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CIRCLES           PRESENTED BY          ADAMYA SHYAM
CIRCLE                         DEFINITIONA Circle is a simple shape of Euclidean geometry that is the setof points in the ...
CIRCLE                            HISTORYThe word "circle" derives from the Greek, kirkos "a circle," from thebase Ker- wh...
CIRCLE                  TERMINOLOGYChord:    A line segment whose endpoints lie on the circle.Diameter: A line segment who...
CIRCLE                             CHORDChords are equidistant from the centre of a circle if and only ifthey are equal in...
CIRCLE                        DIAMETERIn geometry, the diameter of a circle is any straight linesegment that passes throug...
CIRCLE                           RADIUSIn classical geometry, a radius of a circle is any line segmentfrom its center to i...
CIRCLE                      ARC & SECTORIn geometry, an arc is a closed segment of a differentiable curvein the two-dimens...
CIRCLE                         SEGMENTIn geometry, a circular segment is an area of a circle informallydefined as an area ...
CIRCLE                         THEOREMSThe chord theorem states that if two chords, CD and EB, intersectat A, then CA × DA...
Circles
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Circles

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Transcript of "Circles"

  1. 1. CIRCLES PRESENTED BY ADAMYA SHYAM
  2. 2. CIRCLE DEFINITIONA Circle is a simple shape of Euclidean geometry that is the setof points in the plane that are equidistant from a givenpoint, the centre. The distance between any of the points on thecircle and the centre is called the radius. A circle is a simpleclosed curve which divides the plane into 3 regions:Interior, Exterior and On The Circle . In everyday use, the term"circle" may be used interchangeably to refer to either theboundary of the figure, or to the whole figure including itsinterior; in strict technical usage, the circle is the former and thelatter is called a disk. A circle can be defined as the curve tracedout by a point that moves so that its distance from a given pointis constant. A circle may also be defined as a special ellipse inwhich the two foci are coincident and the eccentricity is 0.Circles are conic sections attained when a right circular cone isintersected by a plane perpendicular to the axis of the cone.
  3. 3. CIRCLE HISTORYThe word "circle" derives from the Greek, kirkos "a circle," from thebase Ker- which means to turn or bend. The origins of the words"circus" and "circuit" are closely related. The circle has been knownsince before the beginning of recorded history. Natural circleswould have been observed, such as the Moon, Sun, and a shortplant stalk blowing in the wind on sand, which forms a circle shapein the sand. The circle is the basis for the wheel, which, with relatedinventions such as gears, makes much of modern civilizationpossible. In mathematics, the study of the circle has helped inspirethe development of geometry, astronomy, and calculus. Earlyscience, particularly geometry and astrology and astronomy wasconnected to the divine for most medieval scholars, and manybelieved that there was something intrinsically "divine" or "perfect"that could be found in circles.
  4. 4. CIRCLE TERMINOLOGYChord: A line segment whose endpoints lie on the circle.Diameter: A line segment whose endpoints lie on the Circle and which passes through the centre.Radius: Half of DiameterArc: Any connected part of the circles circumference.Sector: A region bounded by two radii and an arc lying between the radii.Segment: A region bounded by a chord and an arc lying between the chords endpoints.
  5. 5. CIRCLE CHORDChords are equidistant from the centre of a circle if and only ifthey are equal in length. If a central angle and an inscribed angleof a circle are subtended by the same chord and on the same sideof the chord, then the central angle is twice the inscribed angle. Iftwo angles are inscribed on the same chord and on the same sideof the chord, then they are equal. If two angles are inscribed onthe same chord and on opposite sides of the chord, then they aresupplemental. An inscribed angle subtended by a diameter is aright angle. The diameter is the longest chord of the circle. If theintersection of any two perpendicular chords divides one chordinto lengths a and b and divides the other chord into lengths cand d, then a2 + b2 + c2 + d2 equals the square of the diameter.The distance from a point on the circle to a given chord times thediameter of the circle equals the product of the distances fromthe point to the ends of the chord.
  6. 6. CIRCLE DIAMETERIn geometry, the diameter of a circle is any straight linesegment that passes through the center of the circle andwhose endpoints are on the boundary of the circle. Thediameters are the longest chords of the circle. In this senseone speaks of diameter rather than a diameter, because alldiameters of a circle have the same length, this being twicethe radius. For a convex shape in the plane, the diameter isdefined to be the largest distance that can be formedbetween two opposite parallel lines tangent to itsboundary, and the width is defined to be the smallest suchdistance. For a curve of constant width such as the Reuleauxtriangle, the width and diameter are the same because allsuch pairs of parallel tangent lines have the same distance.See also Tangent lines to circles.
  7. 7. CIRCLE RADIUSIn classical geometry, a radius of a circle is any line segmentfrom its center to its perimeter. By extension, the radius of acircle or sphere is the length of any such segment, which is halfthe diameter. If the object does not have an obvious center, theterm may refer to its circum radius, the radius of itscircumscribed circle . In either case, the radius may be morethan half the diameter, which is usually defined as themaximum distance between any two points of the figure. The inradius of a geometric figure is usually the radius of the largestcircle or sphere contained in it. The inner radius of a ring, tubeor other hollow object is the radius of its cavity. For regularpolygons, the radius is the same as its circumradius.The namecomes from Latin radius, meaning "ray" but also the spoke of achariot wheel.
  8. 8. CIRCLE ARC & SECTORIn geometry, an arc is a closed segment of a differentiable curvein the two-dimensional plane; for example, a circular arc is asegment of the circumference of a circle. If the arc is part of agreat circle , it is called a great arc.A circular sector or circle sector, is the portion of a diskenclosed by two radii and an arc, where the smaller area isknown as the minor sector and the larger being the majorsector. In the diagram, θ is the central angle in radians, theradius of the circle, and is the arc length of the minor sector. Asector with the central angle of 180° is called a semicircle.Sectors with other central angles are sometimes given specialnames, these include quadrants (90°), sextants (60°) andoctants (45°).
  9. 9. CIRCLE SEGMENTIn geometry, a circular segment is an area of a circle informallydefined as an area which is "cut off" from the rest of the circleby a chord. The circle segment constitutes the part betweenthe secant and an arc, excluding of the circles center. SEGMENT CHORD RADIUS CENTRE DIAMETER SECTOR
  10. 10. CIRCLE THEOREMSThe chord theorem states that if two chords, CD and EB, intersectat A, then CA × DA = EA × BA. If a tangent from an external point Dmeets the circle at C and a secant from the external point D meetsthe circle at G and E respectively, then DC2 = DG × DE. If twosecants, DG and DE, also cut the circle at H and F respectively, thenDH × DG = DF × DE. The angle between a tangent and chord isequal to one half the subtended angle on the opposite side of thechord. If the angle subtended by the chord at the centre is 90degrees then l = r√2, where l is the length of the chord and r is theradius of the circle. If two secants are inscribed in the circle asshown at right, then the measurement of angle A is equal to onehalf the difference of the measurements of the enclosed arcs.
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