Algebra Period 3

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

Polygons 4-5

A polygon is a closed figure formed by joining segments—the sides of the polygons at that endpoints—the vertices of the polygon. Polygons are names according to the number of sides they have.

Triangle 3 sides
Quadrilateral 4 sides
Pentagon 5 sides
Hexagon 6 sides
Octagon 8 sides
Decagon 10 sides

A polygon is REGULAR if all its sides and all its angles are congruent.
A regular triangle is the equilateral triangle
A regular quadrilateral is the square.

To name a polygon we name its consecutive vertices in order.
A diagonal of a polygon is a segment joining two non consecutive vertices.

To find the perimeter of a polygon add all the lengths of its sides. The perimeter is the distance around the figure. Finding the perimeter of a parallelogram can be done by computing the sum of the lengths or by using the distributive property to obtain
For instance a parallelogram with sides 9 cm and 6 cm
has a perimeter of 9 + 6 + 9 + 6 = 30 cm
but you could calculate that by 2(9) + 2(6) = 18 + 12 = 30 cm or using the distributive property, even 2(9+6) = 2(15) = 30 cm

If you have a regular polygon you can simple multiple the side by the number of sides in the polygon
For example,
a quadrilateral with side 16.5 m has a perimeter of 4(16.5) = 66 m

The sum of the measures of the angles of any pentagon is 540 degrees. If it is a regular pentagon, what must be the measure of each angle of the regular pentagon? 540/5 = 108 degrees.

The sum of the measures of the angles of any pentagon is 540 degrees. How can you prove that? Draw your pentagon and then draw all the diagonals from ONE of the vertices. Count the number of triangles created. Three. How many degrees does a triangle have? 180. Multiply the number of triangles created by 180… 540 is your answer. IT works every time. So How could you create a general rule or formula for the sum of the measures of the angles of any polygon with n sides?

Practice drawing various polygons—now practice drawing all the diagonals for each of them. Can you determine a general rule for the number of diagonals that can be drawn for any polygon?


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 segment 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 and this length is called the radius of the circle.

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.

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 3.14159 is used. In our book 3.14 is concerned a fairly good approximation.

C ÷ d = π

Formulas
Let C = circumference d = diameter, and r = radius

Then
C = πd d = C ÷ π
C = 2π r 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.

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

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 from the textbook.

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 m angle A = m angle B
We say that angle A and angle B are congruent angles


If two lines intersect so that the angles they form are all congruent, the lines are perpendicular. We use the symbol that looks like an upside down capital T to mean “is perpendicular to.”



Angles formed by perpendicular lines each have measure of 90° . A 90° 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°. An obtuse angle has measure between 90° and 180°



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

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

Triangle ABC ΔABC
Each of the Points A, B, C is called a vertex
(plural: Vertices) of ΔABC

Each of the angles angle A. angle B. and angle C is called an angle of ΔABC

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

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 sides 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

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!!!