-r)}{b-2r}}.

Characterizations

A triangle $\triangle ABC$ with sides $a\leq b<c$ ,semiperimeter ${\textstyle s={\tfrac {1}{2}}(a+b+c)}$ ,area $T, {\displaystyle T,}$ altitude $h_{c}$ opposite the longest side,circumradius $R, {\displaystyle R,}$ inradius $r, {\displaystyle r,}$ exradii $r_{a},r_{b},r_{c}$ tangent to $a, b, c {\displaystyle a,b,c}$ respectively, andmedians $m_{a},m_{b},m_{c}$ is a right triangleif 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

$a^{2}+b^{2}=c^{2}\quad ({\text{Pythagorean theorem}})$
$(s-a)(s-b)=s(s-c)$
$s=2R+r.$ ^[8]
$a^{2}+b^{2}+c^{2}=8R^{2}.$ ^[9]

Angles

$A {\displaystyle A}$ and $B {\displaystyle B}$ arecomplementary.^[10]
$\cos {A}\cos {B}\cos {C}=0.$ ^[9]^[11]
$\sin ^{2}{A}+\sin ^{2}{B}+\sin ^{2}{C}=2.$ ^[9]^[11]
$\cos ^{2}{A}+\cos ^{2}{B}+\cos ^{2}{C}=1.$ ^[11]
$\sin {2A}=\sin {2B}=2\sin {A}\sin {B}.$

Area

$T={\frac {ab}{2}}$
$T=r_{a}r_{b}=rr_{c}$
$T=r(2R+r)$
$T={\frac {(2s-c)^{2}-c^{2}}{4}}=s(s-c)$
$T=|PA|\cdot |PB|,$ where $P {\displaystyle P}$ is the tangency point of theincircle at the longest side $A B . {\displaystyle AB.}$ ^[12]

Inradius and exradii

$r=s-c=(a+b-c)/2$
$r_{a}=s-b=(a-b+c)/2$
$r_{b}=s-a=(-a+b+c)/2$
$r_{c}=s=(a+b+c)/2$
$r_{a}+r_{b}+r_{c}+r=a+b+c$
$r_{a}^{2}+r_{b}^{2}+r_{c}^{2}+r^{2}=a^{2}+b^{2}+c^{2}$
$r={\frac {r_{a}r_{b}}{r_{c}}}.$ ^[13]

Altitude and medians

The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments the hypotenuse is split into. UsingPythagoras' theorem on the 3 triangles of sides(p + q,r,s ),(r,p,h ) and(s,h,q ),
${\begin{aligned}(p+q)^{2}\;\;&=\quad r^{2}\;\;\,+\quad s^{2}\\p^{2}\!\!+\!2pq\!+\!q^{2}&=\overbrace {p^{2}\!\!+\!h^{2}} +\overbrace {h^{2}\!\!+\!q^{2}} \\2pq\quad \;\;\;&=2h^{2}\;\therefore h\!=\!{\sqrt {pq}}\\\end{aligned}}$

{\displaystyle {\begin{aligned}(p+q)^{2}\;\;&=\quad r^{2}\;\;\,+\quad s^{2}\\p^{2}\!\!+\!2pq\!+\!q^{2}&=\overbrace {p^{2}\!\!+\!h^{2}} +\overbrace {h^{2}\!\!+\!q^{2}} \\2pq\quad \;\;\;&=2h^{2}\;\therefore h\!=\!{\sqrt {pq}}\\\end{aligned}}} — The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments the hypotenuse is split into. UsingPythagoras' theorem on the 3 triangles of sides(p + q,r,s ),(r,p,h ) and(s,h,q ),
${\begin{aligned}(p+q)^{2}\;\;&=\quad r^{2}\;\;\,+\quad s^{2}\\p^{2}\!\!+\!2pq\!+\!q^{2}&=\overbrace {p^{2}\!\!+\!h^{2}} +\overbrace {h^{2}\!\!+\!q^{2}} \\2pq\quad \;\;\;&=2h^{2}\;\therefore h\!=\!{\sqrt {pq}}\\\end{aligned}}$

$h_{c}={\frac {ab}{c}}$
$m_{a}^{2}+m_{b}^{2}+m_{c}^{2}=6R^{2}.$ ^[7]^{: Prob. 954, p. 26}
The length of onemedian is equal to thecircumradius.
The shortestaltitude (the one from the vertex with the biggest angle) is thegeometric mean of theline segments it divides the opposite (longest) side into. This is theright triangle altitude theorem.

Circumcircle and incircle

The triangle can be inscribed in asemicircle, with one side coinciding with the entirety of the diameter (Thales' theorem).
Thecircumcenter is themidpoint of the longest side.
The longest side is adiameter of thecircumcircle $(c=2R).$
The circumcircle istangent to thenine-point circle.^[9]
Theorthocenter lies on the circumcircle.^[7]
The distance between theincenter and the orthocenter is equal to ${\sqrt {2}}r$ .^[7]

Trigonometric ratios

Main article:Trigonometric functions – Right-angled triangle definitions

Thetrigonometric 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 aresimilar. If, for a given angle α, the opposite side, adjacent side and hypotenuse are labeled $O, {\displaystyle O,}$ $A, {\displaystyle A,}$ and $H, {\displaystyle H,}$ respectively, then the trigonometric functions are

\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}}.

For the expression ofhyperbolic functions as ratio of the sides of a right triangle, see thehyperbolic triangle of ahyperbolic sector.

Special right triangles

Main article: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 ${\tfrac {1}{6}}\pi ,$ and the isosceles right triangle or 45-45-90 triangle which can be used to evaluate the trigonometric functions for any multiple of ${\tfrac {1}{4}}\pi .$

Kepler triangle

Let $H, {\displaystyle H,}$ $G, {\displaystyle G,}$ and $A {\displaystyle A}$ be theharmonic mean, thegeometric mean, and thearithmetic mean of two positive numbers $a {\displaystyle a}$ and $b {\displaystyle b}$ with $a>b.$ If a right triangle has legs $H {\displaystyle H}$ and $G {\displaystyle G}$ and hypotenuse $A, {\displaystyle A,}$ then^[14]

{\frac {A}{H}}={\frac {A^{2}}{G^{2}}}={\frac {G^{2}}{H^{2}}}=\phi ,\qquad {\frac {a}{b}}=\phi ^{3},

where $\phi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}{\bigr )}$ is thegolden ratio. Since the sides of this right triangle are ingeometric progression, this is theKepler triangle.

Thales' theorem

Main article:Thales' theorem

Median of a right angle of a triangle

Thales' theorem states that if $B C {\displaystyle BC}$ is the diameter of a circle and $A {\displaystyle A}$ is any other point on the circle, then $\triangle ABC$ is a right triangle with a right angle at $A . {\displaystyle A.}$ The converse states that the hypotenuse of a right triangle is the diameter of itscircumcircle. As a corollary, the circumcircle has its center at the midpoint of the diameter, so themedian through the right-angled vertex is a radius, and the circumradius is half the length of the hypotenuse.

Medians

The following formulas hold for themedians of a right triangle:

m_{a}^{2}+m_{b}^{2}=5m_{c}^{2}={\frac {5}{4}}c^{2}.

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 $m_{a}$ and $m_{b}$ from the legs satisfy^[7]^{: p.136, #3110}

4c^{4}+9a^{2}b^{2}=16m_{a}^{2}m_{b}^{2}.

Euler line

In a right triangle, theEuler 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 itsperpendicular 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 $({\sqrt {2}}-1).$ ^[15]^: p.281

In a right triangle with legs $a, b {\displaystyle a,b}$ and hypotenuse $c, {\displaystyle c,}$

c\geq {\frac {\sqrt {2}}{2}}(a+b)

with equality only in the isosceles case.^[15]^{: p.282, p.358}

If the altitude from the hypotenuse is denoted $h_{c},$ then

h_{c}\leq {\frac {\sqrt {2}}{4}}(a+b)

with equality only in the isosceles case.^[15]^: p.282

Other properties

If segments of lengths $p {\displaystyle p}$ and $q {\displaystyle q}$ emanating from vertex $C {\displaystyle C}$ trisect the hypotenuse into segments of length ${\tfrac {1}{3}}c,$ then^[3]^{: pp. 216–217}

p^{2}+q^{2}=5\left({\frac {c}{3}}\right)^{2}.

The right triangle is the only triangle having two, rather than one or three, distinct inscribed squares.^[16]

Given any two positive numbers $h {\displaystyle h}$ and $k {\displaystyle k}$ with $h>k.$ Let $h {\displaystyle h}$ and $k {\displaystyle k}$ be the sides of the two inscribed squares in a right triangle with hypotenuse $c . {\displaystyle c.}$ Then

{\frac {1}{c^{2}}}+{\frac {1}{h^{2}}}={\frac {1}{k^{2}}}.

These sides and the incircle radius $r {\displaystyle r}$ are related by a similar formula:

{\frac {1}{r}}=-{\frac {1}{c}}+{\frac {1}{h}}+{\frac {1}{k}}.

The perimeter of a right triangle equals the sum of the radii ofthe incircle and the three excircles:

a+b+c=r+r_{a}+r_{b}+r_{c}.

See also

Acute and obtuse triangles (oblique triangles)
Spiral of Theodorus
Trirectangular spherical triangle

References

^Artmann, Benno (2012) [1999],Euclid: The Creation of Mathematics, Springer, p. 4,doi:10.1007/978-1-4612-1412-0,ISBN 978-1-4612-1412-0.
^Di Domenico, Angelo S., "A property of triangles involving area",Mathematical Gazette 87, July 2003, pp. 323–324.
^^a ^bPosamentier, Alfred S., and Salkind, Charles T.Challenging Problems in Geometry, Dover, 1996.
^Wentworth p. 156
^Voles, Roger, "Integer solutions of $a^{-2}+b^{-2}=d^{-2}$ ,"Mathematical Gazette 83, July 1999, 269–271.
^Richinick, Jennifer, "The upside-down Pythagorean Theorem,"Mathematical Gazette 92, July 2008, 313–317.
^^a ^b ^c ^d ^eInequalities proposed in "Crux Mathematicorum",[1].
^"Triangle right iff s = 2R + r,Art of problem solving, 2011". Archived fromthe original on 2014-04-28. Retrieved2012-01-02.
^^a ^b ^c ^dAndreescu, Titu and Andrica, Dorian, "Complex Numbers from A to...Z", Birkhäuser, 2006, pp. 109–110.
^"Properties of Right Triangles". Archived fromthe original on 2011-12-31. Retrieved2012-02-15.
^^a ^b ^cCTK Wiki Math,A Variant of the Pythagorean Theorem, 2011,[2]Archived 2013-08-05 at theWayback Machine.
^Darvasi, Gyula (March 2005), "Converse of a Property of Right Triangles",The Mathematical Gazette,89 (514):72–76,doi:10.1017/S0025557200176806,S2CID 125992270.
^Bell, Amy (2006),"Hansen's Right Triangle Theorem, Its Converse and a Generalization"(PDF),Forum Geometricorum,6:335–342,archived(PDF) from the original on 2008-07-25
^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.
^^a ^b ^cPosamentier, Alfred S., and Lehmann, Ingmar.The Secrets of Triangles. Prometheus Books, 2012.
^Bailey, Herbert, and DeTemple, Duane, "Squares inscribed in angles and triangles",Mathematics Magazine 71(4), 1998, 278–284.

Weisstein, Eric W."Right Triangle".MathWorld.
Wentworth, G.A. (1895).A Text-Book of Geometry. Ginn & Co.

External links

Wikimedia Commons has media related toRight triangles.

v
t
e

Polygons (List)

By number
of sides

1–10 sides	Monogon (1) Digon (2) Triangle (3) Quadrilateral (4) Pentagon (5) Hexagon (6) Heptagon (7) Octagon (8) Nonagon/Enneagon (9) Decagon (10)
11–20 sides	Hendecagon (11) Dodecagon (12) Tridecagon (13) Tetradecagon (14) Pentadecagon (15) Hexadecagon (16) Heptadecagon (17) Octadecagon (18) Icosagon (20)
>20 sides	Icositrigon (23) Icositetragon (24) Triacontagon (30) 257-gon Chiliagon (1000) Myriagon (10,000) 65537-gon Megagon (1,000,000) Apeirogon (∞)

Classes

Retrieved from "https://en.wikipedia.org/w/index.php?title=Right_triangle&oldid=1329702806"

Hidden categories:

[8]ページ先頭

©2009-2026 Movatter.jp