Special right triangle
Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.
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°.
The sides in this triangle are in the ratio 1 : 1 : √2, which follows immediately from the Pythagorean theorem.
282, p. 358 and the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely √2/4.
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.
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 where m and n are any positive integers such that m > n. 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.
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.
[3] It is known that right angles were laid out accurately in Ancient Egypt; that their surveyors did use ropes for measurement;[3] that Plutarch recorded in Isis and Osiris (around 100 AD) that the Egyptians admired the 3 : 4 : 5 triangle;[3] 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.
"[4] 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.
"[3] 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".
[3] 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: Isosceles right-angled triangles cannot have sides with integer values, because the ratio of the hypotenuse to either other side is √2 and √2 cannot be expressed as a ratio of two integers.
These are right-angled triangles with integer sides for which the lengths of the non-hypotenuse edges differ by one.
[5][6] Such almost-isosceles right-angled triangles can be obtained recursively, an is length of hypotenuse, n = 1, 2, 3, .... Equivalently, where {x, y} are solutions to the Pell equation x2 − 2y2 = −1, with the hypotenuse y being the odd terms of the Pell numbers 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378... (sequence A000129 in the OEIS)..
The smallest Pythagorean triples resulting are:[7] Alternatively, the same triangles can be derived from the square triangular numbers.
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.
be the side length of a regular decagon inscribed in the unit circle, where
be the side length of a regular hexagon in the unit circle, and let
be the side length of a regular pentagon in the unit circle.
