Circles

8 Key excerpts on "Circles"

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  eBook - ePub

  Geometry: A Comprehensive Course

  • Dan Pedoe(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  II

  Circles

  The circle, in various guises, occurs in many branches of mathematics. In this chapter we shall consider those properties which have the habit of appearing in different contexts. Our methods will be mixed. We shall use Euclidean geometry or analytical geometry, according to our needs at the time. We begin with the consideration of certain one-to-one mappings of the plane onto itself, and the product of these mappings. (See 0.11.) Later on we study inversion, which is only one-to-one onto if we cut out a point of the plane, or better still, add a point to the plane.

  12.1   The nine-point circle

  We have already encountered this circle (§7.3 ). It is a circle which passes through nine points intimately connected with any given triangle ABC. (We shall not be using vector methods, and we shall use the customary notation for the points connected with a triangle). The nine-point circle is the first really exciting one to appear in any course on Euclidean geometry which goes far enough.

  Fig. 12.1

  To prove the existence of the nine-point circle of a triangle, we make use of a one-to-one mapping of the Euclidean plane onto itself. This mapping is called a central dilatation. We choose a point V, and call it the center of the dilatation (Fig 12.1 ).

  If P is any point in the plane its image, or map P′ is found by choosing the unique point P′ which lies on VP, is on the same side of V as is P, and is such that where k is a fixed real positive number. If P = V, we define V′ to be V.

  This mapping is evidently one-to-one. It is also called a central similarity. It will occur again later (§41.5 ) as one of the fundamental mappings of the Euclidean plane onto itself. Here we are only interested in the case when k is positive, and when the point P moves on a circle, center O (Fig. 12.2 ). If O′ is the map of O, then the triangles VOP and VO′P′ are similar, so that , Since O is a fixed point, and | OP | = r, the radius of the circle on which P moves, we deduce that P′ moves on a circle, center O′ of radius kr

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  Plain Plane Geometry

  • Amol Sasane(Author)
  • 2015(Publication Date)
  • WSPC

    (Publisher)

  Chapter 5

  Circles

  We had defined the terms circle, its center and its radius in Section 1.6 . In this chapter, we will study Circles. We begin with some terminology. Recall that a circle C (O, r ) with center O and radius r > 0 is the set of points P in the plane such that OP = r . The set of points P in the plane such that OP < r is called the interior of the circle, while the set of points P for which OP > r is called the exterior of the circle. Circles having the same center are said to be concentric .

  A line segment joining any two points on the circle is called a chord .

  A diameter is a special type of chord: it is one which passes through the center of the circle. Clearly the length of any diameter of a circle of radius r is equal to 2r . A diameter divides the circle into two semiCircles .

  Theorem 5.1 . The perpendicular from the center of a circle to a chord bisects the chord .

  Proof . Let OM be the perpendicular dropped from the center O of the circle C (O, r ) to its chord AB . In the right triangles ΔOAM and ΔOBM , we have OA = OB = r , and the side OM is common. By the RHS Congruency Rule, ΔOAM ≃ ΔOBM , giving AM = MB , as wanted.

  Theorem 5.2 . The line joining the center of a circle to the midpoint of a chord is perpendicular to the chord .

  Proof . Let M be the midpoint of the chord AB of the circle C (O, r ). In the two triangles ΔOAM and ΔOBM , we have OA = OB = r , the side OM is common, and AM = BM . By the SSS Congruency Rule, we have ΔOAM ≃ ΔOBM . Hence we obtain ∠OMA = ∠OMB , and being supplementary, they must each equal 90°.

  Corollary 5.1 . The perpendicular bisectors of two chords of a circle contain its center .

  We know that two distinct points determine a unique line passing through them. We can now ask:

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  Mathematical Practices, Mathematics for Teachers

  Activities, Models, and Real-Life Examples

  • Ron Larson, Robyn Silbey(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 12.1 Circumferences and Areas of Circles 459 12.1 Circumferences and Areas of Circles Find the diameter and radius of a circle. Find the circumference of a circle. Find the area of a circle. Parts of a Circle Standards Grades K–2 Geometry Students should identify and describe shapes. Grades 3–5 Measurement and Data Students should understand concepts of angles and angle measurement. Grades 6–8 Geometry Students should solve real-life and mathematical problems involving area. Definition of a Circle A circle is the set of all points in a plane that are the same distance from a point called the center. Center Circle The radius is the distance from the center to any point on the circle. The diameter is the distance across the circle through the center. Radius and Diameter of a Circle Words The diameter d of a circle is twice the radius r. The radius r of a circle is one-half the diameter d. Algebra Diameter: d = 2r Radius: r = d — 2 EXAMPLE 1 Finding a Diameter and a Radius a. The diameter of a circle is 12 feet. b. The radius of a circle is 4.5 meters. Find the radius. Find the diameter. 12 ft 4.5 m SOLUTION a. r = d — 2 Radius of a circle b. d = 2r Diameter of a circle = 12 — 2 Substitute 12 for d. = 2(4.5) Substitute 4.5 for r. = 6 Divide. = 9 Multiply. The radius is 6 feet. The diameter is 9 meters. The tool used to draw a circle is called a compass. To use a compass, place the point at the center of a circle. Adjust the radius. Then rotate the compass so that the pencil draws the circle.

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  Geometry Revisited

  • H. S. M. Coxeter, S. L. Greitzer(Authors)
  • 1967(Publication Date)
  • American Mathematical Society

    (Publisher)

  C H A P T E R 2 Some Properties of Circles Although the Greeks worked fruitfully, not only in geometry but also in the most varied fields of mathematics, neverthe- less we today have gone beyond them everywhere and cer- tainly also in geometry. F. KIcin The circle has been held in highest esteem through the ages. Its perfect form has affected philosophers and astronomers alike. Until Kepler derived his laws, the thought that planets might move in anything but circular paths was unthinkable. Nowadays, the words “square”, “line”, and the like sometimes have derogatory connotations, but the circle- never. Cleared of superstitious nonsense and pseudo-science, it still stands out, as estimable as ever. Limitations of space make it impossible for us to present more than a few of the most interesting properties developed since Euclid of the circle and its relation to triangles and other polygons. 2.1 The power of a point with respect to a circle We begin our investigations by recalling two of Euclid’s theorems: 111.35, about the product of the parts into which two chords of a circle divide each other (that is, in the notation of Figure 2.1A, PA X PA’ = PB X PB’ ), and 111.36, comparing a. secant and a tangent drawn from the same point P outside the circle (in Figure 2.1B, PA X PA’ = PTS ). If we agree to regard a tangent as the limiting form of a secant, we can combine these results as follows: 27 28 PROPERTIES OF Circles THEOREM 2.11. If two lines through a point P meet a circle at points A, A' (possibly coincident) and B, B' (possibly coincident), respec- tively, then PA X PA' = PB X PB'. - Figure 2.1A For a proof we merely have to observe that the similar triangles PAB' and PBA' (with a common angle at P ) yield PA PB PB' PA -= -In Figure 2.1B, we can equally well use the similar triangles PAT and PA PT PTA' to obtain ----PT PA and then say PA X PA' = P P = PB X PB'. - Figure 2.1B Let R denote the radius of the circle, and d the distance from P to the center.

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  Geometry

  A Self-Teaching Guide

  • Steve Slavin, Ginny Crisonino(Authors)
  • 2004(Publication Date)
  • Wiley

    (Publisher)

  C = πd Divide both sides of the equation by π. = d Now we have an equation for the diameter of a circle. = d We’ll substitute the values for C and π. d ≈ 5.1 inches The symbol we’ll use for a circle is . A chord is a line joining any two points on the circumference. Thus AB and AC are chords of the following circle. C A B O 16 3.14 C π 76 GEOMETRY An arc is part of the circumference of a circle. The symbol for arc is . Thus AB refers to arc AB. An arc of 1° is th of a circle. It’s important to distinguish between chords and arcs named with the same letters. In the previous circle, chord AB is a straight line joining points A and B, which are both on the circumference of the circle. But arc AB is actually the curved part of the circumference running from point A to point B. A diameter is a chord that runs through the center of the circle; it’s the longest possi- ble chord. Thus in the last figure, AC is the diameter of O. The diameter divides a circle into two semiCircles. A semicircle is an arc equal to one-half of the circumference of a circle; a semicircle contains 180°. Example 6: Draw a chord, CD, on M. Solution: Here are two of the many possible chords. Are you ready for some congruent Circles? Congruent Circles are Circles having con- gruent radii. Thus if OE = O′G′, then circle O circle O′. O G' F E O' M C D M C D M 1 360 Circles 77 Here’s one last term: central angle. A central angle is an angle formed by two radii. In the following figure, the angle between radii OB and OC is a central angle. Therefore, the central angle between OB and OC is O, which can also be writ- ten BOC. Sometimes angles are not represented in degrees, but in units called radians. If a central angle of a circle intercepts an arc equal in length to the radius of the cir- cle, the central angle is defined as 1 radian. Because the radius can be marked off along the circumference 2π (or about 6.28) times, we see that 2π = 360°, or π = 180°.

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  Geometry with Trigonometry

  • Patrick D Barry(Author)
  • 2001(Publication Date)
  • Woodhead Publishing

    (Publisher)

  7 Circles; their basic properties Hitherto our sets have involved lines and half-planes, and specific subsets of these. Now we introduce Circles and study their relationships to lines. We do not do this just to admire the Circles, and to behold their striking properties of symmetry. They are the means by which we control angles, and simplify our work on them. 7.1 INTERSECTION OF A LINE AND A CIRCLE 7.1.1 Terminology concerning a circle Definition . If O is any point of the plane Π and k is any positive real number, we call the set C ( O ; k ) of all points X in Π which are at a distance k from O , i.e. C ( O ; k ) = { X ∈ Π : | O , X | = k } , the circle with centre O and length of radius k . If X ∈ C ( O ; k ) the segment [ O, X ] is called a radius of the circle. Any point U such that | O , U | < k is said to be an interior point for this circle. Any point V such that | O , V | > k is said to be an exterior point for this circle. For every circle C ( O ; k ) and line l , one of the following holds:-(i) l ∩ C ( O ; k ) = { P } for some point P , in which case every point of l { P } is exterior to the circle. (ii) l ∩C ( O ; k ) = { P, Q } for some points P and Q , with P = Q , in which case every point of [ P, Q ] { P, Q } is interior to the circle, and every point of PQ [ P, Q ] is exterior to the circle. (iii) l ∩ C ( O ; k ) = ∅ , in which case every point of l is exterior to the circle. Proof . Let M = π l ( O ) , and let m be the line which contains M and is perpendicular to l , so that O ∈ m . 105

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  College Geometry

  Using the Geometer's Sketchpad

  • Barbara E. Reynolds, William E. Fenton(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  For example, there is a family of Circles that all have the same center point O, another family of Circles that all pass through a point A, and a family of Circles with the same radius. We have investigated two interesting families of Circles in this chapter: Circles that share a common chord, and Circles that are orthogonal to a given fixed circle. While these two families of Circles are interesting to study for themselves, they will also be important in investigations that we will take up in later chapters. Mathematics is a living study. At every age of history, people have been investi- gating interesting problems. Our investigations of the arbelos and the salinon gave us an opportunity to step back into the early history of mathematics. Although these figures are constructed using just simple arcs of Circles, many interesting CHAPTER OVERVIEW 109 problems can be posed about them. We have investigated just a few of these prob- lems. The nine-point circle comes from the nineteenth century. We will continue our investigations of the nine-point circle in Chapter 5, where we will be able to use methods of analytic geometry—coordinates and equations—as tools to investigate these problems and to prove our conjectures. At the end of Chapter 2, we listed a number of constructions that you should master. These were • construct a perpendicular to a line from any point (on or off the line), • construct the perpendicular bisector of a given line segment, • construct the foot of the perpendicular from a point P to a line , • construct the tangent line to a circle from a point on the circle, • construct the tangent line to a circle from a point not on the circle, • construct the bisector of a given angle. Now, as you are coming to the end of Chapter 4, you should recognize that each of these constructions depends in a fundamental way on some properties of Circles. Each of these constructions can be done by using the intersection points of one, two, or three Circles.

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  Divided Spheres

  Geodesics and the Orderly Subdivision of the Sphere

  • Edward S. Popko(Author)
  • 2012(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  Chapter 4. Circular Reasoning 81 We begin with the fundamental notions on which everything else is based; namely, points and Circles. 4.1 Lesser and Great Circles Any plane cutting through a sphere intersects the sphere in a circle. If the plane goes through the center of the sphere, the circle is called a great circle; and if not, the circle is a small circle, also called a lesser circle. A family of parallel planes in space will cut the sphere in one great circle and infinitely many lesser Circles and some of these Circles are particularly useful in geodesic applications. In Figure 4.1(a), every circle is a lesser circle defined by a plane that intersects the sphere but does not pass through the sphere’s center. In the figure, the plane defining each lesser circle is perpendicular to one of three Cartesian axes. Some Circles have such small radii that they appear as points, but the intersection of the sphere and a plane is a point only when the plane is tangent to the sphere at the point so it does not cut through the sphere. A slice of a sphere also creates something else besides the great or lesser circle, namely two spherical caps. The spherical caps of a great circle are hemispheres or two half-spheres. For a lesser circle, one spherical cap is smaller than a hemisphere and shaped somewhat like a contact lens. A spherical cap has a center on the sphere and that is different from the center of the disc in the plane that cuts out the circle on the sphere. If we choose a collec-tion of points on the sphere and consider a spherical cap centered on each one, and if all the caps have the same size and the largest possible radius without causing any caps to overlap, we can compare the total area of the caps and the area of the sphere to get a measure of how well distributed the points are on the sphere (see Figure 4.1(b)). The dashed caps are on the other side of the sphere. The radii of the cap’s lesser Circles are all the same.

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