Triangle

Origin

Middle English, from Anglo-French, from Latin triangulum, from neuter of triangulus triangular, from tri- + angulus angle

• 14th Century

Definitions

• 1: a polygon having three sides — compare spherical triangle
• 2a : a percussion instrument consisting of a rod of steel bent into the form of a triangle open at one angle and sounded by striking with a small metal rod

b : a drafting instrument consisting of a thin flat right-angled triangle of wood or plastic with acute angles of 45 degrees or of 30 degrees and 60 degrees

• 3: a situation in which one member of a couple is involved in a love affair with a third person

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Description

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted .

In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space).

Triangles can be classified according to the relative lengths of their sides:

• In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.
• In an isosceles triangle, two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the isosceles triangle theorem, which was known by Euclid. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles.
• In a scalene triangle, all sides are unequal, equivalently all angles are unequal. Right triangles are scalene if and only if not isosceles.

In diagrams representing triangles (and other geometric figures), "tick" marks along the sides are used to denote sides of equal lengths – the equilateral triangle has tick marks on all 3 sides, the isosceles on 2 sides. The scalene has single, double, and triple tick marks, indicating that no sides are equal. Similarly, arcs on the inside of the vertices are used to indicate equal angles. The equilateral triangle indicates all 3 angles are equal; the isosceles shows 2 identical angles. The scalene indicates by 1, 2, and 3 arcs that no angles are equal.

Triangles can also be classified according to their internal angles, measured here in degrees.

• A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one of its interior angles measuring 90° (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side of the right triangle. The other two sides are called the legs or cathet (singular: cathetus) of the triangle. Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where . In this situation, 3, 4, and 5 are a Pythagorean Triple. The other one is an isosceles triangle that has 2 angles that each measure 45 degrees.

• Triangles that do not have an angle that measures 90° are called oblique triangles.
• A triangle that has all interior angles measuring less than 90° is an acute triangle or acute-angled triangle.
• A triangle that has one angle that measures more than 90° is an obtuse triangle or obtuse-angled triangle.
• A "triangle" with an interior angle of 180° (and collinear vertices) is degenerate.
• A right degenerate triangle has collinear vertices, two of which are coincident.

A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and such a triangle is therefore equilateral.[1]

See Also

• Triad
• Triodities
• Trinunities