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Trigonometry - Triangles And Their Properties

angles equal similar length

The principles of trigonometry were originally developed around the relationship between the sides of a triangle and its angles. The idea was that the unknown length of a side or size of an angle could be determined if the length or magnitude of some of the other sides or angles were known. Recall that a triangle is a geometric figure made up of three sides and three angles, whose sum is equal to 180°. The three points of a triangle, known as its vertices, are usually denoted by capital letters.

Triangles can be classified by the lengths of their sides or magnitude of their angles. Isosceles triangles have two equal sides and two congruent (equal) angles. Equilateral, or equiangular, triangles have three equal sides and angles. If no sides are equal, the triangle is a scalene triangle. All of the angles in an acute triangle are less than 90° and at least one of the angles in an obtuse triangle is greater than 90°. Triangles, such as these, which do not contain a 90° angle, are generally known as oblique triangles. Right triangles, the most important ones to trigonometry, are those which contain one 90°angle.

Triangles which have proportional sides and congruent angles are called similar triangles. The concept of similar triangles, one of the basic insights in trigonometry, allows us to determine the length of a side of one triangle if we know the length of certain sides of the other triangle. For example, if we wanted to know the height of a tree, we could use the idea of similar triangles to find it without actually having to measure it. Suppose a person is 6 ft (183 cm) tall and casts an 8 ft (2.44 m) long shadow. The tree, whose height is unknown, casts a shadow that is 20 ft (6.1 m) long. The triangles that could be drawn using the shadows and objects as sides are similar. Since the sides of a similar triangles are proportional, the height of the tree is determined by setting up the mathematical equality

By solving this equation, the height of the tree is found to be 15 ft (4.57 m).

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