how do we calculate the area of regular polygons ?

What are regular polygons?
   
Regular polygons are polygons with all sides and all angles equal


Some examples of these are
-Pentagons
-hexagons
-heptagons 


The formula to this :


Area = (½)nas


n = number of sides
a = apothems = length of a side




  
This table shows that the formula is 1/2 times the N(number) of sides, A(apothem) and side length (R)


Example:
   Pentagons - 5 (1/2 x a s)

How do we identify solids ?

First thing we need to know is 


What is a solid ?


A solid is a 3D or three dimensional shape that has a height, width and depth

* There are 2 main types of solids 

     

     - Polyhedra

     - Non - Polyhedra 

Polyhedra 's must have flat faces 

    Examples: Prisms, and Pyramids

Non- Polyhedra's are if any surface isnt flat 

    Examples:

What is isometry ?

An isometry is a transformation where the original figure and its image are congruent.

Types of isometry                    Not a type

        -  Reflection                             -  Dilation 
        -  Rotation   
       -  Translation  

• The reason why dilation isn't a isometry is because the original image is either shrunk or enlarged.

Examples:

    

Reflection

Rotation 

Translation



Practice Question :


What is the type of isometry in the given figure ?

A) Rotation

B) Reflection

C) translation

D) Dilation

Answer : C Translation

How do we solve more complex area problems?

As for example:

 

  Example: 

                    

First you look at your trapezoid & write down the measurements for its bases & its height in order to find your area

Formula: A = 1/2 h (b1+b2)

lets say these were the measurements..

Base1: 3ft

Base2: 8ft

Height:5ft

Area: 1/2h(b1+b2)

Area: 1/2 x 5 (3+8)

Area: 1/2 x 5 (11)

Area: 1/2 x 55

Area: 27.5 ft squared

How do we find the area of a regular polygon?

First we look at our polygon

Example: 
                
Next you look at your formula & start finding the area of the polygon

Formula: A= nas/2


*Make sure you use your formula correctly 

how do we find the area of parallelograms,kites,and rapezoids

Area of a parallelogram 

•  Base x Height 

Area of a trapezoid

• A = 1/2 h (b1+b2)

Area of kites 

• A = d1 d2 / 2

How do we use the other definitions of transformations?

Glide Reflection: Combinations of a reflection in a line and a translation along that line.

Orientation: Refers to the arrangement of points, relative to one another after transformation.


Isometry: A image that the length an size stays the same

Direct Isometry: the orientation letters & length stay the same


Oppisite Isometry: Letters are backwards but same length a reflection.

How do we calculate the area of rectangles & triangles?

Area: The measure of units in total, inside of a figure.

Rectangles: Base * Height 

 8cm * 4cm
 Area = 32cm

Triangles: Area=  1/2 * b * h 

How do we find the locus of points?

The locus is the set of all points that satisfy a given condition.

A locus is a general graph of a given equation.

What is the locus of points equidistant from a single point?
* A circle with the original point in the center

What is the locus of points equidistant from 2 fixed points?
* A line through the middle of the two points.

What is the locus of points from a single line?
* Two parallel lines on opposite sides of original lines

What is the locus of points equidistant from two parallel lines?
* A line through the middle of the two lines

What is he locus of points from two intersecting lines?
* Two intersecting lines halfway between the two original lines

How do we solve logic problems using conditionals?

Example: Two angles are congruent (hyp)
Two angles are both right angles (conclusion)

Conditional: If two angles are congruent then two angles are both right angles

Inverse: if two angles are not congruent then two angles are both right angles

Converse is just switching the hypothesis and conclusion.

Today is Friday - Hyp.
Tomorrow is Saturday - conclusion

Conditional: If today is Friday then tomorrow is sat.
Converse: if tomorrow is sat. than today is Friday.

When the conditional and the converse are both true - Bi conditional.

What is Logic?

Logic is thinking
Logic is the tool to determine between true and false

Inductive Reasoning:
- When your not sure what comes next.

you use logic everyday to:
- To solve geometry problems

What is a mathematical statement?

A Mathematical Statement is a statement that can be judged true or false.

Example: The President of the united states is Barack Obama and the vice president is Joe Biden.

*Both statements must be true for the statement to be true.

4 Conditions can be formed:

• The conditional: "if it is raining then it is cloudy"
• The inverse: "if its raining then its not cloudy"
• The converse: "if it is cloudy then its raining"
• Contrapositive: "if it is not cloudy then it is not raining"

A. The inverse of "if it is cold, then i will take my coat" > "if it is not cold, then i will not take my coat"

B. Let one statement be: Twice Talias age is 10 - Hyp

Let second statement be: Talia is 5 years old - Conclusion

Inverse: If twice talia's age is 10 then talia is 5 years old

How do we solve problems using rotations?

Step 1: Write down the originals shapes coordinates in (x,y) form
 
    Ex: For a triangle you jot down all 3 of its points a pentagon all 5 and so on...

Step 2: Note the rotated shapes coordinates (in (x,y) form as well) 

* Remember it will look similar to the original points of the shape (by turning positive numbers to negative)

Step 3: Compare both sets of coordinates 

* If the coordinates were changed from (x,y) to (-y,x) then that means the shape was rotated by 90 degrees.


* If changed from (x,y) to (-x,-y) then that means the shape was rotated by 180 degrees

* A shape that begins with (x,y) and ends up with (y,-x) then the shape was rotated 270 degrees

* lastly if the shape has been shifted 360 degrees it will give you the same coordinates you had started with

How do we graph dialations?

Step 1: You multiply both coordinates by the scale factor
    
Ex:Point coordinates (2,6)
     Scale Factor: (3)
             
 - 2 * 3 = 6  ; 6 * 3 = 18 


Step 2: Simplify 
      
- (6,18)


Step 3: Graph points if told to do so