Special right triangle

File:Special right triangles for trig.svg of multiples of 30 and 45 degrees.]]

Angle-based special right triangles are specified by the relationships of the angles of which the triangle is composed. The angles of these triangles are such that the larger (right) angle, which is 90 degrees or {{sfrac|{{pi}}|2}} radians, is equal to the sum of the other two angles.

The side lengths are generally deduced from the basis of the unit circle or other geometric methods. This approach may be used to rapidly reproduce the values of trigonometric functions for the angles 30°, 45°, and 60°.

Special triangles are used to aid in calculating common trigonometric functions, as below:

{{multiple image

| width = 120

| image1 = Tile V488 bicolor.svg

| caption1 = 45°–45°–90°

| image2 = Tile V46b.svg

| caption2 = 30°–60°–90°

}}

The 45°–45°–90° triangle, the 30°–60°–90° triangle, and the equilateral/equiangular (60°–60°–60°) triangle are the three Möbius triangles in the plane, meaning that they tessellate the plane via reflections in their sides; see Triangle group.

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File:Squadra 45.jpg shaped as 45°–45°–90° triangle]]

Image:45-45-triangle.svg

File:45° 45° 90° Special Right Triangle.svg of hypotenuse length 1]]

In plane geometry, dividing a square along its diagonal results in two isosceles right triangles, each with one right angle (90°, {{sfrac|{{pi}}|2}} radians) and two other congruent angles each measuring half of a right angle (45°, or {{sfrac|{{pi}}|4}} radians). The sides in this triangle are in the ratio 1 : 1 : {{sqrt|2}}, which follows immediately from the Pythagorean theorem.

Of all right triangles, such 45°–45°–90° degree triangles have the smallest ratio of the hypotenuse to the sum of the legs, namely {{sfrac|{{sqrt|2}}|2}}.Posamentier, Alfred S., and Lehman, Ingmar. The Secrets of Triangles. Prometheus Books, 2012.{{rp|p. 282, p. 358}} and the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely {{sfrac|{{sqrt|2}}|4}}.{{rp|p.282}}

Triangles with these angles are the only possible right triangles that are also isosceles triangles in Euclidean geometry. However, in spherical geometry and hyperbolic geometry, there are infinitely many different shapes of right isosceles triangles.

File:Squadra 30 60.jpg

File:Equilateral triangle with height square root of 3.svg

File:30° 60° 90° Special Right Triangle.svg of hypotenuse length 1]]

This is a triangle whose three angles are in the ratio 1 : 2 : 3 and respectively measure 30° ({{sfrac|{{pi}}|6}}), 60° ({{sfrac|{{pi}}|3}}), and 90° ({{sfrac|{{pi}}|2}}). The sides are in the ratio 1 : Square root of 3 : 2.

The proof of this fact is clear using trigonometry. The geometric proof is:

:Draw an equilateral triangle ABC with side length 2 and with point M as the midpoint of segment BC. Draw an altitude line from A to M. Then ABM is a 30°–60°–90° triangle with hypotenuse of length 2, and base BM of length 1.

:The fact that the remaining leg AM has length Square root of 3 follows immediately from the Pythagorean theorem.

The 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if α, {{nowrap|α + δ}}, {{nowrap|α + 2δ}} are the angles in the progression then the sum of the angles {{nowrap|3α + 3δ}} = 180°. After dividing by 3, the angle {{nowrap|α + δ}} must be 60°. The right angle is 90°, leaving the remaining angle to be 30°.

Right triangles whose sides are of integer lengths, with the sides collectively known as Pythagorean triples, possess angles that cannot all be rational numbers of degrees.{{Cite journal|title=Rational Triangle|last=Weisstein|first=Eric W|journal=MathWorld|url=http://mathworld.wolfram.com/RationalTriangle.html}} (This follows from Niven's theorem.) They are most useful in that they may be easily remembered and any multiple of the sides produces the same relationship. Using Euclid's formula for generating Pythagorean triples, the sides must be in the ratio

:{{nowrap|m{{sup|2}} − n{{sup|2}} : 2mn : m{{sup|2}} + n{{sup|2}}}}

where m and n are any positive integers such that {{nowrap|m > n}}.

{{Main|Pythagorean triple}}

There are several Pythagorean triples which are well-known, including those with sides in the ratios:

:

The 3 : 4 : 5 triangles are the only right triangles with edges in arithmetic progression. Triangles based on Pythagorean triples are Heronian, meaning they have integer area as well as integer sides.

The possible use of the 3 : 4 : 5 triangle in Ancient Egypt, with the supposed use of a knotted rope to lay out such a triangle, and the question whether Pythagoras' theorem was known at that time, have been much debated.{{cite book |last=Cooke |first=Roger L. |title=The History of Mathematics: A Brief Course |url=https://books.google.com/books?id=wOGh7XPowAMC |edition=2nd |year=2011|publisher=John Wiley & Sons |isbn=978-1-118-03024-0 |pages=237–238}} It was first conjectured by the historian Moritz Cantor in 1882. It is known that right angles were laid out accurately in Ancient Egypt; that their surveyors did use ropes for measurement; that Plutarch recorded in Isis and Osiris (around 100 AD) that the Egyptians admired the 3 : 4 : 5 triangle; and that the Berlin Papyrus 6619 from the Middle Kingdom of Egypt (before 1700 BC) stated that "the area of a square of 100 is equal to that of two smaller squares. The side of one is {{sfrac|1|2}} + {{sfrac|1|4}} the side of the other."{{cite book |author=Gillings, Richard J. |title=Mathematics in the Time of the Pharaohs |url=https://archive.org/details/mathematicsintim0000gill |url-access=registration |publisher=Dover |date=1982 |page=[https://archive.org/details/mathematicsintim0000gill/page/161 161]}} The historian of mathematics Roger L. Cooke observes that "It is hard to imagine anyone being interested in such conditions without knowing the Pythagorean theorem." Against this, Cooke notes that no Egyptian text before 300 BC actually mentions the use of the theorem to find the length of a triangle's sides, and that there are simpler ways to construct a right angle. Cooke concludes that Cantor's conjecture remains uncertain: he guesses that the Ancient Egyptians probably did know the Pythagorean theorem, but that "there is no evidence that they used it to construct right angles".

The following are all the Pythagorean triple ratios expressed in lowest form (beyond the five smallest ones in lowest form in the list above) with both non-hypotenuse sides less than 256:

:

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Isosceles right-angled triangles cannot have sides with integer values, because the ratio of the hypotenuse to either other side is {{sqrt|2}} and Square root of 2#Proofs of irrationality. However, infinitely many almost-isosceles right triangles do exist. These are right-angled triangles with integer sides for which the lengths of the non-hypotenuse edges differ by one.{{citation

| last1 = Forget | first1 = T. W.

| last2 = Larkin | first2 = T. A.

| issue = 3

| journal = Fibonacci Quarterly

| pages = 94–104

| title = Pythagorean triads of the form x, x + 1, z described by recurrence sequences

| url = http://www.fq.math.ca/Scanned/6-3/6-3/forget.pdf

| volume = 6

| year = 1968| doi = 10.1080/00150517.1968.12431232

}}.{{citation

| last1 = Chen | first1 = C. C.

| last2 = Peng | first2 = T. A.

| journal = The Australasian Journal of Combinatorics

| mr = 1327342

| pages = 263–267

| title = Almost-isosceles right-angled triangles

| url = http://ajc.maths.uq.edu.au/pdf/11/ajc-v11-p263.pdf

| volume = 11

| year = 1995}}. Such almost-isosceles right-angled triangles can be obtained recursively,

:a₀ = 1, b₀ = 2

:a_n = 2b_n−1 + a_n−1

:b_n = 2a_n + b_n−1

a_n is length of hypotenuse, n = 1, 2, 3, .... Equivalently,

:$(\backslash tfrac\{x-1\}\{2\})^2+(\backslash tfrac\{x+1\}\{2\})^2\; =\; y^2$

where {x, y} are solutions to the Pell equation {{nowrap|x{{sup|2}} − 2y{{sup|2}} {{=}} −1}}, with the hypotenuse y being the odd terms of the Pell numbers 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378... {{OEIS|id=A000129}}.. The smallest Pythagorean triples resulting are:{{OEIS|A001652}}

:

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Alternatively, the same triangles can be derived from the square triangular numbers.{{citation

| last = Nyblom | first = M. A.

| issue = 4

| journal = The Fibonacci Quarterly

| mr = 1640364

| pages = 319–322

| title = A note on the set of almost-isosceles right-angled triangles

| url = http://www.fq.math.ca/Scanned/36-4/nyblom.pdf

| volume = 36

| year = 1998| doi = 10.1080/00150517.1998.12428915

}}.

File:Kepler triangle.svg.]]

{{Main article|Kepler triangle}}

The Kepler triangle is a right triangle whose sides are in geometric progression. If the sides are formed from the geometric progression a, ar, ar² then its common ratio r is given by r = {{sqrt|φ}} where φ is the golden ratio. Its sides are therefore in the ratio {{nowrap|1 : {{sqrt|φ}} : φ}}. Thus, the shape of the Kepler triangle is uniquely determined (up to a scale factor) by the requirement that its sides be in geometric progression.

The 3–4–5 triangle is the unique right triangle (up to scaling) whose sides are in arithmetic progression.{{citation

| last1 = Beauregard | first1 = Raymond A.

| last2 = Suryanarayan | first2 = E. R.

| doi = 10.2307/2691431

| issue = 2

| journal = Mathematics Magazine

| mr = 1448883

| pages = 105–115

| title = Arithmetic triangles

| volume = 70

| year = 1997| jstor = 2691431

}}.

File:Euclid XIII.10.svg circles, form a right triangle.]]

Let $$a=2\backslash sin\backslash frac\{\backslash pi\}\{10\}=\backslash frac\{-1+\backslash sqrt5\}\{2\}=\backslash frac1\backslash varphi\backslash approx\; 0.618$$ be the side length of a regular decagon inscribed in the unit circle, where $\backslash varphi$ is the golden ratio. Let $$b=2\backslash sin\backslash frac\{\backslash pi\}\{6\}=1$$ be the side length of a regular hexagon in the unit circle, and let $$c=2\backslash sin\backslash frac\{\backslash pi\}\{5\}=\backslash sqrt\{\backslash frac\{5-\backslash sqrt5\}\{2\}\}\backslash approx\; 1.176$$ be the side length of a regular pentagon in the unit circle. Then $a^2+b^2=c^2$, so these three lengths form the sides of a right triangle.[http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII10.html Euclid's Elements, Book XIII, Proposition 10]. The same triangle forms half of a golden rectangle. It may also be found within a regular icosahedron of side length $c$: the shortest line segment from any vertex $V$ to the plane of its five neighbors has length $a$, and the endpoints of this line segment together with any of the neighbors of $V$ form the vertices of a right triangle with sides $a$, $b$, and $c$.[http://ncatlab.org/nlab/show/pentagon+decagon+hexagon+identity nLab: pentagon decagon hexagon identity].

• Ailles rectangle, combining several special right triangles
• Integer triangle
• Spiral of Theodorus

• [http://www.mathopenref.com/triangle345.html 3 : 4 : 5 triangle]
• [http://www.mathopenref.com/triangle306090.html 30–60–90 triangle]
• [http://www.mathopenref.com/triangle454590.html 45–45–90 triangle]{{snd}} with interactive animations

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Category:Euclidean plane geometry

Category:Types of triangles