Home Previous Next

MAT081 :: Lecture Note :: Week 14
Assessments | Handouts | Math Resources {MathBabbler: Email | Facebook | Twitter}
GDT::Bits:: Time  |  Weather  |  Populations  |  Special Dates

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°`.

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

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
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.

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

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;

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

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

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.]

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

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

```   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
```

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

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

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

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]

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

Signed Numbers

Numbers that are prefixed with a dash are negative numbers and all negative numbers are less than zero. [Note: zero is neither negative nor positive.]

On the number line, negative numbers are to the left of zero.

Negative numbers are the opposite of positive numbers. For example, `-5` is the opposite of `+5` and vice versa.

The absolute value of a number is its distance from zero on the number line. For example, `-5` is five ones to the left of zero; therefore, its absolute value is `5`.

If both terms have the same sign, then add their respective absolute values and use the common sign as the sign for the sum.

```   -5 + -3
|-5| + |-3|  =  5 + 3  =  8
common sign is -
```

If the terms have differing signs, subtract the smallest absolute value from the largest absolute value. The sum is given the sign of the largest absolute value.

```   -5 + 15
|15| > |-5|
15 - 5 = 10
sign on 15 is +

-5 + 3
|-5| > |3|
5 - 3 = 2
sign on -5 is -
```
Subtracting Signed Numbers

Subtracting signed numbers is performed by converting the problem into an addition problem.

If `a` and `b` are numbers, then `a - b` equals `a + (-b)`. In other words, subtraction equals the first number added to the opposite of the second number.

```   general formula:  a - b = a + (-b)

5 -  3 =  5 + (-3) =  2
5 - -3 =  5 + ( 3) =  8
-5 -  3 = -5 + (-3) = -8
-5 - -3 = -5 + ( 3) = -2
```
Multiplying Signed Numbers

The product of two factors having the same sign is positive. The product of two factors having different signs is negative. In other words, a negative number times a positive number results in a negative number and vice versa. Any number (regardless of sign) times zero is zero.

```   "Minus times minus equals plus; the reason for this we won't discuss."
--anonymous
```
Dividing Signed Numbers

The quotient of two terms having the same sign is positive. The quotient of two terms having different signs is negative. In other words, a negative number divided by a positive number results in a negative number and vice versa. Any number (regardless of sign) cannot be divided by zero.

Exercise
1. Graph the numbers `5, -2, -4.7`.

```   -5   -4   -3   -2  -1     0    1    2    3    4    5
+----+----+----+----+----+----+----+----+----+----+
^            ^                                  ^
-4.7          -2                                  5

-4.7 is less than -2 which is less than 5
-4.7 < -2 < 5
```

PurpleMath.com:: Introduction to Negative Numbers
UGA.edu::Why is a negative number times a negative number a positive number?