How do we find the area of a circle?

For you to find the area of a circle you only need to know the appropriate formula. The formula to find the are of a circle is "pi times the radius of the circle to the second power".

In the circle world, you will also have to find the circumference. the circumference is how long the circle is, around itself. the formula to find the circumference of a circle is "pi times the diameter" of the circle.

Problem1: find the circumference of a circle with radius of 5.
Problem 2: what's the area of a circle with a diameter of 25pi?


google.com/images

3/19 - How Do We Find The Area of Regular Polygons?

A regular polygon is a polygon that is equiangular and equilateral.
To find the area of a regular polygon you'll need to know the formula which is A= 1/2 as, where the variable "a" stands for the length called the apothem. The apothem is the perpendicular segment from the center to the side.

PROBLEM: A 6 sided regular polygon (hexagon) is inscribed in a circle of radius 8 cm, find the length of one side of the hexagon

3/12 - How Do We Calculate The Area of Rectangles & Triangles?

• The area of a plane figure is the measure of the region enclosed by the figure you measure.

For you to find the area of a rectangle you just need to know the formula which is: A= B x H, where A stands for Area, B stands for Base and H stands for height.

For you to find the area of a triangle you just need to know the formula which is A= 1/2 B x H where A stands for Area, B for Base and H for Height.

PROBLEM #1: If the area of a rectangle is 34ft square, and the base is 5ft what is the height of the rectangle?

PROBLEM #2: What's the area of a triangle with a height of 25cm and a base of 15cm?

How Do We Review Logic?

Knowing all from what is logic to the biconditional, we are ready to perform some examples and review what have we learned so far.


Logic
Switch Open: No Current


Mathematical Sentence

• 17 - 5 = 12 ------------> T

• 17 +3 = 42 ------------> F

• 0.1 < 1 -----------------> T

Conditional
REMEMBER:

IF...THEN is only FALSEwhen
T implies F.
All other cases are TRUE.

Inverse

• "If it is not raining, then it is not cloudy"
• "If twice Talia's age is not 10, then Talia is not 5 years old"

REMEMBER: Under negation, TRUE becomes FALSE - or - FALSE becomes TRUE. Converse Conditional:  "If the space shuttle was launched, then a cloud of smoke was seen." Converse:      "If a cloud of smoke was seen, then the space shuttle was launched." HINT:  Try to associate the logical CONVERSE with Converse™ sneakers -- think of the two parts of the sentence "putting on their sneakers" and "running" to their new positions. Contrapositive HINT:  Remember that the contrapositive (a big long word) is really the combining together of the strategies of two other words:  converse and inverse. An important fact to remember about the contrapositive, is that it always has the SAME truth value as the original conditional statement. **If the original statement is TRUE, the contrapositive is TRUE.   If the original statement is FALSE, the contrapositive is FALSE. They are said to be logically equivalent. Biconditional REMEMBER: IF AND ONLY IF is TRUE   when both facts are T  or both facts are F.. Problem 1. Which of the following statements is the converse of"If the moon is full, then the vampires are prowling."? Choose one:  If the vampires are prowling, then the moon is full.  If the moon is not full, then the vampires are  prowling.  If the vampires are not prowling, then the moon is not full. **Images and Examples provided by regentsprep.org and my notes from Geometry class.

How do we Solve Logic Problems using Conditionals?

In this post you will learn about 2 more conditionals and one special conditional.


Remembering that a conditional is when you use "If" in front of the hypothesis of the statement and "then" in front of the conclusion, we can continue our investigation through conditionals.


A Converse is when you switch the hypothesis and the conclusion.
             Ex: If I ate 2 tacos for dinner, then I'm full. The converse to this conditional is "If I'm full, then i ate 2 tacos for dinner"

A Contrapositive must be the hardest one because that's the one the Regents use a lot on their exams. A contrapositive is when you negate and switch the hypothesis and conclusion. So you use both inverse and converse. Contrapositives is very easy, its just that sometimes it might get difficult, depending on the statements which is given.

Ex: If i study, then I'll pass geometry. The contrapositive to this conditional is "If I do not pass Geometry, then I didn't study"

Remember:The contrapositive is the mixing of the inverse and the converse. 

Problem: What is logically equivalent to the statement "If it is not raining, then I'm happy"

**Images and Examples provided by regentsprep.org and google.com/images and also my notes from Geometry class.

What is a Mathematical Statement and What is Logic?

Logic as we know it, means knowledge/thinking. People use logic for everything. You need logic to know how to tie your shoes, to go to school/work, even to cook! We use logic everyday of our lives and that's what makes us different from animals: we have logic.


A Mathematical Statement is a statement that can be said to be true or false. In order to perform a mathematical statement we need logic, and I'm pretty sure all human beings have logic.
                     Ex: Congruent angles are angles that measure the same. T or F

Open Sentences is a mathematical statement where there is a "variable"
                    Ex of Variables: She is pretty. "She" is the variable.


Negations is pretty simple and pretty self explanatory, the name clearly says what is about. Negations is when you deny something and say "not". It always have the opposite truth value. Using negations, you still have to use logic and a mathematical sentence.
                      Ex: An owl is a fish. --> F
                       - Negation: An owl is not a fish. --> T


Conjunction is a word that helps connect two or more sentences.

• "and": both statements must be true for the statement to be true
• "or": either statement (or both) can be true for the statement to be true

There are 4 types of logic in math. In this post we will learn about 2 only.


A Conditional is the most frequently used statement in the construction of an argument or in the study of math.
                      Ex: Today is raining. Tomorrow will be sunny. The conditional of these statements are "If today is raining, then tomorrow will be sunny". So you basically add "If" to the hypothesis and "then" to the conclusion.


An Inverse is formed by negating the hypothesis and the conclusion.
                   Ex: Today is raining. Tomorrow will be sunny. The inverse to these statements are "If today is not raining, then tomorrow will not be cloudy"


Problem:      "A trapezoid is a four sided polygon"
                                   True             or                 False


**Images and some of the examples provided by google.com/images and my notes from Geometry class.

How Do We Use Other Definitions of Transformations?

There are many types of transformations in Geometry. Some of them are Glide Reflections, Orientation, and Isometry, Direct Isometry and Invariant.

• Glide Reflection is the combination of a reflection in a line and a translation along that line

• Orientation is the arrangement of points relative to one another after a transformation has occurred.

• Isometry is a type of transformation of the plane that preserves length. For Example: a transformation that preserves everything except dilation.

• Direct Isometry preserves orientation or order - the letters on the diagram go in  the same clockwise or counterclockwise direction on the figure.

• Invariant is a figure that remains unchanged under a transformation of the plane. No variations have occurred.

How Do We Solve Composition of Transformation Problem?

Knowing the rules of doing the second transformation first and the first transformation second, we are set to solve compositions of transformations.

Rules:


Examples:

 is the image of   under a glide reflection
that is a composition of a reflection over the line l
and a translation through the vector v.

Examples:

*Images and and captions were provided by Google Image

How Do We Identify Composition of Transformation?

We can identify composition of transformation in one way, which for my opinion is the easiest one.

• When two or more transformations are combined to form a new transformation is called the composition of transformations.

For Example:

1.  R(x-axis)  º T(3,4)

Rules:

• Do the coordinates with the "T" first and then do the "R" second.

How Do We Identify Transformations?

A Transformation is when you move a geometric figure. There's four types of transformations:

• Translation
• Rotation
• Reflection
• Dilation

A Translation is when every point is moved the same distance in the same direction.

In this example, every point in the star has moved the same distance in the same distance

A  Rotation is when a figure is turned around a single point (See "How Do We Graph Rotations?" for more info)

In this picture, point A has been rotated 180 degrees.

A Reflection is when a figure is flipped over a line of symmetry.

In this picture, the line of symmetry is X-axis and the mountains are been reflected  on the lake.

A Dilation is an enlargement or reduction in size of an image.

In this example, triangle ABC has been enlarged.

HOW DO WE GRAPH ROTATIONS?

There's 3 steps for you to graph rotations:

• Find the angle of rotation
• Find the direction (clockwise or counterclockwise)
• Find the center of rotation 

A rotation is a type of transformation in which you rotate the figure. This is mostly use in Geometry. You may think rotation is hard, but it really isn't. If you know the rules of rotation you'll be great in a test and/or homework the teacher gives you. The rules are:

• For 90 degrees:

                                  (A, B) -> (-B, A)

• For 180 degrees:

                                    (A, B) -> (-A, -B)

• For 270 degrees:

                                    (A, B) -> (B, -A)

As you can see, you only have to change the signs and/or switch the positions of the coordinates. Its pretty simple.

In the figure above,  the triangle P has been moved 90 degrees counterclockwise.

In this figure, it shows the Triangle in the grid rotating clockwise in every quadrant.