query: what is the opposite side of a right angle triangle

pos: Right triangle Right triangle A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the "hypotenuse" (side "c" in the figure). The sides adjacent to the right angle are called "legs" (or "catheti", singular: "cathetus"). Side "a" may be identified as the side "adjacent to angle B" and "opposed to" (or "opposite") "angle A", while side "b" is the side "adjacent to

pos: Right triangle a right triangle. For a given angle, a right triangle may be constructed with this angle, and the sides labeled opposite, adjacent and hypotenuse with reference to this angle according to the definitions above. These ratios of the sides do not depend on the particular right triangle chosen, but only on the given angle, since all triangles constructed this way are similar. If, for a given angle α, the opposite side, adjacent side and hypotenuse are labeled "O", "A" and "H" respectively, then the trigonometric functions are For the expression of hyperbolic functions as ratio of the sides of a

pos: Trigonometry 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

pos: Cathetus 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

pos: Cathetus 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)

pos: Law of tangents 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

pos: Triangle as follows: The "sine" of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case This ratio does not depend on the particular right triangle chosen, as long as it contains the angle "A", since all those triangles are similar. The "cosine" of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. 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. In our case

pos: Triangle The acronym "SOH-CAH-TOA" is a useful mnemonic for these ratios. The inverse trigonometric functions can be used to calculate the internal angles for a right angled triangle with the length of any two sides. Arcsin can be used to calculate an angle from the length of the opposite side and the length of the hypotenuse. Arccos can be used to calculate an angle from the length of the adjacent side and the length of the hypotenuse. Arctan can be used to calculate an angle from the length of the opposite side and the length of the adjacent side. In introductory

pos: Sine of the Arabic "jiba", which is a transliteration of the Sanskrit word for half the chord, "jya-ardha". To define the sine function of an acute angle "α", start with a right triangle that contains an angle of measure "α"; in the accompanying figure, angle "A" in triangle "ABC" is the angle of interest. The three sides of the triangle are named as follows: Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse, or: The other trigonometric functions of the angle can be

pos: Semiperimeter bisector of the angle opposite the side of length "a" is In a right triangle, the radius of the excircle on the hypotenuse equals the semiperimeter. The semiperimeter is the sum of the inradius and twice the circumradius. The area of the right triangle is formula_9 where "a" and "b" are the legs. The formula for the semiperimeter of a quadrilateral with side lengths "a", "b", "c" and "d" is One of the triangle area formulas involving the semiperimeter also applies to tangential quadrilaterals, which have an incircle and in which (according to Pitot's theorem) pairs of opposite sides have

pos: Hypotenuse formula_9 One may also obtain the value of the angle formula_9 by the equation: in which formula_16 is the other cathetus. Hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 (when squared, 9) and the

pos: Moiré pattern four sides equal to ; (we have a right triangle whose hypotenuse is and the side opposite to the angle is ). The pale lines correspond to the small diagonal of the rhombus. As the diagonals are the bisectors of the neighbouring sides, we can see that the pale line makes an angle equal to with the perpendicular of each pattern's line. Additionally, the spacing between two pale lines is , half of the long diagonal. The long diagonal is the hypotenuse of a right triangle and the sides of the right angle are and . The Pythagorean theorem gives:

pos: Euclidean geometry proposition 47) states that in any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). Thales' theorem, named after Thales of Miletus states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Cantor supposed that Thales proved his theorem by means of Euclid Book I,

pos: Sine architectures this method may offer no advantage. The CORDIC algorithm is commonly used in scientific calculators. Sine In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse). More generally, the definition of sine (and other trigonometric functions) can be extended to any real value in terms of the length of a certain line

pos: Sine Sine In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse). More generally, the definition of sine (and other trigonometric functions) can be extended to any real value in terms of the length of a certain line segment in a unit circle. More modern definitions express the sine as an infinite series or

pos: Right triangle altitude to the hypotenuse is related to the legs of the right triangle by For solutions of this equation in integer values of "a, b, f", and "c", see here. The altitude from either leg coincides with the other leg. Since these intersect at the right-angled vertex, the right triangle's orthocenter—the intersection of its three altitudes—coincides with the right-angled vertex. The Pythagorean theorem states that: In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the

pos: Trigonometric functions 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",

pos: Hypotenuse Hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 (when squared, 9) and the other has a length of 4 (when squared, 16), then their squares add up to 25. The length of the hypotenuse

pos: Triangle the hypotenuse equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length "c", and the legs have lengths "a" and "b", then the theorem states that The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle has a right angle opposite side "c". Some other facts about right triangles: For all triangles, angles and sides are related by the law of cosines and law of sines (also called the "cosine rule" and "sine rule"). The triangle inequality states that the sum

pos: Elementary algebra sign). One of the most well-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle: This equation states that formula_44, representing the square of the length of the side that is the hypotenuse (the side opposite the right angle), is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by formula_45 and formula_46. An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as formula_47); such equations are called

pos: Midpoint inscribed in the triangle. In a right triangle, the circumcenter is the midpoint of the hypotenuse. In an isosceles triangle, the median, altitude, and perpendicular bisector from the base side and the angle bisector of the apex coincide with the Euler line and the axis of symmetry, and these coinciding lines go through the midpoint of the base side. The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides, hence each bisecting two sides. The two bimedians and the line segment joining the midpoints of the diagonals are concurrent at (all intersect

pos: Tangent lines to circles two solutions ("k" = ±1) for the two external tangent lines: Geometrically this corresponds to computing the angle formed by the tangent lines and the line of centers, and then using that to rotate the equation for the line of centers to yield an equation for the tangent line. The angle is computed by computing the trigonometric functions of a right triangle whose vertices are the (external) homothetic center, a center of a circle, and a tangent point; the hypotenuse lies on the tangent line, the radius is opposite the angle, and the adjacent side lies on the line of

pos: Pythagorean theorem side "AB". Point "H" divides the length of the hypotenuse "c" into parts "d" and "e". The new triangle "ACH" is similar to triangle "ABC", because they both have a right angle (by definition of the altitude), and they share the angle at "A", meaning that the third angle will be the same in both triangles as well, marked as "θ" in the figure. By a similar reasoning, the triangle "CBH" is also similar to "ABC". The proof of similarity of the triangles requires the triangle postulate: the sum of the angles in a triangle is two right angles, and

pos: Sine defined similarly; for example, the cosine of the angle is the ratio between the adjacent side and the hypotenuse, while the tangent gives the ratio between the opposite and adjacent sides. As stated, the value formula_2 appears to depend on the choice of right triangle containing an angle of measure "α". However, this is not the case: all such triangles are similar, and so the ratio is the same for each of them. In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system. Let a line through the

pos: Pythagorean theorem the lengths of any two sides are known the length of the third side can be found. Another corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of the other sides, but less than their sum. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation.

pos: Pentagon determine the length of this side, the two right triangles "DCM" and "QCM" are depicted below the circle. Using Pythagoras' theorem and two sides, the hypotenuse of the larger triangle is found as formula_11. Side "h" of the smaller triangle then is found using the half-angle formula: where cosine and sine of "ϕ" are known from the larger triangle. The result is: With this side known, attention turns to the lower diagram to find the side "s" of the regular pentagon. First, side "a" of the right-hand triangle is found using Pythagoras' theorem again: Then "s" is found using Pythagoras'

pos: Trigonometric functions summarized in the following table and described in more detail below. The angle is the angle between the hypotenuse and the adjacent line – the angle at "A" in the accompanying diagram. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. The word comes from the Latin "sinus" for gulf or bay, since, given a unit circle, it is the side of the triangle on which the angle "opens". In our case: The cosine ("sine complement", Latin: "cosinus", "sinus complementi") of an angle is the ratio of the

pos: Pythagorean theorem 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

pos: Pythagorean theorem Pythagoras personally, [...] one can safely assume that it belongs to the very oldest period of Pythagorean mathematics." Around 400 BC, according to Proclus, Plato gave a method for finding Pythagorean triples that combined algebra and geometry. Around 300 BC, in Euclid's "Elements", the oldest extant axiomatic proof of the theorem is presented. 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

pos: Pythagorean theorem and the remaining square. Putting the two rectangles together to reform the square on the hypotenuse, its area is the same as the sum of the area of the other two squares. The details follow. Let "A", "B", "C" be the vertices of a right triangle, with a right angle at "A". Drop a perpendicular from "A" to the side opposite the hypotenuse in the square on the hypotenuse. That line divides the square on the hypotenuse into two rectangles, each having the same area as one of the two squares on the legs. For the formal proof, we require

pos: Altitude (triangle) foot. Also the altitude having the incongruent side as its base will be the angle bisector of the vertex angle. It is common to mark the altitude with the letter "h" (as in "height"), often subscripted with the name of the side the altitude is drawn to. In a right triangle, the altitude drawn to the hypotenuse "c" divides the hypotenuse into two segments of lengths "p" and "q". If we denote the length of the altitude by "h", we then have the relation For acute and right triangles the feet of the altitudes all fall on the triangle's sides

pos: History of mathematical notation back to any school or period before that of the Ionian Greeks, but the subsequent history may be divided into periods, the distinctions between which are tolerably well marked. Greek mathematics, which originated with the study of geometry, tended from its commencement to be deductive and scientific. Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of

pos: Congruence (geometry) 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

pos: Villarceau circles a quartic equation. The cross-section of the swept surface in the "xz" plane now includes a second circle. This pair of circles has two common internal tangent lines, with slope at the origin found from the right triangle with hypotenuse "R" and opposite side "r" (which has its right angle at the point of tangency). Thus "z"/"x" equals ±"r" / ("R" − "r"), and choosing the plus sign produces the equation of a plane bitangent to the torus. By symmetry, rotations of this plane around the "z" axis give all the bitangent planes through the center. (There are also horizontal

neg: Triangle 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

neg: Pythagorean theorem 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

neg: Triangle inequality angles satisfy: Therefore, and so, in particular, That means side opposite angle is shorter than side opposite the larger angle . But . Hence: A similar construction shows , establishing the theorem. An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point : (i) as depicted (which is to be proven), or (ii) coincident with (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle , which would violate the triangle postulate), or lastly, (iii) interior to the right triangle between points and (in which case angle

neg: Bisection The interior angle bisectors of a triangle are concurrent in a point called the incenter of the triangle, as seen in the diagram at right. The bisectors of two exterior angles and the bisector of the other interior angle are concurrent. Three intersection points, each of an external angle bisector with the opposite extended side, are collinear (fall on the same line as each other). Three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear. The angle bisector theorem

neg: Pythagorean theorem angle θ opposite side "b" and a side with length "s" along "c", as shown in the figure. Thābit ibn Qurra stated that the sides of the three triangles were related as: As the angle θ approaches /2, the base of the isosceles triangle narrows, and lengths "r" and "s" overlap less and less. When θ = /2, "ADB" becomes a right triangle, "r" + "s" = "c", and the original Pythagorean theorem is regained. One proof observes that triangle "ABC" has the same angles as triangle "ABD", but in opposite order. (The two triangles share the angle at vertex

neg: Extended side a reference triangle. If the point is outside the triangle, the perpendicular from the point to the sideline may meet the sideline outside the triangle—that is, not on the actual side of the triangle. In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear. In a triangle, three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear. An ex-tangential quadrilateral is a quadrilateral for which there exists a circle that

neg: Pythagorean theorem cosines: where θ is the angle between sides "a" and "b". When θ is 90 degrees (/2 radians), then cos"θ" = 0, and the formula reduces to the usual Pythagorean theorem. At any selected angle of a general triangle of sides "a, b, c", inscribe an isosceles triangle such that the equal angles at its base θ are the same as the selected angle. Suppose the selected angle θ is opposite the side labeled "c". Inscribing the isosceles triangle forms triangle "ABD" with angle θ opposite side "a" and with side "r" along "c". A second triangle is formed with

neg: Acute and obtuse triangles in which the Euler line is parallel to one side are acute. This property holds for side BC if and only if formula_2 If angle "C" is obtuse then for sides "a", "b", and "c" we have with the left inequality approaching equality in the limit only as the apex angle of an isosceles triangle approaches 180°, and with the right inequality approaching equality only as the obtuse angle approaches 90°. If the triangle is acute then If C is the greatest angle and "h" is the altitude from vertex "C", then for an acute triangle with the opposite inequality

neg: Bisection 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

neg: Dimensional analysis orientation as the identity element, independent of the "symmetry of the problem". Physical quantities that are vectors have the orientation expected: a force or a velocity in the z-direction has the orientation of . For angles, consider an angle that lies in the z-plane. Form a right triangle in the z-plane with being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation and the side opposite has an orientation . Then, since we conclude that an angle in the xy-plane must have an orientation , which is not unreasonable. Analogous

neg: Acute and obtuse triangles Acute and obtuse triangles An acute triangle is a triangle with all three angles acute (less than 90°). An obtuse triangle is one with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's angles must sum to 180°, no triangle can have more than one obtuse angle. Acute and obtuse triangles are the two different types of oblique triangles—triangles that are not right triangles because they have no 90° angle. In all triangles, the centroid—the intersection of the medians, each of which connects a vertex with the midpoint of the opposite side—and the incenter—the center of

neg: Altitude (triangle) 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

neg: Altitude (triangle) formula_30 we have If is any point on an altitude of any triangle , then For any point within an equilateral triangle, the sum of the perpendiculars to the three sides is equal to the altitude of the triangle. This is Viviani's theorem. In a right triangle the three altitudes , and (the first two of which equal the leg lengths and respectively) are related according to 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

neg: Trigonometric functions or equivalently, In this formula the angle at is opposite to the side . This theorem can be proven by dividing the triangle into two right ones and using the Pythagorean theorem. The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known. The following all form the law of tangents The explanation of the formulae in words would be cumbersome,

neg: Altitude (triangle) (not extended). In an obtuse triangle (one with an obtuse angle), the foot of the altitude to the obtuse-angled vertex falls in the interior of the opposite side, but the feet of the altitudes to the acute-angled vertices fall on the opposite extended side, exterior to the triangle. This is illustrated in the adjacent diagram: in this obtuse triangle, an altitude dropped perpendicularly from the top vertex, which has an acute angle, intersects the extended horizontal side outside the triangle. The three (possibly extended) altitudes intersect in a single point, called the orthocenter of the triangle, usually denoted by .

neg: Law of cosines the right triangle to which the Pythagorean theorem is applied moves "outside" the triangle . The only effect this has on the calculation is that the quantity is replaced by As this quantity enters the calculation only through its square, the rest of the proof is unaffected. However, this problem only occurs when is obtuse, and may be avoided by reflecting the triangle about the bisector of . Referring to Fig. 6 it is worth noting that if the angle opposite side is then: This is useful for direct calculation of a second angle when two sides and an included

neg: Triangle length of the other sides and the angle opposite to the unknown side. As per the law: For a triangle with length of sides "a", "b", "c" and angles of α, β, γ respectively, given two known lengths of a triangle "a" and "b", and the angle between the two known sides γ (or the angle opposite to the unknown side "c"), to calculate the third side "c", the following formula can be used: If the lengths of all three sides of any triangle are known the three angles can be calculated: The law of tangents, or tangent rule, can

neg: Law of cosines Consider a triangle with sides of length , , , where is the measurement of the angle opposite the side of length . This triangle can be placed on the Cartesian coordinate system by plotting the following points, as shown in Fig. 4: By the distance formula, we have Now, we just work with that equation: An advantage of this proof is that it does not require the consideration of different cases for when the triangle is acute, right, or obtuse. Drop the perpendicular onto the side to get (see Fig. 5) (This is still true if or is obtuse,

neg: Converse (logic) 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

neg: Symmedian Symmedian In geometry, symmedians are three particular geometrical lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the corresponding angle bisector (the line through the same vertex that divides the angle there in half). The angle formed by the symmedian and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector. The three symmedians meet at a triangle center

neg: Law of cosines Law of cosines In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states where denotes the angle contained between sides of lengths and and opposite the side of length . The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle is a right angle (of measure 90 degrees, or radians), then , and thus the law of cosines reduces to

neg: Altitude (triangle) The orthocenter lies inside the triangle if and only if the triangle is acute (i.e. does not have an angle greater than or equal to a right angle). If one angle is a right angle, the orthocenter coincides with the vertex at the right angle. Let denote the vertices and also the angles of the triangle, and let be the side lengths. The orthocenter has trilinear coordinates formula_2 and barycentric coordinates Since barycentric coordinates are all positive for a point in a triangle's interior but at least one is negative for a point in the exterior, and two of the

neg: Hyperbolic triangle end at an ideal vertex represented as an "omega point". Such a pair of sides may also be said to form an angle of zero. A triangle with a zero angle is impossible in Euclidean geometry for straight sides lying on distinct lines. However, such zero angles are possible with tangent circles. A triangle with one ideal vertex is called an omega triangle. Special Triangles with ideal vertices are: A triangle where one vertex is an ideal point, one angle is right: the third angle is the angle of parallelism for the length of the side between the right and

neg: Angle bisector theorem Angle bisector theorem In geometry, the angle bisector theorem 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. Consider a triangle "ABC". Let the angle bisector of angle "A" intersect side "BC" at a point "D" between "B" and "C". The angle bisector theorem states that the ratio of the length of the line segment "BD" to the length of segment "DC" is equal to the ratio

neg: Right triangle 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"

neg: Skinny triangle the triangle is equal to the length of the arc of circle of radius "r" subtended by angle "θ". The error is 10% or less for angles less than about 43°, and improves quadratically: when the angle decreases by a factor of , the error decreases by . The side-angle-side formula for the area of the triangle is; Applying the small angle approximations results in; The approximated solution to the right skinny triangle, referring to figure 3, is; This is based on the small-angle approximation; which when substituted into the exact solution; yields the desired result. The error of this

neg: Isosceles triangle two equal sides, and for isosceles sets, sets of points every three of which form an isosceles triangle. In an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the "vertex angle" and the angles that have the base as one of their sides are called the "base angles". The vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base. Whether an isosceles triangle

neg: Triangle form or its self-intersecting form, the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle. Every acute triangle has three inscribed squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two "distinct" inscribed

neg: Integer triangle harmonic progression. This is because such angles have to be rational angles of the form with rational 0 < < 1. But all the angles of integer triangles must have rational cosines and this will occur only when = i.e. the integer triangle is equilateral. The square of each internal angle bisector of an integer triangle is rational, because the general triangle formula for the internal angle bisector of angle "A" is formula_2 where "s" is the semiperimeter (and likewise for the other angles' bisectors). Any altitude dropped from a vertex onto an opposite side or its extension will split

neg: Square in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two "distinct" inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side. The fraction of the triangle's area that is filled by

neg: Pythagorean theorem B, both contain the angle θ, and so also have the same third angle by the triangle postulate.) Consequently, "ABC" is similar to the reflection of "ABD", the triangle "DBA" in the lower panel. Taking the ratio of sides opposite and adjacent to θ, Likewise, for the reflection of the other triangle, Clearing fractions and adding these two relations: the required result. The theorem remains valid if the angle formula_47 is obtuse so the lengths "r" and "s" are non-overlapping. Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the

neg: Schwarz triangle function of the upper half plane. To see how to handle the case when an angle of Δ is a right angle, note that the inequality implies that if one of the angles is a right angle, say "a" = 2, then both "b" and "c" are greater than 2 and one of them, "b" say, must be greater than 3. In this case, reflecting the triangle across the side AB gives an isosceles hyperbolic triangle with angles /"c", /"c" and 2/"b". If 2/"b" ≤ /3, i.e. "b" is greater than 5, then all the angles of the doubled triangle are

neg: Triangle it relatively easy to measure the length of each side, but relatively difficult to construct a 'height'. Various methods may be used in practice, depending on what is known about the triangle. The following is a selection of frequently used formulae for the area of a triangle. The height of a triangle can be found through the application of trigonometry. "Knowing SAS": Using the labels in the image on the right, the altitude is . Substituting this in the formula formula_26 derived above, the area of the triangle can be expressed as: (where α is the interior angle at "A",

neg: Triangle with angles α, β and γ is similar to a triangle with side lengths equal to sin α, sin β and sin γ. This triangle can be constructed by first constructing a circle of diameter 1, and inscribing in it two of the angles of the triangle. The length of the sides of that triangle will be sin α, sin β and sin γ. The side whose length is sin α is opposite to the angle whose measure is α, etc. The law of cosines, or cosine rule, connects the length of an unknown side of a triangle to the

neg: Symmedian This is also the point that minimizes the squared distance from the faces of the tetrahedron. Symmedian In geometry, symmedians are three particular geometrical lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the corresponding angle bisector (the line through the same vertex that divides the angle there in half). The angle formed by the symmedian and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is

neg: Law of sines information it is impossible to decide which is the triangle being asked for. The following are examples of how to solve a problem using the law of sines. Given: side , side , and angle . Angle A is desired. Using the law of sines, we conclude that Note that the potential solution is excluded because that would necessarily give . If the lengths of two sides of the triangle and are equal to , the third side has length , and the angles opposite the sides of lengths , , and are , , and respectively then In the

neg: Trigonometric functions gives the -component (the opposite to the angle or the rise) of the triangle, the cosine gives the -component (the adjacent of the angle or the run), and the tangent function gives the slope (-component divided by the -component). For angles less than a right angle, trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and their values can be found in the lengths of various line segments around a unit circle. Modern definitions express trigonometric functions as infinite series or as solutions of certain differential equations, allowing the extension of the

neg: Pythagorean theorem 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

neg: Ceyuan haijing of 12 volumes. The monography begins with a master diagram called the Diagram of Round Town(圆城图式). It shows a circle inscribed in a right angle triangle and four horizontal lines, four vertical lines. C: Center of circle: The North, South, East and West direction in Li Zhi's diagram are opposite to our present convention. There are a total of fifteen right angle triangles formed by the intersection between triangle TLQ, the four horizontal lines, and four vertical lines. The names of these right angle triangles and their sides are summarized in the following table In problems from Vol 2 to

neg: Acute and obtuse triangles the opposite sides. These altitudes fall entirely outside the triangle, resulting in their intersection with each other (and hence with the extended altitude from the obtuse-angled vertex) occurring in the triangle's exterior. Likewise, a triangle's circumcenter—the intersection of the three sides' perpendicular bisectors, which is the center of the circle that passes through all three vertices—falls inside an acute triangle but outside an obtuse triangle. The right triangle is the in-between case: both its circumcenter and its orthocenter lie on its boundary. In any triangle, any two angle measures "A" and "B" opposite sides "a" and "b" respectively are related

neg: Special right triangle 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 an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles

neg: Zephyr (sculpture) cylinders is a right triangle that measures tall and on the shortest side. It lies at an angle so that the right angle rests between the two cylinders. A third cylinder rests on top of the triangle and it measures in diameter and long. On top of this cylinder and the triangle is a large hoop with a diameter of approximately . There is a pole going through the hoop from the proper bottom right to the proper top left. It passes behind the hoop on the proper left and in front of the hoop on the proper right, where

neg: Solution of triangles side length. Assume that two sides and the angle are known. The equation for the angle can be implied from the law of sines: We denote further (the equation's right side). There are four possible cases: Once is obtained, the third angle . The third side can then be found from the law of sines: or The known characteristics are the side and the angles . The third angle . Two unknown sides can be calculated from the law of sines: or The procedure for solving an AAS triangle is same as that for an ASA triangle: First, find the

neg: Triangle and the sides are related by and and equivalently for "m" and "m". For angle A opposite side "a", the length of the internal angle bisector is given by for semiperimeter "s", where the bisector length is measured from the vertex to where it meets the opposite side. The interior perpendicular bisectors are given by where the sides are formula_78 and the area is formula_79 The altitude from, for example, the side of length "a" is The following formulas involve the circumradius "R" and the inradius "r": where "h" etc. are the altitudes to the subscripted sides; and The product

neg: Pappus's area theorem rather than only for right angled ones and secondly it uses parallelograms rather than squares. For squares on two sides of an arbitrary triangle it yields a parallelogram of equal area over the third side and if the two sides are the legs of a right angle the parallelogram over the third side will be square as well. For a right-angled triangle, two parallelograms attached to the legs of the right angle yield a rectangle of equal area on the third side and again if the two parallelograms are squares then the rectangle on the third side will be a

neg: Parsec 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

neg: Tangent half-angle formula gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers. The above descriptions of the tangent half-angle formulae (projection the unit circle and standard hyperbola onto the -axis) give a geometric interpretation of this function. The tangent of half of an acute angle of a right triangle whose sides are a Pythagorean triple will necessarily be a rational number in the interval . Vice versa, when a half-angle tangent is a rational number in the interval , there is a right triangle that has the full angle and that has side lengths

neg: Golden ratio called a golden pyramid. The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of size apothem by semi-base). The height of this pyramid is formula_34 times the semi-base, and the medial right triangle of the pyramid is a Kepler triangle, with sides formula_35. This is the only right triangle proportion with edge lengths in geometric progression. The angle with tangent formula_34 corresponds to the angle that the side of the pyramid makes with respect to the ground, 51.827... degrees (51° 49' 38"). According to various

neg: Perpendicular with lengths in the ratio 3:4:5. These can be laid out to form a triangle, which will have a right angle opposite its longest side. This method is useful for laying out gardens and fields, where the dimensions are large, and great accuracy is not needed. The chains can be used repeatedly whenever required. If two lines ("a" and "b") are both perpendicular to a third line ("c"), all of the angles formed along the third line are right angles. Therefore, in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other,

neg: Congruence (geometry) and the corresponding angles are acute and the length of the side opposite the angle is greater than the length of the adjacent side multiplied by the sine of the angle (but less than the length of the adjacent side), then the two triangles cannot be shown to be congruent. This is the ambiguous case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence. In Euclidean geometry, AAA (Angle-Angle-Angle) (or just AA, since in Euclidean geometry the angles of a triangle add up to 180°) does

neg: Area of a circle on a diameter. By Thales' theorem, this is a right triangle with right angle at B. Let the length of A′B be "c", which we call the complement of "s"; thus "c"+"s" = (2"r"). Let C bisect the arc from A to B, and let C′ be the point opposite C on the circle. Thus the length of CA is "s", the length of C′A is "c", and C′CA is itself a right triangle on diameter C′C. Because C bisects the arc from A to B, C′C perpendicularly bisects the chord from A to B, say at P. Triangle C′AP

neg: Stadiametric rangefinding easier to use, electronic rangefinders can give away the shooter's position to a well-equipped adversary, and the need for accurate range estimation existed for much longer than electronic rangefinders small and rugged enough to be suitable for military use. The stadia method is based upon the principle that in similar triangles homologous sides are proportional. This means that, for a right triangle with a given angle, the ratio of opposite side length to adjacent side length (see tangent) is constant. By using a reticle with marks of a known angular spacing, the principle of similar triangles can be used to

neg: Right angle to vectors. The presence of a right angle in a triangle is the defining factor for right triangles, making the right angle basic to trigonometry. A rectangle is a quadrilateral with four right angles. A square has four right angles, in addition to equal-length sides. The Pythagorean theorem states how to determine when a triangle is a right triangle. In Unicode, the symbol for a right angle is . It should not be confused with the similarly shaped symbol . Related symbols are , , and . In diagrams, the fact that an angle is a right angle is usually

neg: Transversal (geometry) then the lines are parallel. Euclid proves this by contradiction: If the lines are not parallel then they must intersect and a triangle is formed. Then one of the alternate angles is an exterior angle equal to the other angle which is an opposite interior angle in the triangle. This contradicts Proposition 16 which states that an exterior angle of a triangle is always greater than the opposite interior angles. Euclid's Proposition 28 extends this result in two ways. First, if a transversal intersects two lines so that corresponding angles are congruent, then the lines are parallel. Second, if a

neg: Isosceles triangle axis of the triangle, this distance above the base. An isosceles triangle has the largest possible inscribed circle among the triangles with the same base and apex angle, as well as also having the largest area and perimeter among the same class of triangles. The radius of the circumscribed circle is: The center of the circle lies on the symmetry axis of the triangle, this distance below the apex. For any isosceles triangle, there is a unique square with one side collinear with the base of the triangle and the opposite two corners on its sides. The Calabi triangle is

neg: Hyperbolic triangle the third angle. The triangle where two vertices are ideal points and the remaining angle is right, one of the first hyperbolic triangles (1818) described by Ferdinand Karl Schweikart. The triangle where all vertices are ideal points, an ideal triangle is the largest possible triangle in hyperbolic geometry because of the zero sum of the angles. The relations among the angles and sides are analogous to those of spherical trigonometry; the length scale for both spherical geometry and hyperbolic geometry can for example be defined as the length of a side of an equilateral triangle with fixed angles. The length

neg: Spherical trigonometry right spherical triangle is (Todhunter, Art.62) A quadrantal spherical triangle is defined to be a spherical triangle in which one of the sides subtends an angle of "π"/2 radians at the centre of the sphere: on the unit sphere the side has length "π"/2. In the case that the side "c" has length "π"/2 on the unit sphere the equations governing the remaining sides and angles may be obtained by applying the rules for the right spherical triangle of the previous section to the polar triangle "A'B'C' " with sides "a',b',c' " such that "A' "= "π"−"a", "a' " =

neg: Solution of triangles and the angles . The side can be found from the law of sines: If the angle for the side is acute and , another solution exists: We can find other characteristics by using Napier's analogies: Known: the angles . From the law of cosines we infer: The above algorithms become much simpler if one of the angles of a triangle (for example, the angle ) is the right angle. Such a spherical triangle is fully defined by its two elements, and the other three can be calculated using Napier's Pentagon or the following relations. If one wants to measure

neg: Integer triangle and its sides are integers: where formula_134 and "p" and "q" are any relatively prime integers such that formula_135. With angle A opposite side formula_136 and angle B opposite side formula_137, some triangles with B=2A are generated by with integers "m", "n" such that 0 < "n" < "m" < 2"n". Note that all triangles with "B" = 2"A" (whether integer or not) have formula_141. The equivalence class of similar triangles with formula_142 are generated by with integers formula_146 such that formula_147, where formula_148 is the golden ratio formula_149. Note that all triangles with formula_142 (whether with integer sides or

neg: Integer triangle common divisor. Integer triangles with a 60° angle can also be generated by with coprime integers "m", "n" with 0 < "n" < "m" (the angle of 60° is opposite to the side of length "a"). From here, all primitive solutions can be obtained by dividing "a", "b", and "c" by their greatest common divisor (e.g. an equilateral triangle solution is obtained by taking "m" = 2 and "n" = 1, but this produces "a" = "b" = "c" = 3, which is not a primitive solution). See also More precisely, If formula_109, then formula_110, otherwise formula_111. Two different pairs

neg: Tarski's axioms axioms to Euclid's parallel postulate, has an advantage over the others: Let a line segment join the midpoint of two sides of a given triangle. That line segment will be half as long as the third side. This is equivalent to the interior angles of any triangle summing to two right angles. Given any triangle, there exists a circle that includes all of its vertices. Given any angle and any point "v" in its interior, there exists a line segment including "v", with an endpoint on each side of the angle. Begin with two triangles, "xuz" and "x'u'z'." Draw the

neg: Triangle 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

neg: Exterior angle theorem segments). Only one of these angles contains the third side of the triangle in its interior, and this angle is called an "interior angle" of the triangle. In the picture below, the angles "∠ABC", "∠BCA" and "∠CAB" are the three interior angles of the triangle. An "exterior angle" is formed by extending one of the sides of the triangle; the angle between the extended side and the other side is the exterior angle. In the picture, angle "∠ACD" is an exterior angle. The proof of Proposition 1.16 given by Euclid is often cited as one place where Euclid gives a

neg: Cartan–Hadamard theorem and for any point "z" in "U" and constant speed geodesic γ in "U", one has This inequality may be usefully thought of in terms of a geodesic triangle Δ = "z"γ(0)γ(1). The left-hand side is the square distance from the vertex "z" to the midpoint of the opposite side. The right-hand side represents the square distance from the vertex to the midpoint of the opposite side in a Euclidean triangle having the same side lengths as Δ. This condition, called the CAT(0) condition is an abstract form of Toponogov's triangle comparison theorem. The assumption of non-positive curvature can be

neg: Equilateral triangle have simple relationships to the altitude ("h") of each vertex from the opposite side: In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, and the medians to each side coincide. A triangle "ABC" that has the sides "a", "b", "c", semiperimeter "s", area "T", exradii "r", "r", "r" (tangent to "a", "b", "c" respectively), and where "R" and "r" are the radii of the circumcircle and incircle respectively, is equilateral if and only if any one of the statements in the following nine categories is true. Thus these are properties that are unique to equilateral triangles, and