"cathetus. The ratio of the lengths of the catheti defines the trigonometric functions tangent and cotangent of the acute angles in the triangle: the ratio formula_1 is the tangent of the acute angle adjacent to formula_2 and is also the cotangent of the acute angle adjacent to formula_3. In a right triangle, the length of a cathetus is the geometric mean of the length of the adjacent segment cut by the altitude to the hypotenuse and the length of the whole hypotenuse. By the Pythagorean theorem, the sum of the squares of the lengths of the catheti is equal to"
"both the length of the hypotenuse and the angle the hypotenuse makes with the base line (""c"" above) at the same time when given ""x"" and ""y"". The angle returned is normally given by atan2(""y"",""x""). Orthographic projections: By means of trigonometric ratios, one can obtain the value of two acute angles, formula_3 and formula_4, of the right triangle. Given the length of the hypotenuse formula_5 and of a cathetus formula_6, the ratio is: The trigonometric inverse function is: in which formula_9 is the angle opposite the cathetus formula_6. The adjacent angle of the catheti formula_6 is formula_3 = 90° –"
"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"
"and ""G"" and hypotenuse ""A"", then and where formula_40 is the golden ratio formula_41 Since the sides of this right triangle are in geometric progression, this is the Kepler triangle. Thales' theorem states that if ""A"" is any point of the circle with diameter ""BC"" (except ""B"" or ""C"" themselves) ""ABC"" is a right triangle where ""A"" is the right angle. The converse states that if a right triangle is inscribed in a circle then the hypotenuse will be a diameter of the circle. A corollary is that the length of the hypotenuse is twice the distance from the right"
"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"
"angle across to make a right angle with one side, the right angle triangle's hypotenuse is length one and the sides are of length and . From this the trigonometric function tangent of 60° equals , and the sine of 60° and the cosine of 30° both equal . The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°. It is the distance between parallel sides of a regular hexagon with sides of length"
"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"
"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"
"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"
"≥ b ≥ c"" and ""formula_8"" is the golden ratio, as In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides, and less than their sum. The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle . An isosceles triangle is constructed with equal sides . From the triangle postulate, the angles in the right triangle satisfy: Likewise, in the isosceles triangle , the"
"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"
"and the midpoint of the side opposite that vertex. This is because the right triangle's orthocenter, the intersection of its altitudes, falls on the right-angled vertex while its circumcenter, the intersection of its perpendicular bisectors of sides, falls on the midpoint of the hypotenuse. In any right triangle the diameter of the incircle is less than half the hypotenuse, and more strongly it is less than or equal to the hypotenuse times formula_44 In a right triangle with legs ""a"", ""b"" and hypotenuse ""c"", with equality only in the isosceles case. If the altitude from the hypotenuse is denoted ""h"","
": , which follows immediately from the Pythagorean theorem. Of all right triangles, the 45°–45°–90° degree triangle has the smallest ratio of the hypotenuse to the sum of the legs, namely . Of all right triangles, the 45°–45°–90° degree triangle has the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely . Triangles with these angles are the only possible right triangles that are also isosceles triangles in Euclidean geometry. However, in spherical geometry and hyperbolic geometry, there are infinitely many different shapes of right isosceles triangles. This is a triangle whose three angles"
"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,"
"be made quickly and without technical instruments. The geometric law behind the measurement is the Pythagorean theorem (""The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides""). Thales' theorem states that an angle inscribed in a semicircle (with a vertex on the semicircle and its defining rays going through the endpoints of the semicircle) is a right angle. Two application examples in which the right angle and the Thales' theorem are included (see animations). Right angle In geometry and trigonometry, a right angle is an angle of exactly"
"form of the equation becomes So if a right triangle is constructed with side ""r"" and hypotenuse ""a"" then the angle between them will be θ/3. It is straightforward to generate a third method from this. Limaçon trisectrix In geometry, a limaçon trisectrix (called simply a trisectrix by some authors) is a member of the limaçon family of curves which has the trisectrix, or angle trisection, property. It can be defined as locus of the point of intersection of two lines, each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is"
"Geometric mean theorem The right triangle altitude theorem or geometric mean theorem is a result in elementary geometry that describes a relation between the lengths of the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that the geometric mean of the two segments equals the altitude. If ""h"" denotes the altitude in a right triangle and ""p"" and ""q"" the segments on the hypotenuse then the theorem can be stated as: or in term of areas: The latter version yields a method to square a rectangle with ruler"
"of its actual size: the ratios depend upon the three angles, not the lengths of the sides. Thus for either of the similar right triangles in the figure, the ratio of its horizontal side to its hypotenuse is the same, namely cos θ. The elementary definitions of the sine and cosine functions in terms of the sides of a right triangle are: The Pythagorean identity follows by squaring both definitions above, and adding; the left-hand side of the identity then becomes which by the Pythagorean theorem is equal to 1. This definition is valid for all angles, due to the"
"angles, the enclosed angle and the form corresponding to Pythagoras's theorem is regained: formula_34 The Pythagorean theorem, valid for right triangles, therefore is a special case of the more general law of cosines, valid for arbitrary triangles. In a right triangle with sides ""a"", ""b"" and hypotenuse ""c"", trigonometry determines the sine and cosine of the angle ""θ"" between side ""a"" and the hypotenuse as: From that it follows: where the last step applies Pythagoras's theorem. This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity. In similar triangles, the ratios of the sides are the"
"than 256: Isosceles right-angled triangles cannot have sides with integer values, because the ratio of the hypotenuse to either other side is , but cannot be expressed as a ratio of two integers. However, infinitely many ""almost-isosceles"" right triangles do exist. These are right-angled triangles with integral sides for which the lengths of the non-hypotenuse edges differ by one. Such almost-isosceles right-angled triangles can be obtained recursively, ""a"" is length of hypotenuse, ""n"" = 1, 2, 3, ... Equivalently, where {""x"", ""y""} are the solutions to the Pell equation , with the hypotenuse ""y"" being the odd terms of the"
"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"
"is the square root of 25, that is, 5. The word ""hypotenuse"" is derived from Greek (sc. or ), meaning ""[side] subtending the right angle"" (Apollodorus), The Greek term was loaned into Late Latin, as ""hypotēnūsa"". Adoption as a learned Latinism used in modern languages dates to the 16th century. The spelling in ""-e"", as ""hypotenuse"", is French in origin (Estienne de La Roche 1520). The length of the hypotenuse is calculated using the square root function implied by the Pythagorean theorem. Using the common notation that the length of the two legs of the triangle (the sides perpendicular to"
"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"
"hypotenuse to the length of the adjacent side, so called because it represents the secant line that ""cuts"" the circle (from Latin: ""secare"", to cut): The cosecant (""secant complement"", Latin: ""cosecans"", ""secans complementi"") of an angle is the reciprocal of its sine, that is, the ratio of the length of the hypotenuse to the length of the opposite side, so called because it is the secant of the complementary or co-angle: The cotangent (""tangent complement"", Latin: ""cotangens"", ""tangens complementi"") of an angle is the reciprocal of its tangent, that is, the ratio of the length of the adjacent side to"
"two legs (the two sides that meet at a right angle). This can be stated in equation form as where ""c"" is the length of the hypotenuse, and ""a"" and ""b"" are the lengths of the remaining two sides. Pythagorean triples are integer values of ""a, b, c"" satisfying this equation. The radius of the incircle of a right triangle with legs ""a"" and ""b"" and hypotenuse ""c"" is The radius of the circumcircle is half the length of the hypotenuse, Thus the sum of the circumradius and the inradius is half the sum of the legs: One of the"
"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)"
"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"
"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""."
"then with equality only in the isosceles case. If segments of lengths ""p"" and ""q"" emanating from vertex ""C"" trisect the hypotenuse into segments of length ""c""/3, then The right triangle is the only triangle having two, rather than one or three, distinct inscribed squares. Let ""h"" and ""k"" (""h"" > ""k"") be the sides of the two inscribed squares in a right triangle with hypotenuse ""c"". Then These sides and the incircle radius ""r"" are related by a similar formula: The perimeter of a right triangle equals the sum of the radii of the incircle and the three excircles:"
"the angle approaches zero, it is clear that the gap between the approximation and the original function quickly vanishes. The red section on the right, , is the difference between the lengths of the hypotenuse, , and the adjacent side, . As is shown, and are almost the same length, meaning is close to 1 and helps trim the red away. The opposite leg, , is approximately equal to the length of the blue arc, . Gathering facts from geometry, , from trigonometry, and , and from the picture, and leads to: Simplifying leaves, can also be expressed this way,"
"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"
"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:"
"Hypot Hypot is a mathematical function defined to calculate the length of the hypotenuse of a right-angle triangle. It was designed to avoid errors arising due to limited-precision calculations performed on computers. Calculating the length of the hypotenuse of a triangle is possible using the square root function on the sum of two squares, but hypot(""x"", ""y"") avoids problems that occur when squaring very large or very small numbers. The magnitude of the hypotenuse from (0, 0) to (""x"", ""y"") can be calculated using: This operation is also known as Pythagorean addition. However the squares of very large or small"