Math 6 H Periods 1, 6 & 7 (Tuesday)

Geometry PreView: Continued

Polygons: 4-5

Polygon - closed figure with at least 3 sides- no curves - no overlapping lines
Named by the number of sides (called laterals) or their number of angles
Polygon literally means MANY (poly) ANGLES (gon)

Quadrilaterals - literally means 4 laterals (sides) and angles - 360 degrees (2 triangles!)
Trapezoids - Only 1 set of parallel lines
Kite - no parallel lines - 2 adjacent sides are congruent
Parallelogram - Opposite sides parallel and congruent
Types of parallelograms:
Rhombus - all sides congruent
Rectangle - 4 right angles
Square - Rectangle with all sides congruent (so it's also a rhombus)


Regular polygons - all sides and angles congruent
2 famous ones - equilateral triangle and squares

To name a polygon-- we name its consecutive vertices IN ORDER.

A diagonal of a polygon is a segment joining two NONCONSECUTIVE vertices.

The PERIMETER of a figure is the distance around it. Thus, the perimeter of a polygon is the sum of the lengths of its sides. So to find the PERIMETER of a regular figure, you just need to know one side and multiply by the total number of sides.

Circles: 4-6
A circle is the set of all points in a plane at a given distance from a given point O called the center
A segement joining the center to a point on the circle is called a radius (plural: radii) of the circle. All radii of a given circle have the same length
A segment joining two points on a circle is called a chord, and a chord passing through the center is a diameter of the circle. The ends of a diameter divide the circle into two semicircles. The length of a diameter is called the diameter of the circle.
The perimeter of a circle is called the circumference. “Sir Cumference” – it is the ‘fence’ around it!!

The quotient Circumference ÷ diameter can be showed to be the same for all circles-- Regardless of their size. This quotient is denoted by a Greek letter π
No decimal gives π exactly!! It is non-terminating (it never ends) and non repeating!! I like to use 3.14159. In our book 3.14 is concerned a fairly good approximation.
C ÷ d = π

Formulas you need to know:
Let C = circumference d = diameter, and r = radius

Then
C = πd d = C ÷ π

C = 2πr r = C ÷ (2π)


A polygon is inscribed in a circle if all of its vertices are on the circle.
It can be shown that three points not on a line determine a circle.
There is one and only one circle that passes through the three given points.

Algebra Period 3 (Tuesday)

SOLVING SYSTEMS OF EQUATIONS: 8-1 TO 8-3
(2 equations with 2 variables)
You cannot solve an equation with 2 variables - you can find multiple coordinates that work
TO SOLVE MEANS THE
ONE COORDINATE THAT WORKS FOR BOTH EQUATIONS

There are 3 ways to find that point:
1. Graph both equations: Where the 2 lines intersect is the solution
2. Substitution method: Solve one of the equations for either x or y and plug in to the other equation
3. Addition method: Eliminate one of the variables by multiplying the equations by that magical number that will make one of the variables the ADDITIVE INVERSE of the other

Example solved all 3 ways:
Find the solution to the following system:
2x + 3y = 8 and 5x + 2y = -2

1. GRAPH BOTH LINES: Put both in y = mx + b form and graph
Read the intersection point (you should get (-2, 4))

2. SUBSTITUTION:
Isolate whatever variable seems easiest
I will isolate y in the second equation: 2y = -5x - 2
y = -5/2 x - 1
Plug this -5/2 x - 1 where y is in the other equation
2x + 3(-5/2 x - 1) = 8
2x - 15/2 x - 3 = 8
4/2 x - 15/2 x - 3 = 8
-11/2 x - 3 = 8
-11/2 x = 11
-2/11(-11/2 x) = 11(-2/11)
x = -2
Plug into whichever equation is easiest to find y

3. ADDITION:
Multiply each equation so that one variable will "drop out" (additive inverse)
I will eliminate the x, but could eliminate the y if I wanted to
5(2x + 3y) = (8)5
-2(5x + 2y) = (-2)-2

10x + 15y = 40
-10x - 4y = 4

ADD TO ELIMINATE THE x term:
11y = 44
y = 4
Plug into whichever equation is easiest to find x

NOTICE THAT FOR ALL 3 METHODS, THE SOLUTION IS THE SAME!
THEREFORE, USE WHATEVER METHOD SEEMS EASIEST!!!

Math 6 H Periods 1, 6 & 7 (Monday)

Geometry PreView

(Some of the symbols and notations cannot be displayed here. You will need to find notes from another student or even check your textbook-- what a thought!!)

Points, Lines Planes: 4-1

All the figures that we study in geometry are made up of points. We usually picture a single point by making a dot and labeling it with a capital letter

The words point, line, and plane are undefined words in geometry- we can only describe them.

Among the most important geometric figures that we study are straight lines- or simply –lines.

You probably know this important fact about lines:

Two points determine exactly one line

This means that through two points P and Q we can draw one line and only one line,

Notice the use of arrowheads to show that a line extends without end in either direction

Three points my or may not line on the same line. Three or more points that do lie on the same line are called collinear. Points not on the same line are called noncollinear.

If we take a point P on a line and all the points on the line that lie on one side of P we have a ray with endpoint P. We name a ray by naming first its endpoint and then any other point on it.

It is important to remember that the endpoint is always named first.

Ray PQ is not the same ray as Ray QP ( check your textbook)

If we take two points P and Q on a line and all the points that lie between P and Q we have a segment denoted by the points P and Q are called the endpoints of

Just as two points determine a line, three noncollinear points in space determine a flat surface called a plane. We can name a plane by naming any three noncollinear points in it. Because a plane extends without limit in all directions of the surface, we can show only part of it



Lines in the same plane that do not intersect are called parallel lines. Two segments or rays are parallel if they are part of parallel lines.

is parallel to may be written as

Planes that do not intersect are called parallel planes


Two nonparallel lines that do not intersect are called skew lines

Angles and Angle Measure: 4-3
An angle is a figure formed by two rays with the same endpoints. The common endpoint is called the vertex. The rays are called the sides.
We may name an angle by giving its vertex letter if this is the only angle with that vertex, or my listing letters for points on the two sides with the vertex letter in the middle. We use the symbol that looks like a 'less than ' symbol for angle.


To measure segments we use a rule to mark off unit lengths. To measure angles, we use a protractor that is marked off in units of angle measure called degrees.

To use a protractor, place its center point at the vertex of the angle to be measured and one of its zero points on the side.


We often label angels with their measures. When angles have equal measures we can write
We say that Angle A and Angle B are congruent angles and we write Angle A equal sign with a ~ over it.


If two lines intersect so that the angles they form are all congruent, the lines are perpendicular. We use the symbol (Upside down T to mean “is perpendicular to.”



Angles formed by perpendicular lines each have measure of 90 degrees . A 90 degree angle is called a right angle. A small square is often used to indicate a right angle in a diagram

An acute angle is an angle with measure less than 90 degrees . An obtuse angle has measure between 90 degrees and 180 degrees



Two angles are complementary if the sum of the measures is 90 degrees
Two angles are supplementary if the sum of their measures is 180 degrees




Triangles: 4-4
A triangle is the figure formed when three points, not on a line are jointed by segments.


Triangle ABC
Having segments as its sides
Each of the Points A, B, C is called a vertex
(plural: Vertices)

Each of the angles Angle A ,Angle B , AngleC is called an angle of the triangle ABC
In any triangle- the sum of the lengths of any two sides is greater than the length of the third side

The sum of the measures of the angles is 180 degrees



There are several ways to name triangles. One way is by angles


Acute Triangle
3 acute angles


Right Triangle
1 right angle

Obtuse Triangle
1 obtuse angle

Triangles can be classified by their sides



Scalene Triangle
no 2 sides congruent


Isosceles Triangle
at least 2 side
congruent

Equilateral Triangle
all 3 sides
congruent

The longest side of a triangle is opposite the largest angle and the shortest side is opposite the smallest angle. Two angles are congruent if and only if the sides opposite them are congruent.

Algebra Period 3 (Monday)

Simplify, Multiply, and Divide RATIONAL EXPRESSIONS 10-1, 10-2, and 10-3

Rational Expressions = Expressions in fraction format (division) with a variable in the denominator

You have already been simplifying, multiplying and dividing these throughout this year!

SIMPLIFY: 10-1

You will need to FACTOR (Chapter 6) both the numerator and denominator and "cross out" common factors in both (their quotient is 1!)
EXAMPLE: Simplify
y² + 3y + 2 =
y² - 1


(y + 2)(y + 1) =
(y - 1)(y + 1)

y + 2
y - 1

MULTIPLY: 10-2

FACTOR if possible, cross cancel if possible, multiply numerators, then denominators, simplify
EXAMPLE:
(y + 4)³[y² + 4y + 4] =
[(y + 2) ³(y² + 8y + 16)]

(y + 4) ³][ (y + 2) ² =
(y + 2) ³(y + 4) ²

y + 4
y + 2



DIVIDE: 10-3


Same as the previous example, only this time you will need to

“FLIP the SECOND” fraction,

FACTOR then
MULTIPLY!!!!!
EXAMPLE:
x + 1 ÷ x + 1 =
x² - 1 x² - 2x + 1


( x + 1) ( x2 - 2x + 1) =
(x + 1)( x - 1) (x + 1)

( x + 1)( x - 1)(x - 1) =
(x + 1)( x - 1)(x + 1)


x - 1
x + 1







Add and subtract rational expressions with LIKE DENOMINATORS: 10-4
Add and subtract with UNLIKE DENOMINATORS: 10-5
When adding with LIKE DENOMINATORS,
simply add the numerators,
simplify

When subtracting with LIKE DENOMINATORS,
CHANGE THE SIGNS
OF EACH TERM IN THE NUMERATOR AFTER THE SUBTRACTION SIGN,
THEN ADD (double check!!!)

When adding or subtracting with UNLIKE DENOMINATORS,
find the common denominator (the least common multiple of all denominators),
then use equivalent fractions to restate each numerator using the new common denominator.
EXAMPLE:
2 + x
x² - 16 x – 4

First, factor the denominators if possible



2x + x
(x + 4)(x - 4) x – 4

Find the LCM = (x + 4)(x - 4);

2x + (x + 4)(x)
(x + 4)(x - 4) (x + 4)(x - 4)


Restate both fractions with the LCM

2x + x² + 4x
(x + 4)(x - 4)


Now add the numerators

x² + 6x =
(x + 4)(x - 4)

Simplify and re-factor numerator, if possible
x(x + 6)
(x + 4)(x - 4)