Construction and Loci

5 Key excerpts on "Construction and Loci"

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  eBook - ePub

  College Geometry

  An Introduction to the Modern Geometry of the Triangle and the Circle

  • Nathan Altshiller-Court(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  The nature of the loci obtained depends upon the condition omitted. In elementary geometry these conditions must be such that the loci shall consist of straight lines and circles. The neatness and simplicity of a solution depend very largely upon the judicious choice of the geometric loci.

  Knowledge about a considerable number of geometric loci may often enable one to discover immediately where the required point is to be located.

  FIG. 8

  The following are the most important and the most frequently useful geometric loci:

  Locus 1. The locus of a point in a plane at a given distance from a given point is a circle having the given point for center and the given distance for radius.

  Locus 2. The locus of a point from which tangents of given length can be drawn to a given circle is a circle concentric with the given circle.

  Locus 3. The locus of a point at a given distance from a given line consists of two lines parallel to the given line.

  Locus 4. The locus of a point equidistant from two given points is the perpendicular bisector of the segment determined by the two points.

  For the sake of brevity the expression perpendicular bisector may be replaced conveniently by the term mediator.

  Locus 5. The locus of a point equidistant from two intersecting lines indefinitely produced consists of the two bisectors of the angles formed by the two given lines.

  Locus 6. The locus of a point such that the tangents from it to a given circle form a given angle, or, more briefly, at which the circle subtends a given angle, is a circle concentric with the given circle.

  Locus 7. The locus of a point, on one side of a given segment, at which this segment subtends a given angle is an arc of a circle passing through the ends of the segment.

  Let M (Fig. 8 ) be a point of the locus so that the angle AMB is equal to the given angle. Pass a circle through the three points A, B, M. At any point M ′ of the arc AMB the segment AB subtends the same angle as at M; hence every point of the arc AMB belongs to the required locus. On the other hand, any point N not on the arc AMB will lie either inside or outside this arc. In the first case the angle ANB will be larger, and in the second case smaller, than the angle AMB. Hence N

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  Engineering Graphics & Technology NQF2 SB

  TVET FIRST

  • Sparrow Consulting(Author)
  • 2013(Publication Date)
  • Macmillan

    (Publisher)

  When you have to do compass and straightedge constructions (geometric constructions), there are five basic constructions that are used repeatedly, using the points, lines and circles that have already been constructed. These are: Figure 5.1: Geometry in nature Geometric construction: A special type of drawing technique used in geometry. The process of drawing lines, shapes and angles accurately is limited to using only a compass, a straightedge and a pencil. This means that you may not measure angles with a protractor or measure lengths with a graduated ruler in geometric construction. Words & Terms Geometry: The study of shapes and their properties. Graduated ruler: A ruler that is marked off in centimetres or millimetres so that it can be used to measure distances. Words & Terms 79 Module 5: Use construction procedures for geometric applications Did you know? Geometric constructions are also called Euclidean constructions, named after the ancient Greek mathematician Euclid who is considered to be the father of geometry. Figure 5.2: Euclid, the father of geometry • Creating the line through two existing points • Creating the circle through one point with the centre of another point • Creating the point that is the intersection of two existing, non-parallel lines • Creating the one or two points in the intersection of a line and a circle (if they intersect) • Creating the one or two points in the intersection of two circles (if they intersect). Unit 5.1: Explain basic concepts related to simple line applications in geometrical constructions As stated in the definition of geometric construction, the only tools required are a straightedge, a compass and a pencil (Figure 5.3). An eraser may be added to the list of required instruments. Because these instruments were discussed in detail in Module 1, they will not be dealt with again here. However, make sure that you know what they are, and what they are used for.

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  Exploring Geometry

  • Michael Hvidsten(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  C H A P T E R 4 Constructions Geometry is the science of correct reasoning on incorrect figures. – George Pólya (1887–1985) (from [35, page 208]) 4.1 EUCLIDEAN CONSTRUCTIONS The quote by Pólya is somewhat tongue-in-cheek, but contains an im-portant nugget of wisdom that is at the heart of how the Greeks viewed geometric constructions. For Euclid a geometric figure drawn on paper was only an approximate representation of the abstract, exact geometric relationship described by the figure and established through the use of axioms, definitions, and theorems. When we think of drawing a geometric figure, we typically imag-ine using some kind of straightedge (perhaps a ruler) to draw segments and a compass to draw circles. Euclid, in his first three axioms for pla-nar Euclidean geometry, stipulates that there are exact , ideal versions of these two tools that can be used to construct perfect segments and circles. Euclid is making an abstraction of the concrete process of draw-ing geometric figures so that he can provide logically rigorous proofs of geometric results. But Euclid was paradoxically quite concrete in his notions of what constituted a proof of a geometric result. It was not enough to show that a figure or result could be constructed—the actual construction had to be demonstrated, using an ideal straightedge and compass or other constructions that had already been proved valid. In this section we will follow Euclid and assume the ability to con-struct a segment connecting two given points and the ability to construct a circle with a given point as the center and a constructed segment as 151 152 � Exploring Geometry the radius. From these two basic constructions, we will develop some of the more useful constructions that appear in Euclidean geometry. Many of these constructions are well known from high school geom-etry and proofs of validity will be developed in the exercises.

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  Teaching and Learning Geometry

  • Doug French(Author)
  • 2004(Publication Date)
  • Continuum

    (Publisher)

  Moreover the diagram shows readily that the diagonals are equal in length and that their point of intersection at (3, 4) divides each in the ratio 2:1, the same as the ratio of AB to CD. This agrees, of course, with the result that would be obtained using the formulae discussed above for finding a point that divides a line segment in a given ratio. Figure 10.6 is an accurately plotted diagram, but in practice problems like this can be solved by referring to a diagram that has been sketched by hand so that it looks approximately right. This section has focused on the link between the coordinates of points on a straight line and its equation together with some basic procedures related to position, distance and gradient. Understanding of these important ideas is reinforced when they are illustrated with simple diagrams using small integers for the coordinates of the points involved. Familiar ideas like Pythagoras' theorem, gradient and the properties of similar triangles can then be readily applied. This understanding is hindered if there is too early an emphasis on remembering and attempting to apply formulae whose derivation and purpose is often not fully appreciated. LOCUS One of the powerful features of coordinate geometry is its use in determining algebraically the locus of a set of points subject to some conditions. The conditions lead to an equation involving a general point (x,y) which is the equation of the curve that forms the locus. Many equations of standard curves can be obtained in this way as well as solutions to a variety of locus problems. Two simple examples are illustrated by Figure 10.7. In the first case the problem is to find the locus of a set of points that are equidistant from two given points. This is the perpendicular bisector of the line segment joining the points.

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  The Mathematical Career of Pierre de Fermat, 1601-1665

  Second Edition

  • Michael Sean Mahoney(Author)
  • 2018(Publication Date)
  • Princeton University Press

    (Publisher)

  It is the latter property of algebraic analysis that Descartes takes pains to render explicit in the opening paragraphs of the Geometry, as Viete had done earlier in his Canonical Recension of Geometric Construc-tions. 19 For the algebraic analysis oflocus problems to succeed, Fermat must provide a technique which relates the algebra to the construction of the curve and the framework within which that construction takes place. With the fol-lowing statement, he fulfills that requirement and states the specific theorem that the rest of the Introduction is designed to demonstrate: The equations can easily be set up, if we arrange the two unknown quanti-ties at a given angle-which we will usually take as a right angle-and if one endpoint of one of these [quantities] given in position is given. Provided that neither of the unknown quantities exceeds the square, the locus will be plane or solid, as will be made clear from what is said. (p. 92) Again, much prior mathematical development lies implicit in Fermat's state-ment. 20 Where Descartes justifies in detail the use of line lengths as algebraic variables, that is, by showing that geometric magnitudes constitute an alge-braic field, Fermat assumes it as part of the Vietan algebra he employs.

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