Distance and Midpoints

3 Key excerpts on "Distance and Midpoints"

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  eBook - ePub

  Barron's Math 360: A Complete Study Guide to Geometry with Online Practice

  • Lawrence S. Leff, Elizabeth Waite, Barron's Educational Series(Authors)
  • 2021(Publication Date)
  • Barrons Educational Services

    (Publisher)

  Theorem 16.1 .

  THEOREM 16.1 THE MIDPOINT FORMULA

  The coordinates of the midpoint M(x

  m

  , y

  m

  ) of a segment whose endpoints are A(x1 , y1 ) and B(x2 , y2 ) may be found using the formulas:

  and

  EXAMPLE 16.4

  Find the coordinates of the midpoint of a segment whose endpoints are H(4, 9) and K(–10, 1).

  SOLUTION

  EXAMPLE 16.5

  M(7, –1) is the midpoint of WL. If the coordinates of endpoint W are (5, 4), find the coordinates of endpoint L.

  SOLUTION

  EXAMPLE 16.6

  The coordinates of quadrilateral ABCD are A(–3, 0), B(4, 7), C(9, 2), and D(2, –5). Prove that ABCD is a parallelogram.

  SOLUTION

  If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. In the diagram, diagonals and intersect at point E. If the coordinates of point E are the coordinates of the midpoints of and , then the diagonals bisect each other, and the quadrilateral is a parallelogram.

  To find the midpoint of , let (x1 , y1 ) = (–3, 0) and (x2 , y2 ) = (9, 2):

  Hence, the coordinates of the midpoint of are (3, 1).

  To find the midpoint of , let (x1 , y1 ) = (4, 7) and (x2 , y2 ) = (2, –5):

  The coordinates of the midpoint of are (3, 1).

  Since the diagonals have the same midpoint, they bisect each other, thereby establishing that the quadrilateral is a parallelogram.

  FINDING THE DISTANCE BETWEEN TWO POINTS

  The distance between two points in the coordinate plane is the length of the segment that joins the two points. Figures 16.3 and 16.4 illustrate how to find the distance between two points that determine a horizontal or a vertical line. In each case, the distance between the two given points is found by subtracting a single pair of coordinates. The length of a horizontal segment (Figure 16.3 ) is obtained by calculating the difference in the x-coordinates of the two points. The length of a vertical segment (Figure 16.4 ) is found by subtracting the y

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  eBook - PDF

  College Algebra

  • Cynthia Y. Young(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  You will calculate the distance between two points and find the midpoint of a line segment joining two points. You will then apply point-plotting techniques to sketch graphs of equations. Special attention is given to two types of equations: lines and circles. 2.1 Basic Tools: Cartesian Plane, Distance, and Midpoint SKILLS OBJECTIVES • Plot points on the Cartesian plane. • Calculate the distance between two points in the Cartesian plane. • Find the midpoint of a line segment joining two points in the Cartesian plane. CONCEPTUAL OBJECTIVES • Expand the concept of a one-dimensional number line to a two-dimensional plane. • Derive the distance formula using the Pythagorean theorem. • Conceptualize the midpoint as the average of the x- and y-coordinates. 2.1.1 Cartesian Plane 2.1.1 Skill Plot points on the Cartesian plane. 2.1.1 Conceptual Expand the concept of a one-dimensional number line to a two-dimensional plane. HIV infection rates, stock prices, and temperature conversions are all examples of relationships between two quantities that can be expressed in a two-dimensional graph. Because it is two dimensional, such a graph lies in a plane. Two perpendicular real number lines, known as the axes in the plane, intersect at a point we call the origin. Typically, the horizontal axis is called the x-- axis, and the vertical axis is denoted as the y-- axis. The axes divide the plane into four quadrants, numbered by Roman numerals and ordered counterclockwise. LEARNING OBJECTIVES • Calculate the distance between two points and the midpoint of a line segment joining two points. • Sketch the graph of an equation using intercepts and symmetry as graphing aids. • Find the equation of a line. • Graph circles. • Find the line of best fit for a given set of data.* *Optional Technology Required Section 174 CHAPTER 2 Graphs Points in the plane are represented by ordered-pairs, denoted (x, y).

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  Dictionary of Distances

  • Michel-Marie Deza, Elena Deza(Authors)
  • 2006(Publication Date)
  • Elsevier Science

    (Publisher)

  Part I Mathematics of Distances Chapter 1. General Definitions Chapter 2. Topological Spaces Chapter 3. Generalizations of Metric Spaces Chapter 4. Metric Transforms Chapter 5. Metrics on Normed Structures Chapter 1 General Definitions 1.1. BASIC DEFINITIONS • Distance Let X be a set. A function d : X × X → R is called distance (or dissimilarity ) on X if, for all x, y ∈ X , it holds: 1. d(x, y) 0 ( non-negativity ); 2. d(x, y) = d(y, x) ( symmetry ); 3. d(x, x) = 0. In Topology, it is also called symmetric . The vector from x to y having the length d(x, y) is called displacement . A distance which is a squared metric, is called quad-rance . For any distance d , the function, defined for x = y by D(x, y) = d(x, y) + c , where c = max x,y,z ∈ X (d(x, y) − d(x, z) − d(y, z)) , and D(x, x) = 0, is a metric. • Distance space A distance space (X, d) is a set X equipped with a distance d . • Similarity Let X be a set. A function s : X × X → R is called similarity (or proximity ) on X if s is non-negative , symmetric , and if s(x, y) s(x, x) holds for all x, y ∈ X , with equality if and only if x = y . Main transforms used to obtain a distance (dissimilarity) d from a similarity s are: d = 1 − s , d = 1 − s s , d = √ 1 − s , d = 2 ( 1 − s 2 ) , d = − ln s , d = arccos s . • Semi-metric Let X be a set. A function d : X × X → R is called semi-metric (or écart , pseudo-metric ) on X if d is non-negative , symmetric , if d(x, x) = 0 holds for all x ∈ X , and if d(x, y) d(x, z) + d(z, y) holds for all x, y, z ∈ X ( triangle inequality ). For any distance d , the equality d(x, x) = 0 and the strong triangle inequality d(x, y) d(x, z) + d(y, z) imply that d is a semi-metric. 2 Chapter 1: General Definitions [ • Metric] 3 • Metric Let X be a set. A function d : X × X → R is called metric on X if, for all x, y, z ∈ X , it holds: 1. d(x, y) 0 ( non-negativity ); 2. d(x, y) = 0 if and only if x = y ( separation or self-identity axiom ); 3.

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