## Triangles

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## Triangles

# Triangles

A triangle is a closed three-sided, three-angled figure, and is the simplest example of what mathematicians call polygons (figures having many sides). Triangles are among the most important objects studied in mathematics owing to the rich mathematical theory built up around them in **Euclidean geometry** and **trigonometry** , and also to their applicability in such areas as astronomy, architecture, engineering, physics, navigation, and surveying.

In Euclidean geometry, much attention is given to the properties of triangles. Many theorems are stated and proved about the conditions necessary for two triangles to be similar and/or congruent. Similar triangles have the same shape but not necessarily the same size, whereas congruent triangles have both the same shape and size.

One of the most famous and useful theorems in mathematics, the Pythagorean Theorem, is about triangles. Specifically, the Pythagorean Theorem is about right triangles, which are triangles having a 90° or "right" angle. The theorem states that if the length of the sides forming the right angle are given by *a* and *b,* and the side opposite the right angle, called the **hypotenuse** , is given by *c,* then *c* ^{2} = *a* ^{2} + *b* ^{2}. It is almost impossible to over-state the importance of this theorem to mathematics and, specifically, to trigonometry.

## Triangles and Trigonometry

Trigonometry literally means "triangle measurement" and is the study of the properties of triangles and their ramifications in both pure and applied mathematics. The two most important functions in trigonometry are the **sine** and **cosine** functions, each of which may be defined in terms of the sides of right triangles, as shown below.

These functions are so important in calculating side and angle measures of triangles that they are built into scientific calculators and computers. Although the sine and cosine functions are defined in terms of right triangles, their use may be extended to any triangle by two theorems of trigonometry called the Law of Sines and the Law of Cosines (see page 108). These laws allow the calculation of side lengths and angle measures of triangles when other side and angle measurements are known.

## Ancient and Modern Applications

Perhaps the most ancient use of triangles was in astronomy. Astronomers developed a method called **triangulation** for determining distances to far away objects. Using this method, the distance to an object can be calculated by observing the object from two different positions a known distance apart, then measuring the angle created by the apparent "shift" or **parallax** of the object against its background caused by the movement of the observer between the two known positions. The Law of Sines may then be used to calculate the distance to the object.

The Greek mathematician and astronomer, Aristarchus (310 b.c.e.–250 b.c.e.) is said to have used this method to determine the distance from the Earth to the Moon. Eratosthenes (c. 276 b.c.e.–195 b.c.e.) used triangulation to calculate the circumference of Earth.

Modern Global Positioning System (GPS) devices, which allow earth-bound travelers to know their longitude and latitude at any time, receive signals from orbiting satellites and utilize a sophisticated triangulation **algorithm** to compute the position of the GPS device to within a few meters of accuracy.

see also Astronomy, Measurements in; Global Positioning System; Polyhedrons; Trigonometry.

*Stephen Robinson*

## Bibliography

Foerster, Paul A. *Algebra and Trigonometry.* Menlo Park, CA: Addison-Wesley Publishing Company, 1999.

Serra, Michael. *Discovering Geometry.* Emeryville, CA: Key Curriculum Press, 1997.

Narins, Brigham, ed. *World of Mathematics,* Detroit: Gale Group, 2001.

## Triangles (Network)

# Triangles (Network)

Network funded by the Lucis Trust, formed to propagate the teachings of **Alice A. Bailey** (1880-1949), former Theosophist who founded her own **Arcane School.**

The Triangles Program was inaugurated in 1937 by Bailey in which she called upon people to form groups of three who would daily unite to channel spiritual energy to the world. Address: 120 Wall St., 24th Fl., New York, NY 10005. British headquarters are at Ste. 54, 3 Whitehall Ct., London, SW1A 2EF, England. Website: http://www.lucistrust.org/.

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