Segment Length

3 Key excerpts on "Segment Length"

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  eBook - PDF

  Elementary Geometry for College Students

  • Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  DEFINITION The distance between two points A and B is the length of the line segment AB that joins the two points. In construction, a string joins two stakes. The line determined is described in Postulate 1 on the previous page. Geometry in the Real World Recall from Section P.3 that the symbol for line segment AB, named by its endpoints, is AB . Omission of the bar from AB , as in AB, means that we are considering the length of the segment. These symbols are summarized in Table 1.1. 1.1 ■ Early Definitions and Postulates 35 EXAMPLE 3 In Figure 1.5, a) find AB if AX 5 7.32 and XB 5 6.19 b) find AX if AB 5 12.7 and XB 5 5.9 c) find AB if AX 5 2x 1 3 and XB 5 3x 2 7 SOLUTION a) AB 5 AX 1 XB 5 7.32 1 6.19, so AB 5 13.51. b) AB 5 AX 1 XB, so 12.7 5 AX 1 5.9 and AX 5 6.8 c) AB 5 AX 1 XB 5 (2x 1 3) 1 (3x 2 7), so AB 5 5x 2 4. As we saw in Section P.3, there is a relationship between the lengths of the line segments determined in Figure 1.5. This relationship is stated in the third postulate. The title and meaning of the postulate are equally important! The title “Segment-Addition Postulate” will be cited frequently in later sections. POSTULATE 3 ■ Segment-Addition Postulate If X is a point on AB and A-X-B, then AX 1 XB 5 AB. The laser distance measure and digital tape measure calculate distances and midpoints quite accurately. In construc- tion and manufacturing technology, these devices replace the ruler and tape measure. Geometry in the Real World Technology Exploration Use software if available. 1. Draw line segment XY . 2. Choose point P on XY . 3. Measure XP , PY , and XY . 4. Show that XP 1 PY 5 XY. Congruent ( _ ) line segments are two line segments that have the same length. DEFINITION The midpoint of a line segment is the point that separates the line segment into two congruent parts. DEFINITION In general, geometric figures that can be made to coincide (fit perfectly one on top of the other) are said to be congruent.

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  eBook - ePub

  Learning and Teaching Early Math

  The Learning Trajectories Approach

  • Douglas H. Clements, Julie Sarama(Authors)
  • 2020(Publication Date)
  • Routledge

    (Publisher)

  Length is a characteristic of an object found by quantifying how far it is between the endpoints of the object. “Distance” is often used similarly to quantify how far it is between any two points in space. The discussion of the number line (see Chapter 4) is critical here because this defines the number line used to measure length. Measuring length or distance consists of two aspects, then: identifying a unit of measure and subdividing (mentally and physically) the object by that unit, placing that unit end to end (iterating) alongside the object. Subdividing and unit iteration are complex mental accomplishments that are too often ignored in traditional measurement curriculum materials and instruction. Therefore, many researchers go beyond the physical act of measuring to investigate children’s understandings of measuring as covering space (one-dimensional for length; see Chapter 11 for two and three dimensions) and quantifying that covering. Let’s lay out the concepts that children must learn (adapted from Clements & Stephan, 2004; Stephan & Clements, 2003). Understanding of the attribute of length includes understanding that lengths span fixed distances. Conservation of length includes understanding that as a rigid object is moved, its length does not change. Transitivity is the understanding that, if the length of a white strip is greater than the length of a gray strip, and the length of the gray strip is greater than that of the black strip, then the white strip is longer than the black strip (even if you could not compare those two— Figure 10.1). A child with this understanding can use a third object to compare the lengths of two other objects. Figure  10.1 An illustration of the transitivity of length Equal partitioning is the mental activity of slicing up an object into the same-sized units. This idea is not obvious to children

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  eBook - ePub

  Figures of Thought

  Mathematics and Mathematical Texts

  • David Reed(Author)
  • 2003(Publication Date)
  • Routledge

    (Publisher)

  It should be superfluous to add that Euclid is not here attempting to describe (well or poorly) the visual appearance of a point, nor his, our or the ideal mathematician's intuition or perception of what a point is. Any reading of Euclid along these lines (Heath's notes contain a number of examples) omits all of the aspects of this definition which differentiate it from other similar definitions used by mathematicians ancient and modern. While it may have been traditional then as now to begin geometry texts with the definition of a point, whatever the tradition demanded Euclid has provided a definition which provides its own justification for commencing his argument. It is sufficient to note that he begins his argument with that which indicates the beginning of discourse; to refer to extraneous reasons for the formulation or positioning of this definition is unnecessary.

  The orientation provided by Definition 1 clarifies the sequence which follows. Unlike a point, a line does have parts, one part in particular which is labelled length.6 By referring to something with one part Euclid shows clearly that he does not have in mind a sense of ‘part’ which has to do with division or with ‘sub-objects’. Parts are the ways in which things may be known or described. As far as their definition is concerned, lines are things about which one can know only one thing, their length. For convenience this type of definition will be referred to as definition by specification of a measureable (in this case length). It is inappropriate to think of length in this context in geometric, metric or measure theoretic terms. All of these approaches require previous specification of some type of measure or some kind of line. Definition 2 merely states that lines (a) are distinguished from points by having parts, (b) are distinguished from other geometric things by having only one part and (c) can be compared amongst themselves by this part, their length. To repeat, anything can be considered to be a line, provided it is considered as having only one characteristic or term of comparison. Clearly there is no distinguishing (at this stage in the argument) between ‘two’ lines of the ‘same’ length. The grammar of singular and plural is limited precisely to distinctions of length.

  It is also inappropriate to think of this definition as embodying some (primitive) type of ‘dimensional’ analysis. As can be seen from Table 1.1 , each item in the sequence ‘point’, ‘line’, ‘surface’, ‘solid’ is defined in a (slightly) different manner. The distinguishing characteristic of dimensional definitions is that the things to be defined are defined in the same way except for the number of dimensions involved. Euclid's sequence by contrast has a form or shape and each term occupies the place it does for a particular reason. It therefore cannot have a ‘dimensional’ character, even a primitive one. The Elements

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