Definitions

Special right triangles

Two types of special right triangles appear commonly in geometry, the "angle based" and the "side based" (or Pythagorean) triangles. The former are characterised by integer ratios between the triangle angles, and the latter by integer ratios between the sides. Knowing the ratios of the sides of these special right triangles allows one to quickly calculate various lengths in geometric problems.

Angle-based

"Angle-based" special right triangles are specified by the integer ratio of the angles of which the triangle is composed. The integer ratio of the angles of these triangles are such that the larger (right) angle equals the sum of the smaller angles: $m:n:\left(m+n\right),$. The side lengths are generally deduced from the basis of the unit circle or other geometric methods. This form is most interesting in that it may be used to rapidly reproduce the values of trigonometric functions for the angles 30°, 45°, & 60°.

45-45-90 triangle

Constructing the diagonal of a square results in a triangle whose three angles are in the ratio $1:1:2,$. With the three angles adding up to 180°, the angles respectively measure 45°, 45°, and 90°. The sides are in the ratio

$1:1:sqrt\left\{2\right\}.,$

A simple proof. Say you have such a triangle with legs a and b and hypotenuse c. Suppose that a = 1. Since two angles measure 45°, this is an isosceles triangle and we have b = 1. The fact that $c=sqrt\left\{2\right\}$ follows immediately from the Pythagorean theorem.

30-60-90 triangle

This is a triangle whose three angles are in the ratio $1:2:3,$, and respectively measure 30°, 60°, and 90°. The sides are in the ratio

$1:sqrt\left\{3\right\}:2.,$

The proof of this fact is clear using trigonometry. Although the geometric proof is less apparent, it is equally trivial:

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 $sqrt\left\{3\right\}$ follows immediately from the Pythagorean theorem.

Side-based

All of the special side based right triangles possess angles which are not necessarily rational numbers, but whose sides are always of integer length and form a Pythagorean triple. They are most useful in that they may be easily remembered and any multiple of the sides produces the same relationship.

Common Pythagorean triples

There are several Pythagorean triples which are very well known, including:

$3:4:5,$

$5:12:13,$

$6:8:10,$ (a multiple of the 3:4:5 triple)

$8:15:17,$

$7:24:25,$

The smallest of these (and its multiples, 6:8:10, 9:12:15, ...) is the only right triangle with edges in arithmetic progression. Triangles based on Pythagorean triplets are Heronian and therefore have integer area.

Fibonacci triangles

Starting with 5, every other Fibonacci number {0,1,1,2,3,5,8,13,21,34,55,89,...} is the length of the hypotenuse of a right triangle with integral sides, or in other words, the largest number in a Pythagorean triple. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle.

The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely and approaches a limiting triangle with edge ratios:

$sqrt\left\{5\right\}:2:1$.

This right triangle is sometimes referred to as a dom, a name suggested by Andrew Clarke to stress that this is the triangle obtained from dissecting a domino along a diagonal.

Almost-isosceles Pythagorean triples

Isosceles right-angled triangles can not have integral sides. However, infinitely many almost-isosceles right triangles do exist. These are right-angled triangles with integral sides for which the lengths of the non-hypotenuse edges differ by one. Such almost-isosceles right-angled triangles can be obtained recursively using Pell's equation:

a0 = 1, b0 = 2
an = 2bn-1 + an-1
bn = 2an + bn-1

an is length of hypotenuse, n=1, 2, 3, .... The smallest Pythagorean triples resulting are:

$3:4:5,$

$20:21:29,$

$119:120:169,$

$696:697:985,$