"rational formula_14 where formula_15 reduces the generated Heronian triangle to its primitive and formula_16 scales up this primitive to the required size. A Pythagorean triangle is right angled and Heronian. Its three integer sides are known as a Pythagorean triple or Pythagorean triplet or Pythagorean triad. All Pythagorean triples formula_17 with hypotenuse formula_18 which are primitive (the sides having no common factor) can be generated by where ""m"" and ""n"" are coprime integers and one of them is even with ""m"" > ""n"". Every even number greater than 2 can be the leg of a Pythagorean triangle (not necessarily primitive)"
"the accompanying figure: The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle and one of the two sides adjacent to angle ""A"". The adjacent leg is the other side that is adjacent to angle ""A"". The opposite side is the side that is opposite to angle ""A"". The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively.(see below under Mnemonics). The reciprocals of these functions are named the cosecant (csc), secant (sec), and cotangent (cot), respectively: The inverse functions are called"
"angle A"" and ""opposed to angle B"". If the lengths of all three sides of a right triangle are integers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple. As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the base then the other is height, so the area of a right triangle is one half the product of the two legs. As a formula the area ""T"" is where ""a"""
"Cathetus In a right triangle, a cathetus (originally from the Greek word ; plural: catheti), commonly known as a leg, is either of the sides that are adjacent to the right angle. It is occasionally called a ""side about the right angle"". The side opposite the right angle is the hypotenuse. In the context of the hypotenuse, the catheti are sometimes referred to simply as ""the other two sides"". If the catheti of a right triangle have equal lengths, the triangle is isosceles. If they have different lengths, a distinction can be made between the minor (shorter) and major (longer)"
"within a regular icosahedron of side length ""c"": the shortest line segment from any vertex ""V"" to the plane of its five neighbors has length ""a"", and the endpoints of this line segment together with any of the neighbors of ""V"" form the vertices of a right triangle with sides ""a"", ""b"", and ""c"". Special right triangle A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called"
"according to This implies that the longest side in an obtuse triangle is the one opposite the obtuse-angled vertex. An acute triangle has three inscribed squares, each with one side coinciding with part of a side of the triangle and with the square's other two vertices on the remaining two sides of the triangle. (In a right triangle two of these are merged into the same square, so there are only two distinct inscribed squares.) However, an obtuse triangle has only one inscribed square, one of whose sides coincides with part of the longest side of the triangle. All triangles"
"the square of the length of the hypotenuse. The term leg, in addition to referring to a cathetus of a right triangle, is also used to refer to either of the equal sides of an isosceles triangle or to either of the non-parallel sides of a trapezoid. Cathetus In a right triangle, a cathetus (originally from the Greek word ; plural: catheti), commonly known as a leg, is either of the sides that are adjacent to the right angle. It is occasionally called a ""side about the right angle"". The side opposite the right angle is the hypotenuse. In the"
"is 90°, leaving the remaining angle to be 30°. Right triangles whose sides are of integer lengths, with the sides collectively known as Pythagorean triples, possess angles that cannot all be rational numbers of degrees. (This follows from Niven's theorem.) They are most useful in that they may be easily remembered and any multiple of the sides produces the same relationship. Using Euclid's formula for generating Pythagorean triples, the sides must be in the ratio where ""m"" and ""n"" are any positive integers such that . There are several Pythagorean triples which are well-known, including those with sides in the"
"product of the two shorter sides of the triangle, at least one of which must be even. An example of a Heronian triangle which is not right-angled is the isosceles triangle with sidelengths 5, 5, and 6, whose area is 12. This triangle is obtained by joining two copies of the right-angled triangle with sides 3, 4, and 5 along the sides of length 4. This approach works in general, as illustrated in the adjacent picture. One takes a Pythagorean triple (""a"", ""b"", ""c""), with ""c"" being largest, then another one (""a"", ""d"", ""e""), with ""e"" being largest, constructs the"
"the circumcenter is located on a side of the triangle, then the opposite angle is a right one. If the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse. An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. This opposite side is called the ""base"" of the altitude, and the point where the altitude intersects the base (or its extension) is called the ""foot"" of the altitude. The length of"
"its sides be in a geometric progression. The 3–4–5 triangle is the unique right triangle (up to scaling) whose sides are in an arithmetic progression. Let be the side length of a regular decagon inscribed in the unit circle, where ""φ"" is the golden ratio. Let be the side length of a regular hexagon in the unit circle, and let {2}}</math>}} be the side length of a regular pentagon in the unit circle. Then , so these three lengths form the sides of a right triangle. The same triangle forms half of a golden rectangle. It may also be found"
"extremely elongated imaginary right triangle in space. The two dimensions on which this triangle is based are its shorter leg, of length one astronomical unit (the average Earth-Sun distance), and the subtended angle of the vertex opposite that leg, measuring one arc second. Applying the rules of trigonometry to these two values, the unit length of the other leg of the triangle (the parsec) can be derived. One of the oldest methods used by astronomers to calculate the distance to a star is to record the difference in angle between two measurements of the position of the star in the"
"a triangle is right, obtuse, or acute, as follows. Let ""c"" be chosen to be the longest of the three sides and (otherwise there is no triangle according to the triangle inequality). The following statements apply: Edsger W. Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language: where ""α"" is the angle opposite to side ""a"", ""β"" is the angle opposite to side ""b"", ""γ"" is the angle opposite to side ""c"", and sgn is the sign function. A Pythagorean triple has three positive integers ""a"", ""b"", and ""c"", such that In other words, a"
"Altitude (triangle) In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the ""extended base"" of the altitude. The intersection of the extended base and the altitude is called the ""foot"" of the altitude. The length of the altitude, often simply called ""the altitude"", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as"
"with sides ""a"", ""b"" and ""c"", and every such triangle has a right angle between the sides of lengths ""a"" and ""b"". An alternative statement is: For any triangle with sides ""a"", ""b"", ""c"", if then the angle between ""a"" and ""b"" measures 90°. This converse also appears in Euclid's ""Elements"" (Book I, Proposition 48): It can be proven using the law of cosines or as follows: Let ""ABC"" be a triangle with side lengths ""a"", ""b"", and ""c"", with Construct a second triangle with sides of length ""a"" and ""b"" containing a right angle. By the Pythagorean theorem, it"
"""c"" is the length of the hypotenuse of a right triangle having sides ""a"" and ""b"". For instance, in the illustration to the left, the two squares in the Pythagorean tiling have side lengths 5 and 12 units long, and the side length of the tiles in the overlaying square tiling is 13, based on the Pythagorean triple (5,12,13). By overlaying a square grid of side length ""c"" onto the Pythagorean tiling, it may be used to generate a five-piece dissection of two unequal squares of sides ""a"" and ""b"" into a single square of side ""c"", showing that the"
"as shown in the upper part of the diagram, with ""BC"" the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length ""y"", the side ""AC"" of length ""x"" and the side ""AB"" of length ""a"", as seen in the lower diagram part. If ""x"" is increased by a small amount ""dx"" by extending the side ""AC"" slightly to ""D"", then ""y"" also increases by ""dy"". These form two sides of a triangle, ""CDE"", which (with ""E"" chosen so ""CE"" is perpendicular to the hypotenuse) is a right triangle approximately similar to ""ABC""."
"it is ""always"" longer when the corresponding angles are right or obtuse. Where the angle is a right angle, also known as the Hypotenuse-Leg (HL) postulate or the Right-angle-Hypotenuse-Side (RHS) condition, the third side can be calculated using the Pythagorean Theorem thus allowing the SSS postulate to be applied. If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is equal to the length of the adjacent side multiplied by the sine of the angle, then the two triangles are congruent. If two triangles satisfy the SSA condition"
"length ""c"" is a right angle, then """"'a"" + ""b"" = ""c"". The converse, which also appears in Euclid's ""Elements"" (Book I, Proposition 48), can be stated as: Given a triangle with sides of length ""a"", ""b"", and ""c"", if ""a"" + ""b"" = ""c"", then the angle opposite the side of length ""c"" is a right angle. If R is a binary relation, formula_1 then the converse relation formula_2 is also called the transpose. In traditional logic, the process of going from ""All S are P"" to its converse ""All P are S"" is called conversion. In the words"
"example) and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is these ratios that the trigonometric functions express. To define the trigonometric functions for the angle ""A"", start with any right triangle that contains the angle ""A"". The three sides of the triangle are named as follows: In ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180° ( radians). Therefore, in a right-angled triangle, the two non-right angles total 90° ( radians), so each of these angles must be in the range"
"Right angle In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a calque of Latin ""angulus rectus""; here ""rectus"" means ""upright"", referring to the vertical perpendicular to a horizontal base line. Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied"
"Trigonometry Trigonometry (from Greek ""trigōnon"", ""triangle"" and ""metron"", ""measure"") is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The 3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be"
"a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices. As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides. Further, every triangle has a unique Steiner circumellipse, which passes through the triangle's vertices and has its center at the triangle's centroid. Of all ellipses going through the triangle's vertices, it has the smallest area. The Kiepert hyperbola is the unique conic which passes through the triangle's three vertices, its"
"sides and the included angle, or two angles and a side, are known. The situation is simplified for right triangles. Consider the right triangle shown. The angle at is a right angle and the angle at is . The lengths of the sides of the triangle will be denoted as , and . can be related to the lengths of the sides as follows: The sine of , written , is defined as the ratio of the side opposite to the hypotenuse, that is, . The cosine of , written , is the side adjacent to over the hypotenuse, that"
"Pythagorean theorem In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides ""a"", ""b"" and ""c"", often called the ""Pythagorean equation"": where ""c"" represents the length of the hypotenuse and ""a"" and ""b"" the lengths of the triangle's other two sides. Although it is"
"Extended side In plane geometry, an extended side or sideline of a polygon is the line that contains one side of the polygon. The extension of a side arises in various contexts. In an obtuse triangle, the altitudes from the acute angled vertices intersect the corresponding extended base sides but not the base sides themselves. The excircles of a triangle, as well as the triangle's inconics that are not inellipses, are externally tangent to one side and to the other two extended sides. Trilinear coordinates locate a point in the plane by its relative distances from the extended sides of"
"length of the adjacent side to the length of the hypotenuse, so called because it is the sine of the complementary or co-angle, the other non-right angle. Because the angle sum of a triangle is radians, the co-angle is equal to ; so . In our case: The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side, so called because it can be represented as a line segment tangent to the circle, i.e. the line that touches the circle, from Latin ""linea tangens"" or touching line (cf. ""tangere"","
"reflects a median in the angle bisector that passes through the same vertex, one obtains a symmedian. The three symmedians intersect in a single point, the symmedian point of the triangle. There are various standard methods for calculating the length of a side or the measure of an angle. Certain methods are suited to calculating values in a right-angled triangle; more complex methods may be required in other situations. In right triangles, the trigonometric ratios of sine, cosine and tangent can be used to find unknown angles and the lengths of unknown sides. The sides of the triangle are known"
"Heronian triangle In geometry, a Heronian triangle is a triangle that has side lengths and area that are all integers. Heronian triangles are named after Hero of Alexandria. The term is sometimes applied more widely to triangles whose sides and area are all rational numbers, since one can rescale the sides by a common multiple to obtain a triangle that is Heronian in the above sense. Any right-angled triangle whose sidelengths are a Pythagorean triple is a Heronian triangle, as the side lengths of such a triangle are integers, and its area is also an integer, being half of the"
"side with the corresponding letter) is strictly less than a straight angle. The difference between the measure of a straight angle and the sum of the measures of a triangle's angles is called the defect of the triangle. The area of a hyperbolic triangle is equal to its defect multiplied by the square of : This theorem, first proven by Johann Heinrich Lambert, is related to Girard's theorem in spherical geometry. In all the formulas stated below the sides , , and must be measured in absolute length, a unit so that the Gaussian curvature of the plane is −1."
"this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Triangles can be classified according to the lengths of their sides: Hatch marks, also called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of ""ticks"", short line segments in the form of tally marks; two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is usually no more than 3"
"angle may be expressed in different units: Throughout history, carpenters and masons have known a quick way to confirm if an angle is a true ""right angle."" It is based on the most widely known Pythagorean triple (3, 4, 5) and so called the ""Rule of 3-4-5."" From the angle in question, running a straight line along one side exactly three units in length, and along the second side exactly four units in length, will create a hypotenuse (the longer line opposite the right angle that connects the two measured endpoints) of exactly 5 units in length. This measurement can"
"is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle. If the side lengths of a triangle are formula_7, the semiperimeter formula_8 and A is the angle opposite side formula_9, then the length of the internal bisector of angle A is or in trigonometric terms, If the internal bisector of angle A in triangle ABC has length formula_12 and if this bisector divides the side opposite A into"