Triangle

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Triangle

A triangle is a closed figure formed by connecting three non-collinear points, called vertices, by line segments to result in three sides and three interior angles (hence the word "tri-angle"). If the three angles all lie in the same plane, the triangle is called a plane, or Euclidean, triangle. If the three angles do not all lie in the same plane, the triangle is called a spherical, or curvilinear, triangle. The word triangle, by itself, is usually taken to mean a plane triangle, while a curvilinear triangle is stated as such. The triangle is the simplest polygon because it is the closed figure having the fewest number of angles and sides. The word triangle can be traced back to the Latin "triangulum", and to the neuter of "triangulus" (three-angled).

Some important facts and definitions pertaining to triangles are:

• Any one of the sides of a triangle may be considered the base.
• The perpendicular distance from the base to the opposite vertex is called the altitude of the triangle.
• A triangle's area is equal to one-half the product of the base and the corresponding altitude.
• The perimeter of a triangle is the sum of its three sides.
• Any triangle has three medians, which are defined to be the line segments joining the midpoint of a particular side to the opposite vertex.
• In Euclidean geometry the sum of the three angles of a triangle is equal to two right angles (180°).

Triangles may be classified by the characteristics that they possess. Some common classifications are:

• Acute triangle: each of its three interior angles is less than 90°,
• Equiangular triangle: all three interior angles are equal,
• Equilateral triangle: all three sides are equal in linear length (equiangular and equilateral triangles are equivalent),
• Isosceles triangle: two of its sides possess equal length,
• Oblique triangle: does not contain a right angle (90°),
• Obtuse triangle: possesses an angle between 90° and 180°,
• Right triangle: has one right angle (90°); the side opposite it is called the hypotenuse,
• Scalene triangle: no two angles are equal.

Trigonometry is the mathematical study of triangles. When trigonometry is restricted to the plane it is called plane trigonometry. One of the aims of plane trigonometry is "solving the triangle", which means finding its unknown parts (sides and angles) from given data. Any triangle may be specified by six parts: three angles and the lengths of its three sides. If three of the six parts of a triangle are specified (where one side and two angles are given or two sides and one angle are given) then the rules of trigonometry can be used to determine the three unknown parts. For a right triangle, one angle (other than the right angle) and the length of a side must be known. Let A, B, and C be the angles of a right triangle and a, b, and c, respectively, be the sides opposite them. If C is the right angle then side c must be the hypotenuse; and the remaining parts of the triangle can then be found using the equation "A + B = 90°", the Pythagorean theorem "a² + b² = c²", and the following trigonometric functions (which relate the two smaller angles of a right triangle to the ratios of particular sides):

• sin(A) = cos(B) = a / c,
• cos(A) = sin(B) = b / c,
• tan(A) = cot(B) = a / b,
• cot(A) = tan(B) = b / a,
• sec(A) = csc(B) = c / b,
• csc(A) = sec(B) = c / a.

For triangles that do not possess a right angle (called oblique), one may "solve the triangle" by first dividing it into two right triangles; then the resulting component triangles can be solved according to the procedures given above. The solution to all oblique triangles can also be found with one or more of the following three equations, plus the equation "180° = A + B + C", where a, b, and c are the sides opposite the angles A, B, and C (respectively):

• Law of Sines: [a / sin(A)] = [b / sin(B)] = [c / sin(C)].
• Law of Cosines:

• a² = b² + c² - 2bc cos(A),
• b² = c² + a² - 2ca cos(B),
• c² = a² + b² - 2ab cos(C).

Law of Tangents:

• [(a - b) / (a + b)] = {tan[1/2(A - B)]} / {tan[1/2(A + B)]},
• [(b - c) / (b + c)] = {tan[1/2(B - C)]} / {tan[1/2(B + C)]},
• [(a - b) / (a + b)] = {tan[1/2(A - B)]} / {tan[1/2(A + B)]}.

These three laws can be used to solve any oblique triangle; that is, the unknown sides or angles can be found when:

• (Two angles and one side are given.) The third angle is found by "180° = A + B + C", and the two other sides are solved using the Law of Sines.
• (Two sides and one angle (opposite one of the known sides) are given, say a, c, and C.) Knowing from the Law of Sines that sin(A) = (a / c) sin(C) (where (a / c) sin(C) 1), B = 180° - A - C, and b = [c sin(B)] / sin(C), several solutions are possible to solve the unknown side and two angles:

• a = c, there is one solution (because A = C),
• a<c, there is one solution (because A<C),
• a>c,

• a is so large that sin(A) 1 is not satisfied, thus no solution exists,
• sin(A) = 1, so C is a right angle and there is one solution,
• sin(A)<1, so angles "A₁" and "A₂ = 180° - A₁" can be calculated and there are two solutions.

• (Two sides and the included angle are given.) To solve for the two unknown angles, use the Law of Tangents and "180° = A + B + C"; and to solve for the unknown side, use the Law of Sines.
• (Three sides are given.) To solve for the three unknown angles, use the Law of Cosines.

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