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MAT081 :: Lecture Note :: Week 13
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Overview
Assignment(s):

Introduction to Geometry

Geometry is a "branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids."

When Moses was alive, these pyramids were a thousand years old.
Here began the history of architecture. Here people learned to
measure time by a calendar, to plot the stars by astronomy and
chart the earth by geometry. And here they developed that most
awesome of all ideas - the idea of eternity.

-- Walter Cronkite (01916-02009) {"the most trusted man in America"; more...} [geometry]

[update::2012.04.16] CBS News marks historic Cronkite anniversary

Terminology

The following terms are used extensively in geometry.

          point:  location (no dimensions, no length, no width, no depth)
          space:  set of all points
          plane:  set of points on one surface
           line:  set of points (at least two) that has only 
                  one dimension, length
   line segment:  line having two end-points
            ray:  line with one end-point
          angle:  two lines that intersect form an angle
         vertex:  point where two lines intersect

According to Euclid, an plane angle is "the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line."

Points

Points are typically labeled.

Lines

A line segment is a line that has a starting and ending point. A ray has a starting point, but no ending point.

Let A and B be two points on the same line.

                                       <-->
   <-----A------------B----->    line:  AB

                                      __
   A---------------B        segment:  AB


                                   -->
   A---------------B----->   ray:  AB

                                      <--
   <-----A--------------B        ray:  AB

Lines are often used in pairs and there are four types of pairs: intersecting, parallel, perpendicular, and skewed. Intersecting lines cross each other; parallel lines never touch and are always equal distance apart; perpendicular lines form right angles; and skewed lines never intersect nor are they parallel (they are not in the same plane). [Railroad tracks provide an example of parallel lines.]

         A
         |
         |
      C--i-----D
         |
         B

      The line segments AB and CD intersect and are perpendicular.  
      The vertex (i.e. point) where they intersect is labeled 'i'.
Angles

An angle is formed when two lines intersect.

There are four types of angles: straight, right angle, acute, and obtuse. Angles are measured using degrees (a fraction of a circle [360°]).

   straight angle ... 180°
   right angle ......  90°
   acute angle ...... less than 90°
   obtuse angle ..... greater than 90° but less than 180°

Jason Kidd, professional basketball player, when traded to a new team, was quoted he was going to help turn the team around 360°.

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Geometry: Polygons

A polygon is a closed plane figure bounded by straight lines (i.e. sides). Common polygons are squares, rectangles, and triangles.

Every polygon contains at two or more angles.

A quadrilateral is a polygon having four sides.

A parallelogram is a quadrilateral where opposite sides are both parallel and equal length.

A rectangle is a parallelogram consisting of four right angles. A square is a rectangle where all four sides are equal length.

   rhombus -- parallelogram having 4 sides of equal length
   trapezoid -- exactly one pair of opposite sides parallel

A triangle is a three-sided polygon.

The following is list of some other polygons.

   type            #sides  #angles
   -------------------------------
   Triangle........   3       3       
   Quadrilateral...   4       4       
   Pentagon........   5       5
   Hexagon.........   6       6
   Heptagon........   7       7
   Octagon.........   8       8
   Nonagon........    9       9
   Decagon........    10      10

Two polygons are congruent if they are the same size and shape (i.e. corresponding angles and sides are equal). Two polygons are similar if they have the same shape, but different sizes.

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Geometry: Circle

A circle is a plane figure consisting of points that are equal distance from a center point.

     radius:  distance from circle center to a circle point
   diameter:  distance across a circle through the center

The diameter of a circle is equal to two times its radius.

   d = 2 * r

   where d is the diameter and r is the radius

The circumference of a circle is its perimeter (i.e. distance around the circle).

   C = 2 * pi * r

   where C is circumference; pi is the value 3.14;
   and r is radius

   C = pi * d

   where d is the diameter

The value of pi approximates the fraction 22/7.

The area of a circle is the number of "square units" needed to cover the circle.

A = pi * r2 where A is area; pi is the value 3.14; and r is radius ["Why do they say 'pie are square' when pies are round?"]
       *   *
    *         *
   *   r       *
   *-----+     *    + is the center of the circle
   *           *
    *         *
       *   * 

GDT::BAB:: Washington Tree May Be Dying

Demon.co.ukPi

http://gwydir.demon.co.uk/jo/numbers/interest/pi.htm

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Geometry: Perimeter and Area for Polygons

A perimeter is a measure of the distance around a region. [Hint: perimeter]

   rectangle    P = 2 * length + 2 * width
   square       P = 4 * side
   triangle     P = side a + side b + side c

An area is a measure of the surface of a region.

   rectangle    A = length * width
   square       A = side2
   triangle     A = 0.5 * base * height
Square Yard Example

[Saturday, 24 April 2004, Arizona Republic]
The U.S. Department of Agriculture reported that most of western Nebraska has at least 15 grasshoppers per square yard. Some areas had as many as 370 grasshoppers in a square yard. [Anything over 40 is considered severe.]

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Continue with Geometry: Plane Figures

Plane Figures

Recall, a plane is a flat surface that spans to infinity. A plane figure is a geometric form that lies on a plane.

Triangles

A triangle has three sides that form three angles. The sum of the three angles total 180°.

The following are special types of triangles.

   equilateral -- all sides have equal length and 
                  all angles are equal
   isoceles -- two sides are same length; angles 
               opposite the equals size are equal
   scalene -- no sides have the same length and no
              angles are equal
   right -- has a right angle; the side opposite the
            right angle is the hypotenuse and the other
            two sides are legs

If you know the length of two sides of a right triangle, then the length of the third side can be calculated using the Pythagorean Theorem.

   a2 + b2 = c2

   where 'a' and 'b' are the legs and 'c' is the hypotenuse

GDT::BAB:: Find X

The perimeter of a triangle is the sum of the length of its sides.

The area of a triangle is one-half the product of its base length and height.

       1
   A = - * b * h
       2

         +
        /|
       / |
      /  | h
     /   |
    /    |
   +-----+
      b

GDT::BAB:: Triangles In Space May Be Signals

Quadrilaterals
   parallelogram -- opposite sides are parallel and
                    have equal length
   rectangle -- parallelogram with 4 right angles
   square -- rectangle with equal length sides
   rhombus -- parallelogram having 4 sides of equal length
   trapezoid -- exactly one pair of opposite sides parallel

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Volume Formulas

Volume is measured in "cubic" units. The volume of a figure is the number of cubes required to completely fill it.

Volumes are used for cubes, rectangular prisms, irregular prisms, cylinders, pyramids, cones, spheres, and ellipsoids.

Math.com::Geometry:: Cylinders, cones and spheres

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Introduction to Statistics

Wikipedia.org offers the following definition for statistics.

   "Statistics is a type of data analysis which practice 
    includes the planning, summarizing, and interpreting 
    of observations of a system possibly possibly mathematical 
    model of future events based on a forecasting followed by 
    predicting or of the system being observed."

According to a t-shirt sold by Computer Gear, "53.7% of all statistics are made up." In other words, caution must be exercised when presented with statistics.

Definitions

A set is a collection of zero or more things. The things in a set are called elements. A set with no elements is the empty set.

In many math worlds, the things in a set are numbers. For example, a set might contain a collection of temperatures or a collection of test scores or a collection of counters and so on.

Set notation uses braces {} to enclose the elements of a set.

A set is typically populated using data obtained from either observation or experimentation.

The following are some of the basic operations performed on a set.

    range: difference between largest and smallest
           values in a set
     mean: the average of a set of numbers
   median: the number found in the middle of sorted set
     mode: most commonly occuring value in a set of numbers;
           a set can have zero or more modes

      Sample data set S:  { 5  10  10  7   3  }
   S sorted (ascending):  { 3  5   7   10  10 }

   range  = 10 - 3 = 7
   mean   = 5 + 10 + 10 + 7 + 3 = 35; 35/5 = 7
   median = 7
   mode   = 10 

   In this example, the range, mean, and median
   are all the same value; however, this is not
   always true.

He uses statistics as a drunken man uses lamp-posts... for support rather than illumination.
-- Andrew Lang (01844-01912) { Scottish poet, novelist, literary critic; more... } [statistics]

External Hyperlink(s)

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