Special right triangle
A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. 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.
Angle-based
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 π/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:
degrees | radians | gons |
turns |
sin |
cos |
tan |
cotan
|
---|---|---|---|---|---|---|---|
0° | 0 | 0g | 0 | √0/2 = 0 | √4/2 = 1 | 0 | undefined |
30° | π/6 | 33+1/3g | 1/12 | √1/2 = 1/2 | √3/2 | 1/√3 | √3 |
45° | π/4 | 50g | 1/8 | √2/2 = 1/√2 | √2/2 = 1/√2 | 1 | 1 |
60° | π/3 | 66+2/3g | 1/6 | √3/2 | √1/2 = 1/2 | √3 | 1/√3 |
90° | π/2 | 100g | 1/4 | √4/2 = 1 | √0/2 = 0 | undefined | 0 |
The 45°–45°–90° triangle, the 30°–60°–90° triangle, and the
45° - 45° - 90° triangle
In
Of all right triangles, the 45° - 45° - 90° degree triangle has the smallest ratio of the hypotenuse to the sum of the legs, namely √2/2.[1]: p. 282, p. 358 and the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely √2/4.[1]: 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.
30° - 60° - 90° triangle
This is a triangle whose three angles are in the ratio 1 : 2 : 3 and respectively measure 30° (π/6), 60° (π/3), and 90° (π/2). The sides are in the ratio 1 : √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 D as the midpoint of segment BC. Draw an altitude line from A to D. Then ABD is a 30°–60°–90° triangle with hypotenuse of length 2, and base BD of length 1.
- The fact that the remaining leg AD has length √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, α + δ, α + 2δ are the angles in the progression then the sum of the angles 3α + 3δ = 180°. After dividing by 3, the angle α + δ must be 60°. The right angle is 90°, leaving the remaining angle to be 30°.
Side-based
Right triangles whose sides are of
- m2 − n2 : 2mn : m2 + n2
where m and n are any positive integers such that m > n.
Common Pythagorean triples
There are several Pythagorean triples which are well-known, including those with sides in the ratios:
3 : 4 : 5 5 : 12 : 13 8 : 15 : 17 7 : 24 : 25 9 : 40 : 41
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
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:
11 : 60 : 61 12 : 35 : 37 13 : 84 : 85 15 : 112 : 113 16 : 63 : 65 17 : 144 : 145 19 : 180 : 181 20 : 21 : 29 20 : 99 : 101 21 : 220 : :221
24 : | 143 : | 145 |
28 : | 45 : | 53 |
28 : | 195 : | 197 |
32 : | 255 : | 257 |
33 : | 56 : | 65 |
36 : | 77 : | 85 |
39 : | 80 : | 89 |
44 : | 117 : | 125 |
48 : | 55 : | 73 |
51 : | 140 : | 149 |
52 : | 165 : | 173 |
57 : | 176 : | 185 |
60 : | 91 : | 109 |
60 : | 221 : | 229 |
65 : | 72 : | 97 |
84 : | 187 : | 205 |
85 : | 132 : | 157 |
88 : | 105 : | 137 |
95 : | 168 : | 193 |
96 : | 247 : | 265 |
104 : | 153 : | 185 |
105 : | 208 : | 233 |
115 : | 252 : | 277 |
119 : | 120 : | 169 |
120 : | 209 : | 241 |
133 : | 156 : | 205 |
140 : | 171 : | 221 |
160 : | 231 : | 281 |
161 : | 240 : | 289 |
204 : | 253 : | 325 |
207 : | 224 : | 305 |
Almost-isosceles Pythagorean triples
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. 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.[5][6] Such almost-isosceles right-angled triangles can be obtained recursively,
- a0 = 1, b0 = 2
- an = 2bn−1 + an−1
- bn = 2an + bn−1
an is length of hypotenuse, n = 1, 2, 3, .... Equivalently,
where {x, y} are solutions to the
3 : 4 : 5 20 : 21 : 29 119 : 120 : 169 696 : 697 : 985
4,059 : | 4,060 : | 5,741 |
23,660 : | 23,661 : | 33,461 |
137,903 : | 137,904 : | 195,025 |
803,760 : | 803,761 : | 1,136,689 |
Alternatively, the same triangles can be derived from the square triangular numbers.[8]
Arithmetic and geometric progressions
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 = √φ where φ is the golden ratio. Its sides are therefore in the ratio 1 : √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.[9]
Sides of regular polygons
Let
See also
- Ailles rectangle, combining several special right triangles
- Integer triangle
- Spiral of Theodorus
References
- ^ a b Posamentier, Alfred S., and Lehman, Ingmar. The Secrets of Triangles. Prometheus Books, 2012.
- ^ Weisstein, Eric W. "Rational Triangle". MathWorld.
- ^ ISBN 978-1-118-03024-0.
- ^ Gillings, Richard J. (1982). Mathematics in the Time of the Pharaohs. Dover. p. 161.
- ^ Forget, T. W.; Larkin, T. A. (1968), "Pythagorean triads of the form x, x + 1, z described by recurrence sequences" (PDF), Fibonacci Quarterly, 6 (3): 94–104.
- MR 1327342.
- ^ (sequence A001652 in the OEIS)
- MR 1640364.
- MR 1448883.
- ^ Euclid's Elements, Book XIII, Proposition 10.
- ^ nLab: pentagon decagon hexagon identity.
External links
- 3 : 4 : 5 triangle
- 30–60–90 triangle
- 45–45–90 triangle – with interactive animations