A chord divides a circle into two what?
"the excess amount. Inscribe a square in the circle, so that its four corners lie on the circle. Between the square and the circle are four segments. If the total area of those gaps, ""G"", is greater than ""E"", split each arc in half. This makes the inscribed square into an inscribed octagon, and produces eight segments with a smaller total gap, ""G"". Continue splitting until the total gap area, ""G"", is less than ""E"". Now the area of the inscribed polygon, ""P"" = ""C"" − ""G"", must be greater than that of the triangle. But this forces a contradiction,"
"mathematicians made use of the chord. Given a circle and an arc on the circle, the chord is the line that subtends the arc. A chord's perpendicular bisector passes through the center of the circle and bisects the angle. One half of the bisected chord is the sine of one half the bisected angle, that is, and consequently the sine function is also known as the ""half-chord"". Due to this relationship, a number of trigonometric identities and theorems that are known today were also known to Hellenistic mathematicians, but in their equivalent chord form. Although there is no trigonometry in"
"a ""chord""."" According to Monath (1984, p. 37); ""A chord is a combination of three or more tones sounded simultaneously,"" and the distances between the tones are called intervals. However sonorities of two pitches, or even single-note melodies, are commonly heard as ""implying"" chords. A simple example of two notes being interpreted as a chord is when the root and third are played, but the fifth is omitted. In the key of C major, if the music comes to rest on the two notes G and B, most listeners will hear this as a G major chord. Since a chord"
"ellipse or parabola, while “two curves in the same plane” is a hyperbola. A chord is a straight line whose two end points are on the figure; i.e., it cuts the figure in two places. If a grid of parallel chords is imposed on the figure, then the diameter is defined as the line bisecting all the chords, reaching the curve itself at a point called the vertex. There is no requirement for a closed figure; e.g., a parabola has a diameter. A parabola has symmetry in one dimension. If you imagine it folded on its one diameter, the two"
"point a ""tangent line"" and in no points an ""exterior line"". A ""chord"" of a circle is the line segment that joins two distinct points of the circle. A chord is therefore contained in a unique secant line and each secant line determines a unique chord. In rigorous modern treatments of plane geometry, results that seem obvious and were assumed (without statement) by Euclid in his treatment, are usually proved. For instance, ""Theorem (Elementary Circular Continuity)"": If formula_1 is a circle and formula_2 a line that contains a point that is inside formula_1 and a point that is outside of"
"Intersecting chords theorem The intersecting chords theorem or just chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords in a circle. It states that the products of the lengths of the line segments on each chord are equal. More precisely for two chords ""AC"" and ""BD"" intersecting in a point ""S"" the following equation holds: The converse is true as well, that is if for two line segments ""AC"" and ""BD"" intersecting in S the equation above holds, then their four endpoints ""A"", ""B"", ""C"" and ""D"""
"""Canon Sinuum"" created at the end of the 16th century. A chord of a circle is a line segment whose endpoints are on the circle. Ptolemy used a circle whose diameter is 120. He tabulated the length of a chord whose endpoints are separated by an arc of ""n"" degrees, for ""n"" ranging from to 180 by increments of . In modern notation, the length of the chord corresponding to an arc of ""θ"" degrees is As ""θ"" goes from 0 to 180, the chord of a ""θ""° arc goes from 0 to 120. For tiny arcs, the chord is"
"a secant line of the circle, and the line segment connecting the intersection points is called a chord of the circle. If formula_33 holds, there exists only one intersection point and the line is tangent to the circle. If the weak inequality does not hold, the line does not intersect the circle. If the circle's midpoint is not the origin, see. The intersection of a line and a parabola or hyperbola may be treated analogously. The determination of the intersection points of two circles can be reduced to the previous case of intersecting a line and a circle. By subtraction"
"it is divided into two parts by the first chord. The length of one part is the sagitta of the arc, ""H"", and the other part is the remainder of the diameter, with length 2""r"" − ""H"". Applying the intersecting chords theorem to these two chords produces whence so Arc (geometry) In Euclidean geometry, an arc (symbol: ⌒) is a closed segment of a differentiable curve. A common example in the plane (a two-dimensional manifold), is a segment of a circle called a circular arc. In space, if the arc is part of a great circle (or great ellipse), it"
"the circle. Since the diameter is twice the radius, the ""missing"" part of the diameter is () in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (. Solving for ""r"", we find the required result. There are many compass-and-straightedge constructions resulting in circles. The simplest and most basic is the construction given the centre of the circle and a point on the circle. Place the fixed leg of the compass on the centre point, the movable leg"
"larger circle. Then the distribution of the chords on that smaller circle needs to be the same as on the larger circle. If the smaller circle is moved around within the larger circle, the probability must not change either. It can be seen very easily that there would be a change for method 3: the chord distribution on the small red circle looks qualitatively different from the distribution on the large circle: The same occurs for method 1, though it is harder to see in a graphical representation. Method 2 is the only one that is both scale invariant and"
"Dyad (music) In music, a dyad (less commonly, doad) is a set of two notes or pitches that, in particular contexts, may imply a chord. To understand dyads it is necessary first to understand the intervals between the notes. Take the notes C and E for example; we know that the interval between these two pitches is a major third, which can imply a C major chord (C E G). The most common two-note ""chord"" is the interval of a perfect fifth, which has been used in nearly all discovered pieces of polyphonic music. The reason fifths are so common"
"circle, then the area of the square on the side of the pentagon equals the sum of the areas of the squares on the sides of the hexagon and the decagon inscribed in the same circle. He used Ptolemy's theorem on quadrilaterals inscribed in a circle to derive formulas for the chord of a half-arc, the chord of the sum of two arcs, and the chord of a difference of two arcs. The theorem states that for a quadrilateral inscribed in a circle, the product of the lengths of the diagonals equals the sum of the products of the two"
"Circular segment In geometry, a circular segment (symbol: ⌓) is a region of a circle which is ""cut off"" from the rest of the circle by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is bounded by an arc (of less than 180°) of a circle and by the chord connecting the endpoints of the arc. Let ""R"" be the radius of the circle, ""θ"" the central angle in radians, α is the central angle in degrees, ""c"" the chord length, ""s"" the arc length, ""h"" the sagitta (height) of the"
"Chord (music) A chord, in music, is any harmonic set of pitches consisting of two or more (usually three or more) notes (also called ""pitches"") that are heard as if sounding simultaneously. (For many practical and theoretical purposes, arpeggios and broken chords, or sequences of chord tones, may also be considered as chords.) Chords and sequences of chords are frequently used in modern West African and Oceanic music, Western classical music, and Western popular music; yet, they are absent from the music of many other parts of the world. In tonal Western classical music (music with a tonic key or"
"is called a chord. Any chord in a circle which has no longer chord is called a diameter, and any segment connecting the circle's center (the midpoint of a diameter) to a point on the circle is called a radius. In an ellipse, the longest chord, which is also the longest diameter, is called the ""major axis"", and a segment from the midpoint of the major axis (the ellipse's center) to either endpoint of the major axis is called a ""semi-major axis"". Similarly, the shortest diameter of an ellipse is called the ""minor axis"", and the segment from its midpoint"
"this line is a chord of the circle. This chord is called a ""jyā"" which in Sanskrit means ""a bow-string"", presumably translating Hipparchus's with the same meaning. The word ""jīvá"" is also used as a synonym for ""jyā"" in geometrical literature. At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs. The half-chords were called ""ardha-jyā""s or ""jyā-ardha""s. These terms were again shortened to ""jyā"" by omitting the qualifier ""ardha"" which meant ""half"
"in radians made by the arc at the centre of the circle. If the value of angle is given in degrees, then we can also use the following formula: The length of a chord formed with the extremal points of the arc is given by where C represents the chord length, R represents the radius of the circle, and θ represents the angular width of the sector in radians. Circular sector A circular sector or circle sector (symbol: ⌔), is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the"
"chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle. The angle ""θ"" is taken in the positive sense and must lie in the interval (radian measure). The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be (), and then using the Pythagorean theorem to calculate the chord length: The last step uses the half-angle formula. Much as modern trigonometry is"
"the two line segments obtained on one chord is equal to the product of the two line segments obtained on the other chord. In the present case: or Substituting into the previous equation gives the law of cosines: Note that the power of the point with respect to the circle has the negative value . Case of obtuse angle . This proof uses the power of a point theorem directly, without the auxiliary triangles obtained by constructing a tangent or a chord. Construct a circle with center and radius (see Figure 9), which intersects the secant through and in and"
"the nephroid, where there is no tangent. For formula_69 one can divide by formula_70 to obtain For formula_76 the chord degenerates to a point. For formula_77 one can divide by formula_78 and gets the equation of the chord: The two angles formula_80 are defined differently (formula_15 is one half of the rolling angle, formula_82 is the parameter of the circle, whose chords are determined), for formula_83 one gets the same line. Hence any chord from the circle above is tangent to the nephroid and The considerations made in the previous section give a proof for the fact, that the caustic"
"would play a C Major chord (the notes C, E and G) and a G Major chord (the notes G, B and D) simultaneously. A simile mark in the middle of an otherwise empty measure tells the musician to repeat the chord or chords of the preceding measure. When seen with two slashes instead of one it indicates that the previous measure's chords should be repeated for two further measures, called a double simile, and is placed on the measure line between the two empty bars. It simplifies the job of both the music reader (who can quickly scan ahead"
"Polychord In music and music theory, a bichord or polychord consists of two or more chords, one on top of the other. In shorthand they are written with the top chord above a line and the bottom chord below, for example F upon C: . The use of polychords may suggest bitonality or polytonality. Harmonic parallelism may suggest bichords. Examples may be found in Igor Stravinsky's ""Petrushka"", p. 15, and ""Rite of Spring"", ""Dance of the Adolescents"" (1921) (see Petrushka chord). They may also be found in the song ""Point of No Return"" from Andrew Lloyd Webber's ""The Phantom of"
"lie on a common circle. Or in other words if the diagonals of a quadrilateral ""ABCD"" intersect in ""S"" and fulfill the equation above then it is a cyclic quadrilateral. The value of the two products in the chord theorem depends only on the distance of the intersection point ""S"" from the circle's center and is called the absolute value of the power of ""S"", more precisely it can be stated that: where ""r"" is the radius of the circle, and ""d"" is the distance between the center of the circle, and the intersection point ""S"". This property follows directly"
"even between players of the same part. Divisi In musical terminology, divisi, or as typically printed ""“div.,”"" is an instruction to divide a single section of instruments into multiple subsections. This usually applies to the violins of the string section in an orchestra, although violas, cellos, and double basses can also be divided. Typically, 4-part French Horn sections include divided sections if Horns 1/2 and/or 3/4 are not playing the same music (""a2""). Other brass instruments can also be divided but it is not as frequent as with the Horn section. Woodwinds - especially Flutes and Clarinets - also utilize"
"a trigonometric table, which he needed when computing the eccentricity of the orbits of the Moon and Sun. He tabulated values for the chord function, which gives the length of the chord for each angle. He did this for a circle with a circumference of 21,600 and a radius (rounded) of 3438 units: this circle has a unit length of 1 arc minute along its perimeter. He tabulated the chords for angles with increments of 7.5°. In modern terms, the chord of an angle equals the radius times twice the sine of half of the angle, i.e.: He described the"
"is perpendicular to the chord. If the intersection of any two perpendicular chords divides one chord into lengths ""a"" and ""b"" and divides the other chord into lengths ""c"" and ""d"", then equals the square of the diameter. The sum of the squared lengths of any two perpendicular chords intersecting at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8""r"" – 4""p"" (where ""r"" is the circle's radius and ""p"" is the distance from the center point to the point of intersection). Thales' theorem states"
"Bridge chord The Bridge chord is a bitonal chord named after its use in the music of composer Frank Bridge (1879–1941). It consists of a minor chord with the major chord a whole tone above (CEG & DFA), as well as a major chord with the minor chord a semitone above (CEG & DFA), which share the same mediant (E/F). () Both form eleventh chords under inversion (DFACEG = D and DFACEG = Dm). Bridge had strong pacifist convictions, and he was deeply disturbed by the First World War, after which his compositions, beginning in 1921–24 with the Piano Sonata,"
"</score> Second inversion The second inversion of a chord is the voicing of a triad, seventh chord, or ninth chord in which the fifth of the chord is the bass note. In this inversion, the bass note and the root of the chord are a fourth apart which traditionally qualifies as a dissonance. There is therefore a tendency for movement and resolution. In notation form, it is referred to with a c following the chord position (For e.g., Ic. Vc or IVc) or as a chord (as in I). In the second inversion of a C-major triad, the bass is"
"Common chord (music) A common chord, in the theory of harmony, is a chord that is diatonic to more than one key or, in other words, is ""common"" to (shared by) two keys. A ""common chord"" may also be defined simply as a triadic chord (e.g., C–E–G), as one of the most commonly used chords in a key (I–IV–V–vi–ii), more narrowly as a triad in which the fifth is perfect (i.e., a major or minor triad), in which sense it is alternatively referred to as a ""perfect chord"" or, more narrowly still (in American practice), as a major triad only."
"(this is called a ""tonic triad""). An arpeggio for the chord of C major going up two octaves would be the notes (C, E, G, C, E, G, C). An arpeggio is a type of broken chord. Other types of broken chords play chord notes out of sequence or more than one note but less than the full chord simultaneously. Arpeggios can rise or fall for more than one octave. Students of musical instruments and singers learn how to play and sing scales and arpeggios. They are often a requirement for music examinations. An ""arpeggiated chord"" means a chord which"
"Scale of chords A scale of chords may be used to set or read an angle in the absence of a protractor. To draw an angle, compasses describe an arc from origin with a radius taken from the 60 mark. The required angle is copied from the scale by the compasses, and an arc of this radius drawn from the sixty mark so it intersects the first arc. The line drawn from this point to the origin will be at the target angle. A chord is a line drawn between two points on the circumference of a circle. Look at"
"section, typically the first violins, is to be divided into two groups for rendering passages that might, for example, include full chords. Thus, in the ""divisi"" first violins the ""outside"" players (nearer the audience) might play the top note of the chord, while the ""inside"" seated players play the middle note, and the second violins play the bottom note. At the point where the first violins no longer play ""divisi"", the score may indicate this with ""unison"" (abbrev. ""unis.""). When several people sing together, as in a chorus, the simplest way for them to sing is to sing in ""one"