Everything about Right Triangle totally explained
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:
. 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
. With the three angles adding up to 180°, the angles respectively measure 45°, 45°, and 90°. The sides are in the ratio
» .
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:
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