Circles Maths

5 Key excerpts on "Circles Maths"

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  eBook - PDF

  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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  College Geometry

  Using the Geometer's Sketchpad

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

    (Publisher)

  At the very least, the experimentation you have done in constructing the diagram will have challenged you to think about the ideas you will need as you develop a proof. In these first several chapters we have been working with Euclid’s axioms. In some later chapters, we will change one of these axioms and observe the effects of that change. This will give us an opportunity to see how much our world view is shaped by our assumptions. (In other words, what we see depends on our axioms.) The geometric content of this chapter has been focused on properties of cir- cles. You have been reminded of a lot of the terminology used in talking about circles. You have probably also encountered some new and challenging ideas in the geometry of circles. A triangle has many circles associated with it, and this provides a bridge back to the work of the previous chapter. Each triangle has one incircle and three excircles, which are associated with the bisectors of the inter- nal and external angles of the triangle. The circumcircle passes through the three vertices of the triangle, and its center can be found by constructing the perpendic- ular bisectors of the sides. The nine-point circle of a triangle passes through many interesting points. We will continue our investigations of the nine-point circle in the next chapter, where we will be able to use methods of analytic geometry to develop some of the proofs. This chapter also explored cyclic quadrilaterals and some of the special properties they possess. We can construct various families of circles—circles that share particular characteristics. 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.

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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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