Segment of a Circle

4 Key excerpts on "Segment of a Circle"

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  eBook - PDF

  College Geometry

  Using the Geometer's Sketchpad

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

    (Publisher)

  DISCUSSION 89 A chord of a circle is a line segment joining two points on the circle. A chord that passes through the center of the circle is called a diameter. Using Sketchpad, we can construct a diameter of a circle by constructing a line through the center of the circle, and then finding the line segment between the points where this line intersects the circle. While a chord intersects a circle at two points, a tangent is a line that intersects a circle at exactly one point. The point where a tangent line touches the circle is called the point of tangency. It is not difficult to prove that a tangent line is perpendicular to a radius at the point of tangency. The circumference of a circle is the length of its perimeter. An arc of a circle is a piece of the circle. A sector of a circle is a pie-shaped portion of the inte- rior of the circle, bounded by an arc of the circle and two radii. If P, Q, and R are three points on a circle with center at O, the angle  POR is called a central angle of the circle, and the angle  PQR is an inscribed angle. The angle  PQR may also be called an angle subtended by the chord PR. We can define the mea- sure of an arc as the measure of the central angle subtended by the arc. This allows us to reword the statement about inscribed angles: The measure of an inscribed angle is half the measure of the arc subtending the inscribed angle. It turns out to be convenient at times to be able to refer to the arc rather than the central angle. Since a tangent to a circle, C , is a line perpendicular to a radius of C at the point of tangency, it is easy to construct a tangent to the circle C at a point A which is on C . To construct a tangent to C from a point B that is exte- rior to the circle, use the idea that an angle inscribed in a semicircle is a right angle. Construct a line segment from the point B to the center of C , then con- struct its midpoint, M. A circle centered at M with radius MB will intersect C in two points, P 1 and P 2 .

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  eBook - ePub

  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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  eBook - PDF

  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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  Elementary Geometry for College Students

  • Daniel C. Alexander, Geralyn M. Koeberlein, , , Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  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. A Look Back at Chapter 6 One goal in this chapter has been to classify angles inside, on, and outside the circle. Formulas for finding the measures of these angles were developed. Line and line segments related to a circle were defined, and some ways of finding the measures of these segments were described. Theorems involving inequalities in a circle were proved. Some constructions that led to tangents of circles were considered. A Look Ahead to Chapter 7 One goal of Chapter 7 is the study of loci (plural of locus), which has to do with point location. In fact, a locus of points is often noth-ing more than the description of some well-known geometric figure. Knowledge of locus leads to the determination of whether certain lines must be concurrent (meet at a common point). Finally, we will extend the notion of concurrence to develop further properties and terminology for regular polygons.

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