Home  Previous  Next 

MAT081 :: Lecture Note :: Week 14
Assessments

Handouts

Math Resources
{MathBabbler:
Email

Facebook

Twitter}
GDT::Bits::
Time

Weather

Populations

Special Dates
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 (0191602009) {"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 endpoints ray: line with one endpoint angle: two lines that intersect form an angle vertex: point where two lines intersectAccording 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
andB
be two points on the same line.<> <AB> line: AB __ AB segment: AB > AB> ray: AB < <AB ray: ABLines 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   CiD  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°
.
{TopOfPage}
{MathWords}
{PurpleMath}
{KhanAcademy}
{SCC math resources... MathAS::id 5778; key 5778 
OER::book 
playbook 
{classroom: quiet 
thinking 
speed eraser 
wave erase}
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 parallelA triangle is a threesided 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 10Two 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.
{TopOfPage}
{MathWords}
{PurpleMath}
{KhanAcademy}
{SCC math resources... MathAS::id 5778; key 5778 
OER::book 
playbook 
{classroom: quiet 
thinking 
speed eraser 
wave erase}
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 centerThe diameter of a circle is equal to two times its radius.
d = 2 * r where d is the diameter and r is the radiusThe 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 diameterThe value of
pi
approximates the fraction22/7
.The area of a circle is the number of "square units" needed to cover the circle.
A = pi * r^{2} 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
{TopOfPage}
{MathWords}
{PurpleMath}
{KhanAcademy}
{SCC math resources... MathAS::id 5778; key 5778 
OER::book 
playbook 
{classroom: quiet 
thinking 
speed eraser 
wave erase}
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 cAn area is a measure of the surface of a region.
rectangle A = length * width square A = side^{2} triangle A = 0.5 * base * heightSquare 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.]
{TopOfPage}
{MathWords}
{PurpleMath}
{KhanAcademy}
{SCC math resources... MathAS::id 5778; key 5778 
OER::book 
playbook 
{classroom: quiet 
thinking 
speed eraser 
wave erase}
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 legsIf 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.
a^{2} + b^{2} = c^{2} where 'a' and 'b' are the legs and 'c' is the hypotenuseGDT::BAB:: Find X
The perimeter of a triangle is the sum of the length of its sides.
The area of a triangle is onehalf the product of its base length and height.
1 A =  * b * h 2 + / /  /  h /  /  ++ bGDT::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
{TopOfPage}
{MathWords}
{PurpleMath}
{KhanAcademy}
{SCC math resources... MathAS::id 5778; key 5778 
OER::book 
playbook 
{classroom: quiet 
thinking 
speed eraser 
wave erase}
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
{TopOfPage}
{MathWords}
{PurpleMath}
{KhanAcademy}
{SCC math resources... MathAS::id 5778; key 5778 
OER::book 
playbook 
{classroom: quiet 
thinking 
speed eraser 
wave erase}
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 tshirt 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 lampposts... for support rather than illumination.
 Andrew Lang (0184401912) { Scottish poet, novelist, literary critic; more... } [statistics]External Hyperlink(s)
 KhanAcademy.org::Statistics intro: mean, median and mode [opens new window]
 YouTube.com:: Mean Median and Mode [opens new window]
 ComputerGear.com::Tshirt:: 53.7% of all statistics are made up [opens new window]
{TopOfPage}
{MathWords}
{PurpleMath}
{KhanAcademy}
{SCC math resources... MathAS::id 5778; key 5778 
OER::book 
playbook 
{classroom: quiet 
thinking 
speed eraser 
wave erase}
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 is5
.Adding Signed Numbers
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  answer is 8If 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 + answer is 10 (+10) 5 + 3 5 > 3 5  3 = 2 sign on 5 is  answer is 2Subtracting Signed Numbers
Subtracting signed numbers is performed by converting the problem into an addition problem.
If
a
andb
are numbers, thena  b
equalsa + (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) = 2Multiplying 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." anonymousDividing 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
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 < 5PurpleMath.com:: Introduction to Negative Numbers
UGA.edu::Why is a negative number times a negative number a positive number?
{TopOfPage}
{MathWords}
{PurpleMath}
{KhanAcademy}
{SCC math resources... MathAS::id 5778; key 5778 
OER::book 
playbook 
{classroom: quiet 
thinking 
speed eraser 
wave erase}
Home  Previous  Next 
